This document summarizes an academic paper that presents an asymptotic analysis of an infectious disease model. The model considers the interaction between a pathogen and an immune system response over time.
The authors first nondimensionalize the model equations to reduce the number of parameters from seven to three. They then perform an asymptotic analysis in the limit where one of the parameters (ε) becomes large. This reveals separate time scales - a short initial layer where the pathogen grows rapidly, and a longer remission region where the immune response dominates.
Approximate solutions are derived for the initial layer and first reinfection peak using singular perturbation theory. The analysis provides simple expressions for key features of the pathogen and immune response dynamics with good quantitative accuracy.
Probability Models for Estimating Haplotype Frequencies and Bayesian Survival...Université de Dschang
M. Kum Cletus Kwa a soutenu une thèse de Doctorat/Phd en mathématiques ce 14 juin 2016 à l'Université de Dschang. Le jury lui a décerné à l'issue des échanges la mention très honorable.
Stability criterion of periodic oscillations in a (1)Alexander Decker
This document summarizes a mathematical model of the interaction between HIV, immune cells (CTLs), and antiretroviral drug treatment. The model accounts for time delays between viral infection of cells and viral production. It consists of 4 equations tracking healthy CD4+ T-cells, latently infected cells, productively infected cells, and free virus over time. The paper analyzes the model's equilibrium states and how time delays impact the stability of periodic oscillations. It finds that oscillations are stable if the time delay is within certain bounds, and drug efficacy can lead to oscillation death at a critical delay value. This threshold could help control HIV dynamics through treatment intervention.
This paper proposes a vaccine-dependent mathematical model to study the transmission dynamics of tuberculosis (TB) epidemics at the population level. The model divides the population into susceptible, latently infected unvaccinated, latently infected vaccinated, actively infected, recovered, and vaccinated classes. The paper proves the existence and uniqueness of a solution to the system of equations that defines the model. It also shows that the infection will die out if the basic reproduction number is less than one. The model could be used to estimate new TB infections and help design prevention and intervention strategies.
An analytic study of the fractional order model of HIV-1 virus and CD4+ T-cel...IJECEIAES
In this article, we study the fractional mathematical model of HIV-1 infection of CD4+ T-cells, by studying a system of fractional differential equations of first order with some initial conditions, we study the changing effect of many parameters. The fractional derivative is described in the caputo sense. The adomian decomposition method (Shortly, ADM) method was used to calculate an approximate solution for the system under study. The nonlinear term is dealt with the help of Adomian polynomials. Numerical results are presented with graphical justifications to show the accuracy of the proposed methods.
Like personalized medicine, personalized vaccinology aims to provide the right vaccine, to the right patient, at the right time, to achieve protection from disease, while being safe (i.e., free from unintended side effects). Starting with these lines, this presentation will provide overall information related to the vaccinomoic along with the suitable examples and thus will be helpful for the students to understand the basics related to the same.
Background: The HIV virus carries projection of significant global population with specific estimations of the mathematical results of evolutionary methods which was presented in Tree Hidden Markov model (HMM).
Materials and Methods: Hidden Markov models used to model the progression of the disease among HIV infected people. The author predicts a Baum Welch Algorithm method through HMM that can assess an unknown state of transition.
Results: The Tree HMM model predicts the break down point starts once patient is infected with the HIV virus as it affects the immune system. The immune system drops more quickly in the initial inter arrival time when compared with the later time interval. The HIV virus length in the nth state within regrouping is uncertain to occur in each state of the given model. A simulation study was done to assess the goodness of fit for the model.
Conclusion: The HIV virus length in the nth state within regrouping is uncertain to occur in each state of the given model. The inter arrival censoring between each state is essential in each infected HIV patients. The outcome of this works states that health care expert can use this model for effective patient cares.
Keywords: expectation, hidden markov model, human immunodeficiency virus, immune system, transition
Projecting ‘time to event’ outcomes in technology assessment: an alternative ...cheweb1
This document discusses alternative methods for projecting survival outcomes in technology assessments beyond what is observed in clinical trials.
The standard method of fitting parametric survival functions to trial data and extrapolating is problematic as it assumes a single mechanism and does not account for trial design or changes in risk over time. LRiG proposes examining trial data to understand risk trajectories and formulating hypotheses based on clinical context rather than selecting a model solely on fit. A case study demonstrates modeling progression-free survival, post-progression survival, and overall survival as separate phases using exponential convolution functions. LRiG advocates understanding empirical data and developing more informative multi-phase models rather than relying on standard projections.
This document provides an overview of survival analysis. It defines survival analysis as statistical methods for analyzing longitudinal data on the occurrence of events over time. Key features include events that may or may not occur for subjects and the length of time until an event can vary. Censoring, where subjects drop out before an event, is accommodated. The objectives, terms, and reasons for using survival analysis are described. Key concepts like hazard rates, survival functions, and the Kaplan-Meier estimate are also introduced.
Probability Models for Estimating Haplotype Frequencies and Bayesian Survival...Université de Dschang
M. Kum Cletus Kwa a soutenu une thèse de Doctorat/Phd en mathématiques ce 14 juin 2016 à l'Université de Dschang. Le jury lui a décerné à l'issue des échanges la mention très honorable.
Stability criterion of periodic oscillations in a (1)Alexander Decker
This document summarizes a mathematical model of the interaction between HIV, immune cells (CTLs), and antiretroviral drug treatment. The model accounts for time delays between viral infection of cells and viral production. It consists of 4 equations tracking healthy CD4+ T-cells, latently infected cells, productively infected cells, and free virus over time. The paper analyzes the model's equilibrium states and how time delays impact the stability of periodic oscillations. It finds that oscillations are stable if the time delay is within certain bounds, and drug efficacy can lead to oscillation death at a critical delay value. This threshold could help control HIV dynamics through treatment intervention.
This paper proposes a vaccine-dependent mathematical model to study the transmission dynamics of tuberculosis (TB) epidemics at the population level. The model divides the population into susceptible, latently infected unvaccinated, latently infected vaccinated, actively infected, recovered, and vaccinated classes. The paper proves the existence and uniqueness of a solution to the system of equations that defines the model. It also shows that the infection will die out if the basic reproduction number is less than one. The model could be used to estimate new TB infections and help design prevention and intervention strategies.
An analytic study of the fractional order model of HIV-1 virus and CD4+ T-cel...IJECEIAES
In this article, we study the fractional mathematical model of HIV-1 infection of CD4+ T-cells, by studying a system of fractional differential equations of first order with some initial conditions, we study the changing effect of many parameters. The fractional derivative is described in the caputo sense. The adomian decomposition method (Shortly, ADM) method was used to calculate an approximate solution for the system under study. The nonlinear term is dealt with the help of Adomian polynomials. Numerical results are presented with graphical justifications to show the accuracy of the proposed methods.
Like personalized medicine, personalized vaccinology aims to provide the right vaccine, to the right patient, at the right time, to achieve protection from disease, while being safe (i.e., free from unintended side effects). Starting with these lines, this presentation will provide overall information related to the vaccinomoic along with the suitable examples and thus will be helpful for the students to understand the basics related to the same.
Background: The HIV virus carries projection of significant global population with specific estimations of the mathematical results of evolutionary methods which was presented in Tree Hidden Markov model (HMM).
Materials and Methods: Hidden Markov models used to model the progression of the disease among HIV infected people. The author predicts a Baum Welch Algorithm method through HMM that can assess an unknown state of transition.
Results: The Tree HMM model predicts the break down point starts once patient is infected with the HIV virus as it affects the immune system. The immune system drops more quickly in the initial inter arrival time when compared with the later time interval. The HIV virus length in the nth state within regrouping is uncertain to occur in each state of the given model. A simulation study was done to assess the goodness of fit for the model.
Conclusion: The HIV virus length in the nth state within regrouping is uncertain to occur in each state of the given model. The inter arrival censoring between each state is essential in each infected HIV patients. The outcome of this works states that health care expert can use this model for effective patient cares.
Keywords: expectation, hidden markov model, human immunodeficiency virus, immune system, transition
Projecting ‘time to event’ outcomes in technology assessment: an alternative ...cheweb1
This document discusses alternative methods for projecting survival outcomes in technology assessments beyond what is observed in clinical trials.
The standard method of fitting parametric survival functions to trial data and extrapolating is problematic as it assumes a single mechanism and does not account for trial design or changes in risk over time. LRiG proposes examining trial data to understand risk trajectories and formulating hypotheses based on clinical context rather than selecting a model solely on fit. A case study demonstrates modeling progression-free survival, post-progression survival, and overall survival as separate phases using exponential convolution functions. LRiG advocates understanding empirical data and developing more informative multi-phase models rather than relying on standard projections.
This document provides an overview of survival analysis. It defines survival analysis as statistical methods for analyzing longitudinal data on the occurrence of events over time. Key features include events that may or may not occur for subjects and the length of time until an event can vary. Censoring, where subjects drop out before an event, is accommodated. The objectives, terms, and reasons for using survival analysis are described. Key concepts like hazard rates, survival functions, and the Kaplan-Meier estimate are also introduced.
A SEIR MODEL FOR CONTROL OF INFECTIOUS DISEASESSOUMYADAS835019
This document presents a SEIR model for controlling infectious diseases with constraints. It begins with an introduction to SEIR models and their use in modeling disease transmission and testing control strategies. It then motivates the study by discussing the COVID-19 pandemic. The document outlines the basic ideas of the SEIR model and describes the compartments and parameters. It presents the optimal control problem formulated to determine vaccination strategies over time. Potential application areas and future research scope are discussed before concluding with references.
Avian Influenza (H5N1) Expert System using Dempster-Shafer TheoryAndino Maseleno
Based on Cumulative Number of Confirmed Human Cases of Avian Influenza (H5N1) Reported to World Health Organization (WHO) in the 2011 from 15 countries, Indonesia has the largest number death because Avian Influenza which 146 deaths. In this research, the researcher built an Avian Influenza (H5N1) Expert System for identifying avian influenza disease and displaying the result of identification process. In this paper, we describe five symptoms as major symptoms which include depression, combs, wattle, bluish face region, swollen face region, narrowness of eyes, and balance disorders. We use chicken as research object. Dempster-Shafer theory to quantify the degree of belief as inference engine in expert system, our approach uses Dempster-Shafer theory to combine beliefs under conditions of uncertainty and ignorance, and allows quantitative measurement of the belief and plausibility in our identification result. The result reveal that Avian Influenza (H5N1) Expert System has successfully identified the existence of avian influenza and displaying the result of identification process.
This document discusses different methods for analyzing survival data in clinical trials, including Kaplan-Meier survival analysis and restricted mean survival time (RMST) analysis. It reviews literature on survival analysis concepts and applications. The document also notes limitations of Kaplan-Meier analysis when data does not satisfy proportional hazards assumptions or when patients are lost to follow up. RMST is presented as an alternative to estimate mean survival times without these limitations. The document then applies different survival analysis methods to a dataset to compare results.
The document discusses frequency measures used in epidemiology, including ratios, proportions, and rates. Ratios compare the occurrence of a variable between two groups, proportions express the size of one group compared to a larger group it belongs to as a percentage, and rates measure the occurrence of an event in a population over a period of time. These measures are calculated in the same way and used to describe disease occurrence and mortality in infectious disease epidemiology. The document also outlines some strengths and limitations of epidemiological study designs, such as bias, misclassification, confounding, and effect modification.
This document provides an introduction and tables for determining sample sizes in various health studies. It covers one-sample situations like estimating a population proportion with absolute or relative precision and hypothesis tests for a population proportion. Two-sample situations covered include estimating the difference between two population proportions and hypothesis tests for two population proportions. It also addresses case-control studies, cohort studies, lot quality assurance sampling, and incidence-rate studies. Tables of minimum sample sizes are provided for each situation.
Hiv Replication Model for The Succeeding Period Of Viral Dynamic Studies In A...inventionjournals
This document presents a new model for HIV replication dynamics. It introduces an exponential distribution to model the rate of HIV multiplication by infected cells. A Bayesian approach is used to estimate the posterior distribution of the rate parameter, using an incomplete gamma function as the prior. The model allows estimating the HIV count in the succeeding period based on viral load and CD4 count data collected periodically.
Use Proportional Hazards Regression Method To Analyze The Survival of Patient...Waqas Tariq
The Kaplan Meier method is used to analyze data based on the survival time. In this paper used Kaplan Meier procedure and Cox regression with these objectives. The objectives are finding the percentage of survival at any time of interest, comparing the survival time of two studied groups and examining the effect of continuous covariates with the relationship between an event and possible explanatory variables. The variables (Age, Gender, Weight, Drinking, Smoking, District, Employer, Blood Group) are used to study the survival patients with cancer stomach. The data in this study taken from Hiwa/Hospital in Sualamaniyah governorate during the period of (48) months starting from (1/1/2010) to (31/12/2013) .After Appling the Cox model and achieve the hypothesis we estimated the parameters of the model by using (Partial Likelihood) method and then test the variables by using (Wald test) the result show that the variables age and weight are influential at the survival of time.
Surveillance of healthcare-associated infections: understanding and utilizing...Evangelos Kritsotakis
Presented at the EUCIC Basic Module for Infection Prevention and Control, Groningen, May 2022.
This module is organised by the European Committee on Infection Control (EUCIC) is taught face-to-face by top experts from different academic centres in Europe, who cover all major aspects of Infection Prevention and Control in the hospital.
Time and dose-dependent risk of pneumococcal pneumonia following influenza- a...Joshua Berus
This document presents a mathematical model of the within-host interaction between influenza virus and Streptococcus pneumoniae bacteria. The model combines existing models of individual influenza and pneumococcal infections through an immune-mediated interaction mechanism. Simulation results from the combined model capture key features observed in animal studies, such as enhanced risk of invasive pneumonia when pneumococcal exposure occurs 4-6 days after influenza infection. The model also predicts that antiviral treatment would only prevent severe pneumococcal disease if administered early in influenza infection. The quantitative model framework provides insights into the clinical and epidemiological consequences of the viral-bacterial interaction.
Running head PROJECT PHASE 4-INFECTIOUS DISEASES1PROJECT PHASE.docxtoltonkendal
Running head: PROJECT PHASE 4-INFECTIOUS DISEASES 1
PROJECT PHASE 4 – INFECTIOUS DISEASES 13
Project Phase 4 – Infectious Diseases
Author Note
This paper is being submitted on
Project Phase 4 – Infectious Diseases
Introduction:
As a healthcare professional, you will work to improve and maintain the health of individuals, families, and communities in various settings. Basic statistical analysis can be used to gain an understanding of current problems. Understanding the current situation is the first step in discovering where an opportunity for improvement exists. This course project will assist you in applying basic statistical principles to a fictional scenario in order to impact the health and wellbeing of the clients being served.
This assignment will be completed in phases throughout the quarter. As you gain additional knowledge through the didactic portion of this course, you will be able to apply your new knowledge to this project. You will receive formative feedback from your instructor on each submission. The final project will be due on week 5.
Scenario:
You are currently working at NCLEX Memorial Hospital in the Infectious Diseases Unit. Over the past few days, you have noticed an increase in patients admitted with a particular infectious disease. You believe that the ages of these patients play a critical role in the method used to treat the patients. You decide to speak to your manager and together you work to use statistical analysis to look more closely at the ages of these patients. You do some research and put together a spreadsheet of the data that contains the following information:
· Client number
· Infection Disease Status
· Age of the patient
You need the preliminary findings immediately so that you can start treating these patients. So, let’s get to work!!!!
Background information on the Data:
The data set consists of 60 patients that have the infectious disease with ages ranging from 35 years of age to 76 years of age for NCLEX Memorial Hospital. Remember this assignment will be completed over the duration of the course.
To begin let’s learn what infectious disease is. Infectious diseases are caused by pathogenic microorganisms, which are bacteria, viruses, parasites or fungi; the diseases can be spread directly or indirectly, through one person to another (WHO, 2017).
This scenario will aim to improve the quality of healthcare services that are provided to individuals, families, and communities at different levels of age. Therefore, the project utilized at NCLEX Memorial Hospital, over the past few days has seen a larger level of infectious disease occurrences. The data set composed was for sixty patients ranging in age from thirty-five to seventy-six.
1)
a) Qualitative infectious: Disease
b) Quantitative: Age
2) Age is a constant variable as it may take any value.
3) A variable is any quantity that can be measured and whose value differs through the
Population and here we se ...
This document discusses survival analysis and Cox regression for cancer clinical trials. It begins with an introduction to Cox regression analysis and how it can be used to analyze the effects of covariates on survival rates in cancer trials. The document then provides examples of Cox regression outputs and how to interpret the results, including checking the proportional hazards assumption. It cautions against some invalid methods of survival analysis that do not properly account for censored or time-dependent data.
A Cox model is a statistical technique used to analyze survival data with several explanatory variables. It allows estimation of the hazard or risk of an event like death for an individual based on prognostic factors. A Cox model expresses the hazard as an exponential function of the explanatory variables. Interpreting a Cox model involves examining the regression coefficients - a positive coefficient means a higher hazard/worse prognosis, while a negative coefficient implies a better prognosis. The model from a study of melanoma patients' survival found age and cancer type increased hazard, while male sex decreased it, and interferon treatment did not significantly impact survival.
Interval observer for uncertain time-varying SIR-SI model of vector-borne dis...FGV Brazil
The issue of state estimation is considered for an SIR-SI model describing a vector-borne disease such as dengue fever, with seasonal variations and uncertainties in the transmission rates. Assuming continuous measurement of the number of new infectives in the host population per unit time, a class of interval observers with estimate-dependent gain is constructed, and asymptotic error bounds are provided. The synthesis method is based on the search for a common linear Lyapunov function for monotone systems representing the evolution of the estimation errors.
Date: 2017
Authors:
Soledad Aronna, Maria
Bliman, Pierre-Alexandre
This document outlines topics related to survival analysis, including its objectives and key methods. Survival analysis is used to analyze longitudinal data on events like death or disease onset over time. It accounts for censoring of data. The Kaplan-Meier method estimates survival rates without dividing time into intervals like life tables do. The log-rank test statistically compares survival curves between groups. Cox regression analysis examines the relationship between covariates and survival while allowing hazards to vary over time.
A gentle introduction to survival analysisAngelo Tinazzi
This document provides an introduction to survival analysis techniques for statistical programmers. It discusses key concepts in survival analysis including censoring, the Kaplan-Meier method for estimating survival probabilities, and assumptions of survival models. Programming aspects like creating time-to-event datasets and using SAS procedures for survival analysis are also covered.
Epidemiological method of research, structure & Maintenance. Eneutron
Epidemiology is the study of disease patterns in populations and uses a systematic method of research to identify risk factors and determine preventive measures. This document discusses the epidemiological method, epidemiological diagnostics, and the system of epidemiological surveillance. The epidemiological method involves descriptive, analytical, and experimental techniques to study disease occurrence and justify prevention. Epidemiological diagnostics provides data to support preventive actions by describing disease manifestations, risk groups, determining causes, and formulating hypotheses. Epidemiological surveillance is the ongoing assessment of disease trends to enable timely intervention through prevention and control programs.
1. The document discusses sampling techniques and sample size calculations for quantitative and qualitative data. It provides formulas to calculate sample size based on population parameters, desired confidence level, and allowable error.
2. Meta-analysis is defined as the statistical analysis of results from multiple studies to integrate findings. Conducting meta-analysis allows for more precise and generalizable treatment estimates compared to single studies.
3. Both advantages and limitations of meta-analysis are discussed. While it provides powerful tools to synthesize evidence, limitations include heterogeneity between studies, publication bias, and potential for poor methodology.
Oseltamivir treatment initiated within 24 hours of symptom onset was associated with shorter duration of self-reported illness symptoms (56% reduction) based on an observational study of 582 influenza patients. However, oseltamivir treatment was not found to be associated with shorter duration of viral shedding by PCR or with reduced risk of household transmission of influenza viruses. The study analyzed data from a community-based study of household transmission of influenza in Hong Kong from 2008-2013.
This document describes the development of a predictive model to identify premature infants born between 33-35 weeks gestational age that are at highest risk of hospitalization due to respiratory syncytial virus (RSV) infection. The model was developed using risk factor data from a Spanish case-control study of 183 infants hospitalized with RSV compared to 371 non-hospitalized controls. Discriminant function analysis identified an initial model using 15 risk factors that discriminated between the two groups with 72% accuracy. Further refinement resulted in a final 7 variable model that predicted risk with 71% accuracy and could help optimize use of RSV prophylaxis for higher risk infants in Europe.
Fast Paper Writing Service, 11 Research Paper Writing IdeasSteven Wallach
This document provides instructions for using a paper writing service called HelpWriting.net. It outlines a 5-step process: 1) Create an account with an email and password. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and choose one. 4) Review the completed paper and authorize payment. 5) Request revisions until satisfied, with a refund option for plagiarism. The service aims to provide original, high-quality content to meet customer needs.
1. The document provides instructions for creating an account and submitting a 'Write My Paper For Me' request on the HelpWriting.net site. It outlines a 5-step process: creating an account, submitting a request form, reviewing writer bids, authorizing payment, and requesting revisions if needed.
2. Writers on the site utilize a bidding system, and customers can choose a writer based on qualifications, order history, and feedback. The site promises original, high-quality content and refunds for plagiarized work.
3. Customers can request multiple revisions to ensure satisfaction with their completed paper.
More Related Content
Similar to Asymptotic Theory Of An Infectious Disease Model
A SEIR MODEL FOR CONTROL OF INFECTIOUS DISEASESSOUMYADAS835019
This document presents a SEIR model for controlling infectious diseases with constraints. It begins with an introduction to SEIR models and their use in modeling disease transmission and testing control strategies. It then motivates the study by discussing the COVID-19 pandemic. The document outlines the basic ideas of the SEIR model and describes the compartments and parameters. It presents the optimal control problem formulated to determine vaccination strategies over time. Potential application areas and future research scope are discussed before concluding with references.
Avian Influenza (H5N1) Expert System using Dempster-Shafer TheoryAndino Maseleno
Based on Cumulative Number of Confirmed Human Cases of Avian Influenza (H5N1) Reported to World Health Organization (WHO) in the 2011 from 15 countries, Indonesia has the largest number death because Avian Influenza which 146 deaths. In this research, the researcher built an Avian Influenza (H5N1) Expert System for identifying avian influenza disease and displaying the result of identification process. In this paper, we describe five symptoms as major symptoms which include depression, combs, wattle, bluish face region, swollen face region, narrowness of eyes, and balance disorders. We use chicken as research object. Dempster-Shafer theory to quantify the degree of belief as inference engine in expert system, our approach uses Dempster-Shafer theory to combine beliefs under conditions of uncertainty and ignorance, and allows quantitative measurement of the belief and plausibility in our identification result. The result reveal that Avian Influenza (H5N1) Expert System has successfully identified the existence of avian influenza and displaying the result of identification process.
This document discusses different methods for analyzing survival data in clinical trials, including Kaplan-Meier survival analysis and restricted mean survival time (RMST) analysis. It reviews literature on survival analysis concepts and applications. The document also notes limitations of Kaplan-Meier analysis when data does not satisfy proportional hazards assumptions or when patients are lost to follow up. RMST is presented as an alternative to estimate mean survival times without these limitations. The document then applies different survival analysis methods to a dataset to compare results.
The document discusses frequency measures used in epidemiology, including ratios, proportions, and rates. Ratios compare the occurrence of a variable between two groups, proportions express the size of one group compared to a larger group it belongs to as a percentage, and rates measure the occurrence of an event in a population over a period of time. These measures are calculated in the same way and used to describe disease occurrence and mortality in infectious disease epidemiology. The document also outlines some strengths and limitations of epidemiological study designs, such as bias, misclassification, confounding, and effect modification.
This document provides an introduction and tables for determining sample sizes in various health studies. It covers one-sample situations like estimating a population proportion with absolute or relative precision and hypothesis tests for a population proportion. Two-sample situations covered include estimating the difference between two population proportions and hypothesis tests for two population proportions. It also addresses case-control studies, cohort studies, lot quality assurance sampling, and incidence-rate studies. Tables of minimum sample sizes are provided for each situation.
Hiv Replication Model for The Succeeding Period Of Viral Dynamic Studies In A...inventionjournals
This document presents a new model for HIV replication dynamics. It introduces an exponential distribution to model the rate of HIV multiplication by infected cells. A Bayesian approach is used to estimate the posterior distribution of the rate parameter, using an incomplete gamma function as the prior. The model allows estimating the HIV count in the succeeding period based on viral load and CD4 count data collected periodically.
Use Proportional Hazards Regression Method To Analyze The Survival of Patient...Waqas Tariq
The Kaplan Meier method is used to analyze data based on the survival time. In this paper used Kaplan Meier procedure and Cox regression with these objectives. The objectives are finding the percentage of survival at any time of interest, comparing the survival time of two studied groups and examining the effect of continuous covariates with the relationship between an event and possible explanatory variables. The variables (Age, Gender, Weight, Drinking, Smoking, District, Employer, Blood Group) are used to study the survival patients with cancer stomach. The data in this study taken from Hiwa/Hospital in Sualamaniyah governorate during the period of (48) months starting from (1/1/2010) to (31/12/2013) .After Appling the Cox model and achieve the hypothesis we estimated the parameters of the model by using (Partial Likelihood) method and then test the variables by using (Wald test) the result show that the variables age and weight are influential at the survival of time.
Surveillance of healthcare-associated infections: understanding and utilizing...Evangelos Kritsotakis
Presented at the EUCIC Basic Module for Infection Prevention and Control, Groningen, May 2022.
This module is organised by the European Committee on Infection Control (EUCIC) is taught face-to-face by top experts from different academic centres in Europe, who cover all major aspects of Infection Prevention and Control in the hospital.
Time and dose-dependent risk of pneumococcal pneumonia following influenza- a...Joshua Berus
This document presents a mathematical model of the within-host interaction between influenza virus and Streptococcus pneumoniae bacteria. The model combines existing models of individual influenza and pneumococcal infections through an immune-mediated interaction mechanism. Simulation results from the combined model capture key features observed in animal studies, such as enhanced risk of invasive pneumonia when pneumococcal exposure occurs 4-6 days after influenza infection. The model also predicts that antiviral treatment would only prevent severe pneumococcal disease if administered early in influenza infection. The quantitative model framework provides insights into the clinical and epidemiological consequences of the viral-bacterial interaction.
Running head PROJECT PHASE 4-INFECTIOUS DISEASES1PROJECT PHASE.docxtoltonkendal
Running head: PROJECT PHASE 4-INFECTIOUS DISEASES 1
PROJECT PHASE 4 – INFECTIOUS DISEASES 13
Project Phase 4 – Infectious Diseases
Author Note
This paper is being submitted on
Project Phase 4 – Infectious Diseases
Introduction:
As a healthcare professional, you will work to improve and maintain the health of individuals, families, and communities in various settings. Basic statistical analysis can be used to gain an understanding of current problems. Understanding the current situation is the first step in discovering where an opportunity for improvement exists. This course project will assist you in applying basic statistical principles to a fictional scenario in order to impact the health and wellbeing of the clients being served.
This assignment will be completed in phases throughout the quarter. As you gain additional knowledge through the didactic portion of this course, you will be able to apply your new knowledge to this project. You will receive formative feedback from your instructor on each submission. The final project will be due on week 5.
Scenario:
You are currently working at NCLEX Memorial Hospital in the Infectious Diseases Unit. Over the past few days, you have noticed an increase in patients admitted with a particular infectious disease. You believe that the ages of these patients play a critical role in the method used to treat the patients. You decide to speak to your manager and together you work to use statistical analysis to look more closely at the ages of these patients. You do some research and put together a spreadsheet of the data that contains the following information:
· Client number
· Infection Disease Status
· Age of the patient
You need the preliminary findings immediately so that you can start treating these patients. So, let’s get to work!!!!
Background information on the Data:
The data set consists of 60 patients that have the infectious disease with ages ranging from 35 years of age to 76 years of age for NCLEX Memorial Hospital. Remember this assignment will be completed over the duration of the course.
To begin let’s learn what infectious disease is. Infectious diseases are caused by pathogenic microorganisms, which are bacteria, viruses, parasites or fungi; the diseases can be spread directly or indirectly, through one person to another (WHO, 2017).
This scenario will aim to improve the quality of healthcare services that are provided to individuals, families, and communities at different levels of age. Therefore, the project utilized at NCLEX Memorial Hospital, over the past few days has seen a larger level of infectious disease occurrences. The data set composed was for sixty patients ranging in age from thirty-five to seventy-six.
1)
a) Qualitative infectious: Disease
b) Quantitative: Age
2) Age is a constant variable as it may take any value.
3) A variable is any quantity that can be measured and whose value differs through the
Population and here we se ...
This document discusses survival analysis and Cox regression for cancer clinical trials. It begins with an introduction to Cox regression analysis and how it can be used to analyze the effects of covariates on survival rates in cancer trials. The document then provides examples of Cox regression outputs and how to interpret the results, including checking the proportional hazards assumption. It cautions against some invalid methods of survival analysis that do not properly account for censored or time-dependent data.
A Cox model is a statistical technique used to analyze survival data with several explanatory variables. It allows estimation of the hazard or risk of an event like death for an individual based on prognostic factors. A Cox model expresses the hazard as an exponential function of the explanatory variables. Interpreting a Cox model involves examining the regression coefficients - a positive coefficient means a higher hazard/worse prognosis, while a negative coefficient implies a better prognosis. The model from a study of melanoma patients' survival found age and cancer type increased hazard, while male sex decreased it, and interferon treatment did not significantly impact survival.
Interval observer for uncertain time-varying SIR-SI model of vector-borne dis...FGV Brazil
The issue of state estimation is considered for an SIR-SI model describing a vector-borne disease such as dengue fever, with seasonal variations and uncertainties in the transmission rates. Assuming continuous measurement of the number of new infectives in the host population per unit time, a class of interval observers with estimate-dependent gain is constructed, and asymptotic error bounds are provided. The synthesis method is based on the search for a common linear Lyapunov function for monotone systems representing the evolution of the estimation errors.
Date: 2017
Authors:
Soledad Aronna, Maria
Bliman, Pierre-Alexandre
This document outlines topics related to survival analysis, including its objectives and key methods. Survival analysis is used to analyze longitudinal data on events like death or disease onset over time. It accounts for censoring of data. The Kaplan-Meier method estimates survival rates without dividing time into intervals like life tables do. The log-rank test statistically compares survival curves between groups. Cox regression analysis examines the relationship between covariates and survival while allowing hazards to vary over time.
A gentle introduction to survival analysisAngelo Tinazzi
This document provides an introduction to survival analysis techniques for statistical programmers. It discusses key concepts in survival analysis including censoring, the Kaplan-Meier method for estimating survival probabilities, and assumptions of survival models. Programming aspects like creating time-to-event datasets and using SAS procedures for survival analysis are also covered.
Epidemiological method of research, structure & Maintenance. Eneutron
Epidemiology is the study of disease patterns in populations and uses a systematic method of research to identify risk factors and determine preventive measures. This document discusses the epidemiological method, epidemiological diagnostics, and the system of epidemiological surveillance. The epidemiological method involves descriptive, analytical, and experimental techniques to study disease occurrence and justify prevention. Epidemiological diagnostics provides data to support preventive actions by describing disease manifestations, risk groups, determining causes, and formulating hypotheses. Epidemiological surveillance is the ongoing assessment of disease trends to enable timely intervention through prevention and control programs.
1. The document discusses sampling techniques and sample size calculations for quantitative and qualitative data. It provides formulas to calculate sample size based on population parameters, desired confidence level, and allowable error.
2. Meta-analysis is defined as the statistical analysis of results from multiple studies to integrate findings. Conducting meta-analysis allows for more precise and generalizable treatment estimates compared to single studies.
3. Both advantages and limitations of meta-analysis are discussed. While it provides powerful tools to synthesize evidence, limitations include heterogeneity between studies, publication bias, and potential for poor methodology.
Oseltamivir treatment initiated within 24 hours of symptom onset was associated with shorter duration of self-reported illness symptoms (56% reduction) based on an observational study of 582 influenza patients. However, oseltamivir treatment was not found to be associated with shorter duration of viral shedding by PCR or with reduced risk of household transmission of influenza viruses. The study analyzed data from a community-based study of household transmission of influenza in Hong Kong from 2008-2013.
This document describes the development of a predictive model to identify premature infants born between 33-35 weeks gestational age that are at highest risk of hospitalization due to respiratory syncytial virus (RSV) infection. The model was developed using risk factor data from a Spanish case-control study of 183 infants hospitalized with RSV compared to 371 non-hospitalized controls. Discriminant function analysis identified an initial model using 15 risk factors that discriminated between the two groups with 72% accuracy. Further refinement resulted in a final 7 variable model that predicted risk with 71% accuracy and could help optimize use of RSV prophylaxis for higher risk infants in Europe.
Similar to Asymptotic Theory Of An Infectious Disease Model (20)
Fast Paper Writing Service, 11 Research Paper Writing IdeasSteven Wallach
This document provides instructions for using a paper writing service called HelpWriting.net. It outlines a 5-step process: 1) Create an account with an email and password. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and choose one. 4) Review the completed paper and authorize payment. 5) Request revisions until satisfied, with a refund option for plagiarism. The service aims to provide original, high-quality content to meet customer needs.
1. The document provides instructions for creating an account and submitting a 'Write My Paper For Me' request on the HelpWriting.net site. It outlines a 5-step process: creating an account, submitting a request form, reviewing writer bids, authorizing payment, and requesting revisions if needed.
2. Writers on the site utilize a bidding system, and customers can choose a writer based on qualifications, order history, and feedback. The site promises original, high-quality content and refunds for plagiarized work.
3. Customers can request multiple revisions to ensure satisfaction with their completed paper.
How To Write A Self Evaluation Essay TelegraphSteven Wallach
The document discusses how to write a self-evaluation essay, outlining 5 steps: 1) create an account, 2) complete an order form providing instructions and deadline, 3) review bids from writers and choose one, 4) review the completed paper and authorize payment, and 5) request revisions to ensure satisfaction and receive a refund for plagiarized work. The process describes how to hire a writer on the HelpWriting.net platform to complete an assignment.
George Washington Papers, Available Online, GeorgeSteven Wallach
The document describes how a student named Nick Allen decides to call pens "frindles" as a joke. His teacher Mrs. Granger does not approve and punishes students for using the made-up word. However, the word catches on and eventually makes it into the dictionary. Years later, Mrs. Granger congratulates Nick and apologizes for opposing the new word, showing how language evolves through common use.
How To Write An Evaluation Essay Types, Steps And Format Of AnSteven Wallach
The document discusses the ancient Greek sculpture The Lansdowne Athlete by Lysippos from 340 BCE. It analyzes how the sculpture portrays the ideal of perfection through a fit body guided by a keen mind, representing the philosophical debates around appropriate forms of perfection. The paper aims to approach the visual essence of the sculpture and analyze how it depicts this ideal of physical and mental perfection.
Law Essay Writing Service Help - Theomnivore.Web.Fc2Steven Wallach
The document provides instructions for requesting assistance with writing law essays from a service called HelpWriting.net. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and select one. 4) Receive the paper and authorize payment if pleased. 5) Request revisions until satisfied. The service promises original, high-quality work with refunds for plagiarism.
Best Photos Of APA Format Example R. Online assignment writing service.Steven Wallach
1. The document discusses the history of Meriwether Lewis and William Clark's westward expedition in 1803-1806 led by Thomas Jefferson to explore the Louisiana Purchase territory and find a route across North America.
2. Lewis and Clark spent months preparing, obtaining scientific equipment, medicines, gifts for Native Americans, and more before departing. Lewis had scientific knowledge and Clark was experienced in mapping and military skills.
3. The expedition made discoveries about the land, geography, native tribes, and wildlife. Lewis and Clark worked as a great team and successfully completed the mission of exploring the new western lands.
How To Write A 6 Page Research Paper -Write MySteven Wallach
The document provides instructions for writing a 6-page research paper through an online service. It outlines 5 steps: 1) Create an account; 2) Complete an order form with instructions and deadline; 3) Review writer bids and choose one; 4) Review the paper and authorize payment; 5) Request revisions until satisfied. It emphasizes getting original, high-quality content and providing a sample for the writer to imitate the writing style.
My Family Essay My Family Essay In English Essay OSteven Wallach
This paper investigates the linear and nonlinear optical susceptibilities and hyperpolarizability of lithium sodium tetraborate (LiNaB4O7) single crystals through both theoretical calculations and experimental measurements in order to evaluate their potential for nonlinear optical applications such as frequency conversion of lasers. Borate materials are well-suited for nonlinear optics and laser engineering due to properties like short growth periods, high damage thresholds, large effective nonlinear coefficients, and good mechanical properties. The results show that LiNaB4O7 crystals exhibit good nonlinear optical performance, making them promising for frequency conversion applications.
FREE 8+ Sample College Essay Templates In MSteven Wallach
1) The document discusses two key human resource management strategies used by Coles Supermarkets: reward management and performance management.
2) It analyzes how reward management is implemented at Coles and identifies areas for improvement to bridge the gap between operational losses and optimal performance.
3) The performance management method used by Coles to efficiently and effectively guide employees' work is also examined. Performance management helps ensure employee activities align with organizational goals.
Argumentative Essay Help – Qu. Online assignment writing service.Steven Wallach
The document discusses a proposed plan for Congress to require all U.S. high school graduates to spend one year in mandatory civil service before attending college. While Congress supports this plan, many others disagree with forcing students to take a year away from potential religious trips or delaying their field of study. The author's high school also opposes the idea, believing it could be counterproductive to have graduates leave their intended career paths.
How To Write A Literary Analysis Essay - Take UsSteven Wallach
1) The document provides instructions for how to request and receive help with writing assignments from the website HelpWriting.net. It outlines a 5-step process: register an account, submit a request form with instructions and deadline, review writer bids and select one, authorize payment after receiving satisfactory work, and request revisions if needed.
2) The website uses a bidding system where writers submit proposals, and clients can ensure original, high-quality work or receive a refund if plagiarized.
3) The process aims to fully meet client needs for assignment writing help.
How To Get Paid To Write Essa. Online assignment writing service.Steven Wallach
Proprietary software has the advantage of being fully controlled by its developers who can closely guard its intellectual property, but it also has the disadvantage of limiting users' freedom since they cannot modify or share the source code. While proprietary software may offer strong technical support from its developers, it can also lock users into certain platforms and force them to pay licensing fees to continue using the software.
Movie Review Example Review Essay Essay TroSteven Wallach
The document provides instructions for requesting writing assistance from HelpWriting.net in 5 steps:
1. Create an account with a password and email.
2. Complete a 10-minute order form with instructions, sources, and deadline. Attach a sample if imitating writing style.
3. Review bids from writers and choose one based on qualifications, history, and feedback. Place a deposit to start.
4. Ensure the paper meets expectations and authorize payment if pleased. Free revisions are provided.
5. Request revisions until satisfied, knowing plagiarized work results in a full refund.
Quoting A Poem How To Cite A Poem All You Need To Know About Citing ...Steven Wallach
The document discusses the multiple causes that led to the Protestant Reformation, including social forces emerging from economic motives, the powerful papacy raking in financial acquisitions while most people were paupers, and Charles V facing a dilemma that allowed Luther to implement his plan and sow seeds of dissent. It argues against identifying a single cause and emphasizes the need to study history with humility and understand the various interconnected events and complex social forces that contributed to the Reformation.
Validity And Reliability Of Research Instrument ExamSteven Wallach
This experiment aims to determine how the rate of reaction between hydrochloric acid and sodium thiosulphate is affected by changing the concentration of the hydrochloric acid. The hypothesis is that increasing the concentration of hydrochloric acid will increase the rate of reaction and decrease the reaction time. This is tested by reacting four different concentrations of hydrochloric acid with a constant amount of sodium thiosulphate, and measuring the time taken for the reaction, marked by disappearance of an X, to complete.
This toy stimulates muscle tone development in children by requiring them to sit upright either in front of or on top of the toy to play with its buttons, keys and other features, engaging their back, core and leg muscles to maintain proper posture. Sitting upright allows children to see and reach the toy at the proper level while developing muscle tension and control. Playing with the toy's dashboard and handlebars keeps children's hands and legs engaged to enhance muscle tone.
This document provides instructions for requesting writing assistance from HelpWriting.net. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and select one based on qualifications. 4) Review the completed paper and authorize payment if satisfied. 5) Request revisions to ensure satisfaction, with a refund offered for plagiarized work.
Science Essay - College Homework Help And OnlinSteven Wallach
Best Buy failed in its expansion into China due to several strategic mistakes including attempting to directly copy its US store model without adapting to the Chinese market, facing strong competition from established domestic retailers, and struggling to build brand awareness and loyalty among Chinese consumers unfamiliar with the Best Buy brand. As a result, Best Buy was forced to exit the Chinese market after several years of unsuccessful operations and financial losses. Best Buy's experience highlights the importance of thorough market research and localization when expanding into foreign markets with very different consumer behaviors and competitive landscapes.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
2. 288 A. M. Whitman, H. Ashrafiuon
response may be inactive for some initial period. Meanwhile, the pathogen
starts multiplying, leading to an extensive increase in its antigenic mass. The
initial delay period of the stimulation of the immune response can be critical
in terms of host mortality. Thus for the more acute infections, estimation of the
time and extent of the critical levels of the pathogenic load may be of great
importance. For example, small levels of sporozoites can lead to clinically sig-
nificant malaria parasitization while large doses can cause death in a matter of
days [8]. In the case of HIV, studies with animals intravenously inoculated with
low level, less stabilized doses revealed a production of neutralizing antibod-
ies and high survival rates, while those inoculated with high level doses rapidly
developed clinical disease [4]. Similarly for measles, a large inoculum which may
occasionally result from airborne transmission has been suggested to increase
vaccine failure risk [11].
Mathematical models can be important tools in not only analyzing the spread
of infectious diseases in a population of individuals [5,12], but also in predict-
ing the timing and extent of infection and possible reinfection processes in
an individual [9]. While the former is generally used for planning, preven-
tion and control strategies, the latter can be effective in therapy/intervention
programs for treating the individuals who have been exposed to the partic-
ular pathogen. Such models can play an important role for treating acute
infections. Understanding the early dynamics of acute infections and antic-
ipating the time of occurrence and magnitude of the maximum pathogenic
load and the immune response can be critical in choosing effective intervention
schemes.
Deterministic mathematical models of infectious diseases are developed as
sets of coupled, first-order ordinary differential equations with given initial-
values [1,5,9,12]. These equations are nonlinear so they are usually solved
numerically; however, analytic approximations written in terms of the model
parameters are also useful due to the insights they provide. Thus, Mohtashemi
and Levins [9] used an averaging technique to derive analytical expressions
for the maximum pathogenic load and its time to peak, in terms of the ini-
tial inoculum, the initial period of immune activity, and other of their model
parameters. Analytical results can be useful even when an exact solution ex-
ists; an example is Kermack and McKendrick’s [7] approximate solution to
the classical epidemic model, which has an exact, though not explicit, solu-
tion [5]. Kermack and McKendrick’s work [7] produced one of the earliest
approximate solutions for a mathematical model of epidemics, and identified
the epidemic threshold, that minimum critical value of the density of sus-
ceptibles, for which an epidemic outbreak will occur. These approximations,
[7,9], depend on mathematical insights that are model specific. By contrast the
approximation we develop in the sequel is an instance of a general method
that has been applied to a host of problems in a wide variety of physical con-
texts.
Linearization of the equations is a first step toward obtaining approxima-
tions, but this method is severely restricted because it eliminates all interesting
nonlinear effects. It has been used to study repeated outbreaks of epidemics [3],
3. Asymptotic theory of an infectious disease model 289
but the results apply only to small fluctuations about equilibrium and so have
limited practical utility. Parameter perturbation is an alternative method for
generating approximate solutions to nonlinear differential equations and has
been widely used for that purpose in engineering [6,10,13]. The key to doing
this type of analysis is to identify a parameter for which, in some limit, the
model has a simple solution. If the solution makes sense, is useful, and can be
improved by recursion we say it is regular; otherwise it is singular. Two basic
singular perturbation methods have been developed, the method of multiple
scales, also known as the method of averaging, and the method of matched
asymptotic expansions. The former has been applied mainly to problems for
which regular perturbation schemes fail over long times (many cycles) due to
an accumulation of small errors incurred in each cycle of motion. The latter is
useful for problems in which each cycle of the motion has slow and fast parts in
which different physical mechanisms are important.
We suggest that singular perturbation theory provides an accurate solution
along with valuable qualitative insights for the transient pathogen-immune
response dynamics in infectious diseases. Here we present such an approximate
asymptotic solution, based on the method of matched asymptotic expansions
[6,10,13], to a model that was developed to describe the transient dynamics of
the interaction between an invading pathogen and its host’s immune system, [9].
In this previous work, the authors presented a model of pathogen and immune
system evolution that has six parameters; the rate of induction of the immune
system, the rate of decay of the immune system, the rate of removal of the path-
ogen by the immune system, the innate immunity, the reproductive rate of the
pathogen, and the initial pathogenic inoculum. They then tried to approximate
the first maximum of the pathogenic load, and the time it occurred, by means
of an averaging process using a hybrid quadratic, and an exponential model, of
the pathogen and immune response interaction.
In this paper, we first make the model equations, [9], dimensionless and find
that the resulting equations depend on three dimensionless parameters. We
then do an asymptotic analysis that is based on the smallness of two of these
parameters. As a result we obtain a solution that depends on only two parame-
ters. This solution yields very accurate values for the first peak of the pathogen
load and the time it occurs. It also gives the immune system peak value and its
time of occurrence, although not as accurately as that for the pathogen. How-
ever, these results are still more accurate than those obtained by the averaging
process in [9].
An interesting feature of this model, like many of this type, is that it predicts
multiple remission and reinfection periods. The asymptotic analysis presented
here is carried out as far as the second pathogen peak, for which we obtain an
estimate. It could be easily extended to include subsequent peaks as well. We
also indicate how the late time dynamics can be determined from the linearized
system. Finally, we present a uniformly valid solution that predicts the time
evolution of the pathogenic load and immune system response from the initial
infection through to the first reinfection time.
4. 290 A. M. Whitman, H. Ashrafiuon
2 Model equations
We analyze the following dynamic model of an infectious disease attacking an
immune system [9];
dĪ
dt̄
= a0 − µĪ + k0H(t̄ − θ)P̄,
dP̄
dt̄
= (r − mĪ)P̄.
Here Ī is the immune system level, P̄ the pathogenic load, and t̄ the time. The
parameters a0, µ, k0, r and m are (positive) characteristic rates, while H denotes
the Heaviside function, which activates the immune system response only after
a delay, θ, from the initial infection time. The initial conditions are taken as
Ī(0) =
a0
µ
P̄(0) = p̄0
in which p̄0 represents the initial pathogenic load while the immune system is
initially at its uninfected level.
Now for times t̄ ≤ θ, the model equations are uncoupled; the solution of the
first is the constant Ī = Ī(0), and therefore the solution of the second is the
exponential P̄ = p̄0 exp{(r − mĪ(0))t̄}. Since the model is only interesting when
this initial pathogen behavior is growth, not decay, we have a condition
r −
ma0
µ
= µ 0 (1)
that applies to our subsequent discussion. With µ 0, this condition means
that 0.
2.1 Dimensionless representation
There are ten quantities in this model; they are the six parameters mentioned
previously (five rates and the delay time), plus p̄0, the two dependent variables,
and the independent variable. There are two dimensional quantities (number
and time) so eight dimensionless ratios can be formed [2]. These are, using 1/µ
for time and p̄0 for number
a0
µp̄0
k0
µ
r
µ
mp̄0
µ
µθ
Ī0
p̄0
P̄0
p̄0
µt̄. (2)
The last three of these are dimensionless variables and the other five are dimen-
sionless parameters (notice that according to the original definitions the dimen-
sions of a0 and m are directly and inversely proportional to number as well as
5. Asymptotic theory of an infectious disease model 291
inversely proportional to time). Since any reduction in the number of param-
eters appearing in a model is a simplification, it is warranted in all but the
simplest case; in particular, it is useful here in reducing the number of indepen-
dent parameters from seven to five.
Defining dimensionless dependent and independent variables by
I =
1
mp̄0
µ
Ī
p̄0
−
a0
µp̄0
P =
1
2
k0
µ
mp̄0
µ
P̄
p̄0
t = µ(t̄ − θ) (3)
the model equations, for t ≥ 0, can be written as
dI
dt
= −I + P, (4)
dP
dt
= (1 − I)P. (5)
The initial conditions, applied at t = 0 (corresponding to t̄ = θ) where the
immune system is activated, are
I(0) = 0 P(0) =
p0α
2
= p. (6)
The dimensionless parameters, p0, and α, are defined by
p0 =
k0
µ
mp̄0
µ
α = eµθ
(7)
while is given by Eq. (1).
An unanticipated benefit of rendering the problem dimensionless in this way
is that of the five dimensionless parameters in the model, only the three, from
Eqs. (1) and (7), appear explicitly in the differential equations and initial condi-
tions; the other two occur only in the definition of the dimensionless variables,
Eq. (3).
2.2 Equilibria
The set of Eqs. (4) and (5) has two equilibrium points. They are the uninfected
equilibrium at I = 0 and P = 0, and an infected equilibrium at I = 1 and
P = −1. Straightforward linearization [3,9], shows that for 0 the unin-
fected equilibrium is stable and the infected equilibrium is unstable, while for
0 the former is unstable and the latter stable. At = 0 there is a bifurcation
analogous to that of the classic endemic SIR model of infectious disease [5]. As
mentioned before, this model is interesting only when the infected equilibrium
is stable, 0.
6. 292 A. M. Whitman, H. Ashrafiuon
3 Large asymptotics
We have already noted that the case 0 corresponds to an initial growth
of the invading pathogen. Moreover, as becomes larger the growth becomes
more rapid. Thus there is a short time scale that characterizes this behavior in an
initial layer, while there is a longer time scale that characterizes the immune sys-
tem behavior after it has reacted to the invasion (the remission region in Fig. 1).
We will see further on that subsequently this process repeats with alternate peri-
ods of short time pathogen growth and long time immune system dominance.
Using singular perturbation theory, we can derive approximate solutions in each
of these regions, starting from the initial layer and moving forward by matching.
We consider the limit ≡ ǫ−1 → ∞, in an attempt to simplify the nonlinear
Eqs. (4), (5) and (6) sufficiently, so that we can obtain an approximation to the
solution that will be valid in this limit. Moreover, investigating this possibility
makes good physical sense given that the parameter value estimates proposed
in [9] produce = 6.99.
3.1 Initial layer (immune system response region)
We first note from Eq. (5), that I = O(1) in order that both terms on its right
hand side are the same order of magnitude. Then t = O(−1) in order that the
derivative term on the left is also of this order of magnitude. Thus the charac-
teristic time in this region is small compared with the characteristic relaxation
time for the immune system, µ−1. Continuing, we make P = O(1) so that the
derivative term in Eq. (4) is the same order of magnitude as the last term on
its right side, and is dominant. These arguments indicate an initial dynamics
2 4 6 8 10 12 14
0
0.5
1
1.5
2
2.5
x 10
4
time (days)
pathogenic
load
initial
layer
remission region
reinfection
layer
Fig. 1 The main regions of pathogenic load response
7. Asymptotic theory of an infectious disease model 293
in which the populations P and I are finite, but vary over a short initial time
interval, t 1; they are consistent with the discussion at the beginning of
Sect. 3. The scaling to a new time variable that is O(1) in this region is then
T =
t
ǫ
. (8)
With this transformation Eqs. (4) and (5) become, with prime denoting differ-
entiation with respect to T
I′
= P − ǫI, (9)
P′
= (1 − I)P. (10)
The initial conditions of Eq. (6) are unchanged
I(0) = 0 P(0) = p
but now we require lim p0 → 0, in addition to lim ǫ → 0, in order to maintain
the initial value, p = O(1). This means that formally we are doing a double
limiting process
lim
ǫ→0
lim
p0→0
p0ǫ2
α1/ǫ
= O(1).
In this event, the problem only depends explicitly on two parameters, p, which
is finite, and ǫ which is small ( big). It does depend on both p0 and α, but only
in the combination indicated by Eq. (6). Recall that p0 and α are defined in Eq.
(7), with ǫ ≡ −1 defined in Eq. (1). Physically by means of this transforma-
tion, we are focusing our attention at the outset on an initial layer of times, near
t = 0, in which the pathogenic load rapidly increases to a maximum and then
decreases as a result of an equally rapid immune system response.
3.1.1 Asymptotic solution
We now expand the dependent variables in powers of ǫ
P = P0 + ǫP1 + O(ǫ2
) I = I0 + ǫI1 + O(ǫ2
) (11)
substitute into Eqs. (9), (10) and (6), and set to zero like powers of ǫ; then to
dominant order
I′
0 = P0 P′
0 = (1 − I0)P0, (12)
I0(0) = 0 P0(0) = p (13)
8. 294 A. M. Whitman, H. Ashrafiuon
while to first order
I′
1 = P1 − I0 P′
1 = (1 − I0)P1 − P0I1, (14)
I1(0) = 0 P1(0) = 0. (15)
Dominant order term: We can obtain a first integral of the system given by
Eqs. (12) by substituting the first into the second, P′
0 = (1 − I0)I′
0, integrating,
and using the initial conditions, Eq. (13). We get
P0 =
s2 − (I0 − 1)2
2
, (16)
where we have written
s2
= 1 + 2p. (17)
Equation (16) indicates that this part of the phase plane trajectory, to dominant
order, is a parabola.
Now, on substituting Eq. (16) into the first of Eqs. (12), integrating and
solving for I0, we get
I0 = 1 + s tanh
s(T − T∗)
2
, (18)
where
T∗
=
2
s
tanh−1
1
s
=
1
s
ln
s + 1
s − 1
. (19)
Substituting Eq. (20) into Eq. (12) and differentiating then gives
P0 =
s2
2
sech2
s(T − T∗)
2
. (20)
Equations (18) and (20) are the dominant order solutions for the model. Clearly,
from Eq. (20), the scaled pathogen level starting from p reaches a maximum of
s2/2 = p + 1/2 at T = T∗ and thereafter subsides. The scaled immune level,
from Eq. (18), increases monotonically from 0 to 1 + s. It passes through 1 at
T∗ causing the pathogen level to decrease beyond that time, according to the
demands of the model [see either Eq. (5) or (10)].
First order term: We can derive a second order equation for I1 from Eqs.
(12) and (14) which, when integrated once, is
I′
1 + (I0 − 1)I1 = f0 ≡ I0 − s2
T + ln P0 − ln p. (21)
9. Asymptotic theory of an infectious disease model 295
An integrating factor of Eq. (21) is P−1
0 = exp{
(I0 − 1) dv} so, using the initial
condition given in Eq. (15), its solution can be written as
I1(T) = P0(T)
T
0
f0(v)
P0(v)
dv. (22)
We can get P1 either from the second of Eqs. (14), noting that it has the same
integrating factor as Eq. (21), or from the first of Eqs. (14) and using Eq. (21)
for I′
1. These results are
P1 = −P0(T)
T
0
I1(v) dv = 2I0 − s2
T + ln
P0
p
− (I0 − 1)I1. (23)
Equations (22) and (23) are the first order corrections for the model.
3.1.2 Large T behavior
In order to continue the solution beyond this initial layer, we need to determine
its behavior at large T. We do this by substituting Eq. (8), T = t/ǫ and doing
the limit ǫ → 0 with t fixed. In this way we obtain from Eqs. (18) and (20)
I0 ∼ 1 + s P0 ∼ 2s2
esT∗
e−st/ǫ
, (24)
where the terms not written are exponentially small. Using these in Eq. (21)
gives the leading two terms as
f0 ∼ −s(1 + s)
t
ǫ
+ C, (25)
where C = 1+s+sT∗ +ln(2s2/p) and again the terms not written are exponen-
tially small. Noting that the value of the integral in Eq. (22) comes principally
from the vicinity of its upper limit, we use the second of Eqs. (24) and (25) in
it, make the substitution x = t/ǫ − v, and evaluate the integral to get
I1
t
ǫ
∼ −(1 + s)
t
ǫ
+
1 + s + C
s
. (26)
Finally, on reexpressing Eq. (26) as a function of T and using it together with
the first of Eq. (24) in Eq. (11), we get the two term representation of the two
term expansion as
Icom ∼ 1 + s + ǫ
1 + s + C
s
− (1 + s)T
. (27)
10. 296 A. M. Whitman, H. Ashrafiuon
Here we have denoted its value in this limit, T → ∞, by the subscript com. We
obtain the expression for P1 in this limit by using Eq. (26) in the integral of Eq.
(23) and integrating. Combining the result with the second of Eqs. (24) in Eq.
(11) then gives
Pcom ∼ 2s2
esT∗
e−sT
1 − ǫ
1 + s + C
s
T −
1 + s
2
T2
. (28)
Equations (27) and (28) give expressions for P and I as they leave the ini-
tial layer in which the pathogen level peaks and then, due to the response of
the immune system, continually subsides. The immune system itself peaks after
the pathogen does and thereafter subsides as well. Clearly in this limit, P is
exponentially small while I = O(1).
A plot of Eqs. (11) using (18) and (22) for I and (20) and (23) for P is shown
in Fig. 2 where they are compared with those of a numerical integration of Eqs.
(6), (9), and (10), for ǫ = 0.1 and p = 1.5. The time scale used in the figure
is t rather than the initial layer time scale T. Clearly the approximate solution
leads to quantitatively accurate values for the populations throughout the initial
layer.
3.2 Main region (immune system relaxation region)
In this region, the time scale is t and P is exponentially small. Therefore, we
return to Eqs. (4) and (5), but make the substitution P = exp{−φ/ǫ}. Denoting
I = ı(t) and P = π(t) to distinguish these quantities here from their expressions
0 0.5 1 1.5 2
10
–1
10
0
10
1
Immune
level
Time
0 0.5 1 1.5 2
10
0
Pathogen
load
Time
numerical
asymptotic
numerical
asymptotic
Fig. 2 Comparison of dimensionless immune and pathogen levels from numerical integration with
the initial layer approximation for ǫ = 0.1( = 10) and p = 1.5
11. Asymptotic theory of an infectious disease model 297
in the initial layer, the equations are
dı
dt
=
1
ǫ
e−φ/ǫ
− ı
dφ
dt
= ı − 1. (29)
The first term on the right in the first of Eqs. (29) is exponentially small as long
as φ 0 and O(1). In that event they are decoupled to all powers in ǫ, and can
therefore be integrated sequentially. Doing this produces
ı = A(ǫ)e−t
+
H(t − tı )
ǫ
t
tı
e−(t−v)−φ(v)/ǫ
dv, (30)
where
φ = B(ǫ) − A(ǫ)e−t
− t. (31)
The behavior indicated by Eqs. (30) and (31) means that initially the immune
system cannot see the pathogen and so it continually relaxes. However, when
its level falls below ı = 1, the pathogen again begins to grow exponentially
thereby producing a reinfection before the immune system can respond.
The constant tı is somewhat arbitrary. It can be any time when π is exponen-
tially small. We have taken it here as the time when ı = 1 so that from either
Eq. (30) or (31), tı = ln(A). The constants of integration A and B are power
series in ǫ that need to be determined. We do this by matching the expressions
we obtain from Eqs. (30) and (31) as they approach the initial layer, lim t → 0,
with those we obtained from the initial layer as they approach this main region,
Eqs. (27) and (28).
3.2.1 Small t behavior
We obtain the small t behavior by substituting Eq. (8), t = ǫT into a two term
expansion of Eq. (30), doing the limit ǫ → 0 with T fixed, and retaining two
terms. In this way we obtain
ı ∼ A0 + ǫ(A1 − A0T) (32)
and
φ ∼ (B0 − A0) + ǫ[(B1 − A1) − (1 − A0)T] + ǫ2
A1T −
A0
2
T2
. (33)
Now Eqs. (27) and (32) are just different representations of the same function in
the transition interval between the two regions, and so must match functionally
12. 298 A. M. Whitman, H. Ashrafiuon
[6,13]. We see that this is indeed so for the following values of the constants
A0 = 1 + s A1 =
(1 + s + C)
s
. (34)
If we write the expression for P in this region, making use of Eq. (33), and
expanding for small ǫ, we obtain
π ∼ e−(B0−A0)/ǫ
e−(B1−A1)+(1−A0)T
1 − ǫ
A1T −
A0
2
T2
. (35)
With the following values of the constants Bk
B0 = A0 B1 = A1 − sT∗
− ln(2s2
). (36)
Equations (35) and (28) are identical.
3.3 Uniformly valid solution
Uniformly valid solutions for I and P can be obtained by adding the solutions
found in each of the two regions and subtracting their common part, [6,13].
Thus, for example,
Iu ∼ I(T) + ı(t) − Icom, (37)
where I(T) is given by Eq. (11) using Eqs. (18) and (22), ı(t) is given by Eqs.
(30) and (31) with Ak and Bk given by Eqs. (34) and (36), and Icom is given by
Eq. (27). The uniformly valid pathogen level is given in similar fashion by
Pu ∼ P(T) + π(t) − Pcom. (38)
These expressions are valid through order ǫ in the initial layer and main region,
but as t increases they ultimately become non-uniform due to the approach of
φ to 0, which results from the continual increase of the last term in Eq. (31).
3.4 Reinfection layer (immune system response region)
It is convenient to measure the recurrence of pathogen growth and the onset of
a second region of immune system response from the time that φ = O(ǫ). Thus,
we write
t = θr + ǫTr, (39)
where tr = ǫTr is the scaled time in this new layer with its origin at θr. Substi-
tuting Eq. (39) into Eq. (31), expanding e−ǫTr and retaining two terms, we find
13. Asymptotic theory of an infectious disease model 299
for π
π = e−(B1−A1 exp{−θr})
esrTr , (40)
where we determine θr and sr by
θr − B0 + A0e−θr = 0 sr = 1 − A0e−θr . (41)
The first of Eqs. (41) is the requirement that φ = O(ǫ), while the second is
a definition of sr. From Eq. (40) it is clear that for sr 0 we again see an
exponential growth of the pathogen.
This layer is similar to the original one, the difference being that here there
are no initial conditions. Instead, we require that the layer solution, when pro-
jected backward in time, matches with the solution coming out of the main
region, an expression that we have just found in Eq. (40). In this new immune
response layer the pathogen population is given to dominant order by Eq. (20),
however with constants of integration sr and T∗
r in place of s and T∗
Pr ∼
s2
r
2
sech2
sr(Tr − T∗
r )
2
. (42)
Here we must determine T∗
r by matching Eq. (42) in the limit Tr → −∞, with
Eq. (40). Substituting Tr = tr/ǫ into Eq. (42) and expanding for lim ǫ → 0 with
tr finite and negative, we find, to dominant order, after reexpressing it back in
terms of Tr
Pr ∼ 2s2
r e−srT∗
r esrTr .
We note that this matches Eq. (40) functionally provided that
T∗
r =
[ln(2s2
r ) + B1 − A1e−θr ]
sr
. (43)
Since sr 1 the second pathogen maximum is less than the first.
4 Results and conclusions
In the previous section we obtained results for the evolution curves for the path-
ogen and immune system levels in each region of interest and a uniformly valid
composite over all the regions. A comparison of this uniformly valid approxi-
mate solution with the corresponding numerical solution is shown in Fig. 3. For
the parameter values that were used, the agreement is excellent, although the
second pathogen peak is slightly low and occurs a little early, and the second
immune peak has not been included. Note in particular that the pathogen levels
are in good agreement over four orders of magnitude. Moreover, the agreement
14. 300 A. M. Whitman, H. Ashrafiuon
0 1 2 3 4 5 6 7 8
10
–1
10
0
10
1
Immune
level
Time
0 1 2 3 4 5 6 7 8
10
0
Pathogen
load
Time
numerical
asymptotic
numerical
asymptotic
Fig. 3 Comparison of dimensionless immune and pathogen levels from numerical integration with
the uniformly valid solution for ǫ = 0.1( = 10) and p = 1.5
in this region can be improved in two ways: first by making larger, although
its actual value must ultimately be determined by experiment, and second by
carrying out the calculations to higher order in . This latter alternative imposes
a tradeoff, because as the expressions become more accurate, they also become
more cumbersome, and thereby lose their ability to provide insight into what
are the important physical (biological) mechanisms.
The main features of these curves are their peak values and the times at
which they occur. We can easily obtain simple expressions for these quantities
from the general results of Sect. 3.
4.1 First pathogen population peak
Since any local extremum in the pathogen population, P′ = 0, occurs for I = 1
[see Eq. (10)], we get the time of the first peak value, TP, implicitly from the
second Eq. (11) as
1 = I0(TP) + ǫI1(TP) = I0(TP0 + ǫTP1) + ǫI1(TP0) + O(ǫ2
),
where we have made an expansion of TP in powers of ǫ. Taylor expanding I0
and solving, we find TP0 ∼ T∗ and TP1 = −2I1(T∗)/s2. Therefore
TP ∼ T∗
− ǫ
2I1(T∗)
s2
. (44)
15. Asymptotic theory of an infectious disease model 301
The maximum value of the pathogen population is then, using the second equal-
ity in Eq. (23)
PP ∼ P0(TP0 + ǫTP1) + ǫP1(TP0) =
s2
2
+ ǫ
2 − s2
T∗
+ ln
s2
2p
. (45)
Equations (44) and (45) are rather simple approximations to the actual values
(although the second term of Eq. (44) is not simple to evaluate it is a small
correction for small epsilon) yet they are quite accurate even for values of ǫ
that are not very small. Moreover, the first order correction terms are small
enough to be negligible as long as p is not too small, a condition for which the
approximation fails in any event.
On using just the dominant order approximations, and writing them in terms
of dimensional variables, we find, with µ = r − ma0/µ
P̄P ∼ p̄0 exp{θµ} +
(µ)2
2k0m
(46)
and
t̄P − θ ∼
1
2k0mP̄P
ln
2k0mP̄P + µ
2k0mP̄P − µ
. (47)
Note that the first term in Eq. (46) is just the value of the pathogen load at the
onset of immune system activity.
4.2 First immune population peak
After the pathogen has peaked and begun to subside the immune population
itself reaches a maximum. The time that this occurs, TI, is obtained by differen-
tiating the second of Eqs. (11) and setting I′ = 0. This gives
P0(TI) + ǫI′
1(TI) + O(ǫ2
) = 0.
Here TI must be large, so we use Eq. (24) for P0 and the derivative of Eq. (26)
for I′
1. Taking −sTI = ln(aǫ), we find a = (s + 1) exp(−sT∗)/2s2, so to dominant
order
TI ∼ T∗
+
1
s
ln
2s2
(1 + s)ǫ
=
1
s
ln
1
ǫ
+
1
s
ln
2s2
s − 1
. (48)
Using this value back in Eq. (11) along with Eqs. (18) and (26) gives the maxi-
mum immune population
II = 1 + s − (1 + s)
ǫ
s
ln
1
ǫ
+
ǫ
s
C + (1 + s)
1 − ln
2s2
s − 1
. (49)
16. 302 A. M. Whitman, H. Ashrafiuon
Equations (48) and (49) are not as accurate as their counterparts for the path-
ogen, because the first neglected term is of a relatively larger order than that
in the former case. Moreover, here both the maximum and the time it occurs
depend on ǫ as well as p, although the variation with ǫ is smaller than that of p.
The accuracy of these approximations can be judged by the results listed
in Table 1. There we have compiled two cases; the first, corresponding to the
values plotted in Figs. 1 and 2, ǫ = 0.1 and p = 1.5, where the approximation is
within its region of validity, and the second, for ǫ = 0.143 and p = 0.312, which
we obtained from using the parameter estimates given in [9]. In this second
case, the value of p is somewhat small [it is approaching O(ǫ)], although the
peak values are still quite accurate.
4.3 Second pathogen population peak
The second time the pathogen peaks, its maximum value is given to dominant
order by Eq. (42) as
PP 2 ∼ s2
r /2 (50)
with sr given by the second Eq. (41), and occurs at a time given by Eq. (39)
tP 2 ∼ θr + ǫT∗
r , (51)
where θr is given by the solution of the first of Eqs. (41), and T∗
r given by Eq.
(43). Accordingly, the maximum does not depend on ǫ. The time however, does
depend on both parameters, although more strongly on p than on ǫ. Results are
listed in Table 2.
Table 1 Numerical and analytical determinations of the first peaks of the pathogen and immune
system levels and times. For the ǫ = 0.1 data, p = 1.5 and in the second case, p = 0.312
ǫ tP PP tI II
Num Asym Num Asym Num Asym Num Asym
0.100 0.06 0.06 2.006 2.007 0.24 0.23 2.60 2.57
0.143 0.25 0.25 0.854 0.854 0.57 0.54 1.79 1.72
Table 2 Numerical and analytical determinations of the second peak of the pathogen level and its
time of occurrence.
ǫ p tP2 PP2
Num Asym Num Asym
0.100 1.500 3.03 2.97 0.366 0.337
0.143 0.312 2.86 2.52 0.273 0.229
17. Asymptotic theory of an infectious disease model 303
4.4 Uniformly valid expansion
Because they match functionally, we can combine our results in each region to
obtain uniformly valid evolution curves. Equations (37) and (38) illustrate how
this is done. In the same way, we can append the result of the reinfection layer,
Eq. (42) to Eq. (38) and subtracting the new common part for Pr. We have
shown a plot of this type in Fig. 2. Results for the peak values and their times,
that are obtained from the uniformly valid approximations are generally less
accurate than those given by the regional solutions.
4.5 Subsequent behavior
The solution described above can be continued by adding a second main region
at the end of the reinfection layer and matching exactly as we did previously,
a second reinfection layer after that, and so on. In this way we obtain the time
evolution of each of these populations as a decaying oscillation about its respec-
tive equilibrium value. Eventually, when I − 1 = O(ǫ) and P − ǫ = O(ǫ3/2), the
nonlinear terms are small and the populations behave like a weakly damped
harmonic oscillator with dominant order natural frequency of ωn = ǫ−1/2. This
means that successive population maxima in this case occur periodically, with
the period given by, to dominant order, tp ∼ 2πǫ1/2.
These results can be obtained from a straightforward, small ǫ, multiple scale
asymptotic analysis, [10], of Eqs. (4) and (5) with the population scalings indi-
cated here, and a time scaling of τ = ωnt.
4.6 Conclusions
We have shown how the method of matched asymptotic expansions provides
a formalism for obtaining an approximate solution to a previously developed
model for the transient dynamics of the interaction between an invading path-
ogen and its host’s immune system in acute infectious diseases. We have shown
that this solution is particularly accurate in predicting the time of occurrence
and amplitude of the maximum pathogenic load and the subsequent remission
periods and reinfection peaks. Not only is this solution quantitatively accurate,
but it is also qualitatively simple and easy to understand.
Due to the similarity of this model to those that characterize infectious dis-
ease, we believe that this formalism could be successfully applied to them as
well. For example, of the various deterministic models presented in [5], the clas-
sic SIR endemic model would seem to be a prime candidate, using the contact
number as the large parameter. This is suggested because the infective-suscep-
tible fraction phase plane resembles a nonlinear oscillator and because of the
similarity of the bifurcation geometry in the two models, as we noted in Sect. 2.2.
18. 304 A. M. Whitman, H. Ashrafiuon
References
1. Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases, 2nd edn, pp. 82–88. Hafner,
NY (1975)
2. Barenblatt, G.I.: Scaling, pp. 22–26. Cambridge University Press, Cambridge (2003)
3. Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model
Building, Analysis, and Interpretation, pp. 43–51. Wiley, NY (2000)
4. Endo, Y., Igarashi, T., Nishimura, Y., Buckler, C., Buckler-White, A., Plishka, R., Dimitrov,
D.S., Martin, M.A.: Short- and long-term clinical outcomes in rhesus monkeys inoculated with
a highly pathogenic chimeric simian/human immunodeficiency virus. J. Virol. 74, 6935–6945
(2000)
5. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev 42, 599–653 (2000)
6. Hinch, E.J.: Perturbation Methods, pp. 52–101. Cambridge University Press, Cambridge (1991)
7. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics
Proc. Roy. Soc. London, Ser. A 115, 700–721 (1927)
8. Marsh, K.: Malaria - a neglected disease? Parasitology 104 (Suppl) pp. S53–S69 (1992)
9. Mohtashemi, M., Levins, R.: Transient dynamics and early diagnosis in infectious disease. J.
Math. Biol. 43, 446–470 (2001)
10. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations, pp. 120–121. Wiley-Interscience, New York
(1979)
11. Paunio, M., Peltola, H., Valle, M., Davidkin, I., Virtanen, M., Heinonen, O.P.: Explosive school-
based measles outbreak: intense exposure may have resulted in high risk, even among revacc-
inees. Am. J. Epidemiol. 148, 1103–1110 (1998)
12. Singer, B.: Mathematical Models of infectious diseases: seeking new tools for planning and
evaluating control programs. Popul. Deve. Rev. 10, 347–365 (1984)
13. Van Dyke, M.: Perturbation Methods in Fluid Mechanics. p. 20. Academic, New York (1966)