This document discusses probability distributions of simple stochastic infection and recovery processes. It begins by introducing common probability distributions like binomial and negative binomial that are relevant for modeling infection and recovery. It then formulates general infection and recovery processes and derives conditions for the existence of a unique stationary probability distribution. Several specific infection and recovery processes are analyzed, including simple infection, simple recovery, simple infection and recovery, and simple infection and recovery with immigration. Explicit formulas are derived for the probability distributions and moment generating functions of each process.
Gamma and inverse Gaussian frailty models: A comparative studyinventionjournals
Frailty models have become very popular during the last three decades and their applications are numerous. The main goal of this manuscript is to compare two frailty models (gamma frailty model and inverse Gaussian frailty model) each of which has a log-logistic distribution to be its baseline hazard function. A real data set is applied for the two considered frailty models in order to deal with models comparison. It has been concluded that the gamma frailty model is the best model fits this data set. Then the inverse Gaussian frailty model, which provides a better fit of the considered data set than the Cox’s model.
A Study on stochastic Models for HIV infection and AidsMangaiK4
Abstract - This paper is review the Epidemiological modeling on threshold measuring measurement using probability and Mathematical Statistics.Mathematical models which would estimate the probable longevity of the infected and the extent of the usefulness of preventive measures Mathematical models that are purely threshold developed with less viability of applications.Some of the important and theoretical models developed include in the review.
A stage-structured delayed advection reaction-diffusion model for single spec...IJECEIAES
In this paper, we derived a delay advection reaction-diffusion equation with linear advection term from a stage-structured model, then the derived equation is used under the homogeneous Dirichlet boundary conditions 푢 푚 ( 0, 푡 ) = 0, 푢 푚 ( 퐿, 푡 ) = 0, and the initial condition 푢 푚 ( 푥, 0 ) = 푢 ( 푥 ) > 0, 푥 ∈ [−휏, 0] with 푢 푚 0 (0) > 0 in order to find the minimum value of domain 퐿 that prevents extinction of the species under the effect of advection reaction-diffusion equation. Finally, for the measurement the time lengths from birth to the development of the species population, time delays are integrated. 푚 0
Life Expectancy Estimate with Bivariate Weibull Distribution using Archimedea...CSCJournals
Archimedean copulas are used to construct bivariate Weibull distributions. Co-movement structures of variables are analyzed through the copulas, where the tail dependence between the variables is explored with more flexibility. Based on the distance between the copula distribution and its empirical version, a copula that may best fit data is selected. With extra computing costs, the adequacy of the copula chosen is then assessed. When multiple myeloma data are considered, it is found that relationship between survival time of a patient and the hemoglobin level is well described by the Clayton copula. The bivariate Weibull distribution constructed by the copula is used to estimate value at risk from which we investigate the anticipated longest life expectancy of a patient with the disease over the treatment period.
Gamma and inverse Gaussian frailty models: A comparative studyinventionjournals
Frailty models have become very popular during the last three decades and their applications are numerous. The main goal of this manuscript is to compare two frailty models (gamma frailty model and inverse Gaussian frailty model) each of which has a log-logistic distribution to be its baseline hazard function. A real data set is applied for the two considered frailty models in order to deal with models comparison. It has been concluded that the gamma frailty model is the best model fits this data set. Then the inverse Gaussian frailty model, which provides a better fit of the considered data set than the Cox’s model.
A Study on stochastic Models for HIV infection and AidsMangaiK4
Abstract - This paper is review the Epidemiological modeling on threshold measuring measurement using probability and Mathematical Statistics.Mathematical models which would estimate the probable longevity of the infected and the extent of the usefulness of preventive measures Mathematical models that are purely threshold developed with less viability of applications.Some of the important and theoretical models developed include in the review.
A stage-structured delayed advection reaction-diffusion model for single spec...IJECEIAES
In this paper, we derived a delay advection reaction-diffusion equation with linear advection term from a stage-structured model, then the derived equation is used under the homogeneous Dirichlet boundary conditions 푢 푚 ( 0, 푡 ) = 0, 푢 푚 ( 퐿, 푡 ) = 0, and the initial condition 푢 푚 ( 푥, 0 ) = 푢 ( 푥 ) > 0, 푥 ∈ [−휏, 0] with 푢 푚 0 (0) > 0 in order to find the minimum value of domain 퐿 that prevents extinction of the species under the effect of advection reaction-diffusion equation. Finally, for the measurement the time lengths from birth to the development of the species population, time delays are integrated. 푚 0
Life Expectancy Estimate with Bivariate Weibull Distribution using Archimedea...CSCJournals
Archimedean copulas are used to construct bivariate Weibull distributions. Co-movement structures of variables are analyzed through the copulas, where the tail dependence between the variables is explored with more flexibility. Based on the distance between the copula distribution and its empirical version, a copula that may best fit data is selected. With extra computing costs, the adequacy of the copula chosen is then assessed. When multiple myeloma data are considered, it is found that relationship between survival time of a patient and the hemoglobin level is well described by the Clayton copula. The bivariate Weibull distribution constructed by the copula is used to estimate value at risk from which we investigate the anticipated longest life expectancy of a patient with the disease over the treatment period.
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.
Logistic Loglogistic With Long Term Survivors For Split Population ModelWaqas Tariq
Split population models are also known as mixture model . The data used in this paper is Stanford Heart Transplant data. Survival times of potential heart transplant recipients from their date of acceptance into the Stanford Heart Transplant program [3]. This set consists of the survival times, in days, uncensored and censored for the 103 patients and with 3 covariates are considered Ages of patients in years, Surgery and Transplant, failure for these individuals is death. Covariate methods have been examined quite extensively in the context of parametric survival models for which the distribution of the survival times depends on the vector of covariates associated with each individual. See [6] for approaches which accommodate censoring and covariates in the ordinary exponential model for survival. Currently, such mixture models with immunes and covariates are in use in many areas such as medicine and criminology. See for examples [4][5][7]. In our formulation, the covariates are incorporated into a split loglogistic model by allowing the proportion of ultimate failures and the rate of failure to depend on the covariates and the unknown parameter vectors via logistic model. Within this setup, we provide simple sufficient conditions for the existence, consistency, and asymptotic normality of a maximum likelihood estimator for the parameters involved. As an application of this theory, the likelihood ratio test for a difference in immune proportions is shown to have an asymptotic chi-square distribution. These results allow immediate practical applications on the covariates and also provide some insight into the assumptions on the covariates and the censoring mechanism that are likely to be needed in practice. Our models and analysis are described in section 5.
Improving predictability and performance by relating the number of events and...Asoka Korale
Many processes require an estimate of the time over which to observe a certain number of events. The applications include queuing models and buffer management in electronics and telecommunications and the characterization of trading patterns in market surveillance. It is common practice in these applications to take a deterministic approach, modeling the events over intervals of time of a particular duration or considering the event inter-arrival times in order to estimate an average rate and a measure of its dispersion.
In this paper however we establish a probabilistic relationship between the number of events and the time over which to observe them. The total time over which to observe a certain number of events is equivalent to the sum of their event inter-arrival times, making the number of events and the number of inter-arrival times in the sum also equivalent. By this sum of random variables, we establish a stochastic relationship between the number of events and the total time interval over which to observe them, allowing greater flexibility in characterizing the relationships between the underlying distributions. We also use this relationship to estimate the uncertainty in the time interval taken to observe a certain number of events and relate it to an uncertainty in the average number of events observed in that interval.
The event inter-arrival times are thus modeled as a sequence of random variables drawn from a single distribution. These random variables could be drawn from a distribution estimated from historical data governing the particular arrival process or from a particular distribution used to model it. The subject of this paper is then to utilize this idea to model the behavior of a queue and server system where each state and the state transition probabilities are also stochastic. Clearer insights in to the performance of such systems is also envisaged with this type of analysis.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
I argue in favour of an elementary analogy that links physical and biological systems. In two words, the interplay between reproduction and cooperation may turn out to be relevant to both physics and biology.
This basic analogy has probably been overlooked as a consequence of the "shut up and calculate" dogma which dominated 20th century physics.
El paquete TestSurvRec implementa las pruebas estadíıticas para comparar dos curvas de supervivencia con eventos recurrentes. Este software ofrece herramientas ´utiles para el an´alisis de la supervivencia en el campo de la biomedicina, epidemiolog´ıa, farmac´eutica y otras áreas. El paquete TestSurvRec contiene dos conjuntos de datos con eventos recurrentes, un conjunto de datos referido al experimento de Byar que contiene los tiempos de recurrencia de tumores de c´ancer de vejiga en los pacientes tratados con piridoxina, tiotepa o considerado como un placebo. Y otro conjunto de datos que contiene los tiempos de rehospitalizaci´on despu´es de la cirug´ıa en pacientes con cáncer colorrectal. Estos datos provienen de un estudio que se llev´o a cabo en el Hospital de Bellvitge, un hospital universitario p´ublico en Barcelona (España).
Mathematical Model for Infection and Removalijtsrd
The mathematical model of infectious diseases is a tool that has been used to study the mechanisms by which disease spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic. We envisaged a community of n individuals comprising at time t, x, susceptible, y infectious in circulation and z individuals who were isolated, dead, or recovered and immune. We further postulated infection and removal rates ßand , so that there wave ßxydt new infections and ydt removals in time tithe simplest way to do this is to introduce a birth parameter µ, so at to give µdt new susceptible in time dt. If the population is to remain stable the arrival of new susceptible must be balanced by an appropriately defined birth rate. The present paper represents the model in special way, in which the infection occurs in human`s body then the resistance of body gradually decays same as motion decays in damped oscillation. On solving the equation of model, we get solution that gives the idea about the seasonal variation in infection. Shukla Uma Shankar "Mathematical Model for Infection and Removal" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-2 , February 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38523.pdf Paper Url: https://www.ijtsrd.com/mathemetics/other/38523/mathematical-model-for-infection-and-removal/shukla-uma-shankar
Este manual es útil e indispensable para el uso del "Package TesSurvRec_1.2.1" de CRAN. Importante para estadístico, médicos, farmacéuticos, seguros, bancos, ingenieros, psicólogos, astrónomos, entre otras profesiones. Son pruebas estadísticas que se utilizan para medir diferencias entre funciones del análisis de supervivencias de grupos de poblaciones que manifiestan eventos recurrentes.
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
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Modeling and qualitative analysis of malaria epidemiologyIOSR Journals
We develop and analyze a mathematical model on malaria dynamics using ordinary differential equations, in order to investigate the effect of certain parameters on the transmission and spread of the disease. We introduce dimensionless variables for time, fraction of infected human and fraction of infected mosquito and solve the resulting system of ordinary differential equations numerically. Equilibrium points are established and stability criteria analyzed. The results reveal that the parameters play a crucial role in the interaction between human and infected mosquito.
Application of Semiparametric Non-Linear Model on Panel Data with Very Small ...IOSRJM
-This research work investigated the behaviour of a new semiparametric non-linear (SPNL) model on
a set of panel data with very small time point (T = 1). The SPNL model incorporates the relationship between
individual independent variable and unobserved heterogeneity variable. Five different estimation techniques
namely; Least Square (LS), Generalized Method of Moments (GMM), Continuously Updating (CU), Empirical
Likelihood (EL) and Exponential Tilting (ET) Estimators were employed for the estimation; for the purpose of
modelling the metrical response variable non-linearly on a set of independent variables. The performances of
these estimators on the SPNL model were examined for different parameters in the model using the Least
Square Error (LSE), Mean Absolute Error (MAE) and Median Absolute Error (MedAE) criteria at the lowest
time point (T = 1). The results showed that the ET estimator which provided the least errors of estimation is
relatively more efficient for the proposed model than any of the other estimators considered. It is therefore
recommended that the ET estimator should be employed to estimate the SPNL model for panel data with very
small time point.
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.
Logistic Loglogistic With Long Term Survivors For Split Population ModelWaqas Tariq
Split population models are also known as mixture model . The data used in this paper is Stanford Heart Transplant data. Survival times of potential heart transplant recipients from their date of acceptance into the Stanford Heart Transplant program [3]. This set consists of the survival times, in days, uncensored and censored for the 103 patients and with 3 covariates are considered Ages of patients in years, Surgery and Transplant, failure for these individuals is death. Covariate methods have been examined quite extensively in the context of parametric survival models for which the distribution of the survival times depends on the vector of covariates associated with each individual. See [6] for approaches which accommodate censoring and covariates in the ordinary exponential model for survival. Currently, such mixture models with immunes and covariates are in use in many areas such as medicine and criminology. See for examples [4][5][7]. In our formulation, the covariates are incorporated into a split loglogistic model by allowing the proportion of ultimate failures and the rate of failure to depend on the covariates and the unknown parameter vectors via logistic model. Within this setup, we provide simple sufficient conditions for the existence, consistency, and asymptotic normality of a maximum likelihood estimator for the parameters involved. As an application of this theory, the likelihood ratio test for a difference in immune proportions is shown to have an asymptotic chi-square distribution. These results allow immediate practical applications on the covariates and also provide some insight into the assumptions on the covariates and the censoring mechanism that are likely to be needed in practice. Our models and analysis are described in section 5.
Improving predictability and performance by relating the number of events and...Asoka Korale
Many processes require an estimate of the time over which to observe a certain number of events. The applications include queuing models and buffer management in electronics and telecommunications and the characterization of trading patterns in market surveillance. It is common practice in these applications to take a deterministic approach, modeling the events over intervals of time of a particular duration or considering the event inter-arrival times in order to estimate an average rate and a measure of its dispersion.
In this paper however we establish a probabilistic relationship between the number of events and the time over which to observe them. The total time over which to observe a certain number of events is equivalent to the sum of their event inter-arrival times, making the number of events and the number of inter-arrival times in the sum also equivalent. By this sum of random variables, we establish a stochastic relationship between the number of events and the total time interval over which to observe them, allowing greater flexibility in characterizing the relationships between the underlying distributions. We also use this relationship to estimate the uncertainty in the time interval taken to observe a certain number of events and relate it to an uncertainty in the average number of events observed in that interval.
The event inter-arrival times are thus modeled as a sequence of random variables drawn from a single distribution. These random variables could be drawn from a distribution estimated from historical data governing the particular arrival process or from a particular distribution used to model it. The subject of this paper is then to utilize this idea to model the behavior of a queue and server system where each state and the state transition probabilities are also stochastic. Clearer insights in to the performance of such systems is also envisaged with this type of analysis.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
I argue in favour of an elementary analogy that links physical and biological systems. In two words, the interplay between reproduction and cooperation may turn out to be relevant to both physics and biology.
This basic analogy has probably been overlooked as a consequence of the "shut up and calculate" dogma which dominated 20th century physics.
El paquete TestSurvRec implementa las pruebas estadíıticas para comparar dos curvas de supervivencia con eventos recurrentes. Este software ofrece herramientas ´utiles para el an´alisis de la supervivencia en el campo de la biomedicina, epidemiolog´ıa, farmac´eutica y otras áreas. El paquete TestSurvRec contiene dos conjuntos de datos con eventos recurrentes, un conjunto de datos referido al experimento de Byar que contiene los tiempos de recurrencia de tumores de c´ancer de vejiga en los pacientes tratados con piridoxina, tiotepa o considerado como un placebo. Y otro conjunto de datos que contiene los tiempos de rehospitalizaci´on despu´es de la cirug´ıa en pacientes con cáncer colorrectal. Estos datos provienen de un estudio que se llev´o a cabo en el Hospital de Bellvitge, un hospital universitario p´ublico en Barcelona (España).
Mathematical Model for Infection and Removalijtsrd
The mathematical model of infectious diseases is a tool that has been used to study the mechanisms by which disease spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic. We envisaged a community of n individuals comprising at time t, x, susceptible, y infectious in circulation and z individuals who were isolated, dead, or recovered and immune. We further postulated infection and removal rates ßand , so that there wave ßxydt new infections and ydt removals in time tithe simplest way to do this is to introduce a birth parameter µ, so at to give µdt new susceptible in time dt. If the population is to remain stable the arrival of new susceptible must be balanced by an appropriately defined birth rate. The present paper represents the model in special way, in which the infection occurs in human`s body then the resistance of body gradually decays same as motion decays in damped oscillation. On solving the equation of model, we get solution that gives the idea about the seasonal variation in infection. Shukla Uma Shankar "Mathematical Model for Infection and Removal" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-2 , February 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38523.pdf Paper Url: https://www.ijtsrd.com/mathemetics/other/38523/mathematical-model-for-infection-and-removal/shukla-uma-shankar
Este manual es útil e indispensable para el uso del "Package TesSurvRec_1.2.1" de CRAN. Importante para estadístico, médicos, farmacéuticos, seguros, bancos, ingenieros, psicólogos, astrónomos, entre otras profesiones. Son pruebas estadísticas que se utilizan para medir diferencias entre funciones del análisis de supervivencias de grupos de poblaciones que manifiestan eventos recurrentes.
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
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Modeling and qualitative analysis of malaria epidemiologyIOSR Journals
We develop and analyze a mathematical model on malaria dynamics using ordinary differential equations, in order to investigate the effect of certain parameters on the transmission and spread of the disease. We introduce dimensionless variables for time, fraction of infected human and fraction of infected mosquito and solve the resulting system of ordinary differential equations numerically. Equilibrium points are established and stability criteria analyzed. The results reveal that the parameters play a crucial role in the interaction between human and infected mosquito.
Similar to The Probability distribution of a Simple Stochastic Infection and Recovery Processes (20)
Application of Semiparametric Non-Linear Model on Panel Data with Very Small ...IOSRJM
-This research work investigated the behaviour of a new semiparametric non-linear (SPNL) model on
a set of panel data with very small time point (T = 1). The SPNL model incorporates the relationship between
individual independent variable and unobserved heterogeneity variable. Five different estimation techniques
namely; Least Square (LS), Generalized Method of Moments (GMM), Continuously Updating (CU), Empirical
Likelihood (EL) and Exponential Tilting (ET) Estimators were employed for the estimation; for the purpose of
modelling the metrical response variable non-linearly on a set of independent variables. The performances of
these estimators on the SPNL model were examined for different parameters in the model using the Least
Square Error (LSE), Mean Absolute Error (MAE) and Median Absolute Error (MedAE) criteria at the lowest
time point (T = 1). The results showed that the ET estimator which provided the least errors of estimation is
relatively more efficient for the proposed model than any of the other estimators considered. It is therefore
recommended that the ET estimator should be employed to estimate the SPNL model for panel data with very
small time point.
Exploring 3D-Virtual Learning Environments with Adaptive RepetitionsIOSRJM
In spatial tasks, the use of cognitive aids reduce mental load and therefore being appealing to trainers and trainees. However, these aids can act as shortcuts and prevents the trainees from active exploration which is necessary to perform the task independently in non-supervised environment. In this paper we used adaptive repetition as control strategy to explore the 3D- Virtual Learning environments. The proposed approach enables the trainee to get the benefits of cognitive support while at the same time he is actively involved in the learning process. Experimental results show the effectiveness of the proposed approach.
Mathematical Model on Branch Canker Disease in Sylhet, BangladeshIOSRJM
Members of the Camellia genus of plants may be affected by many of the diseases, pathogens and pests that mainly affect the tea plant (Camellia sinensis). One of the most common diseases which are found in tea garden is known as Branch Canker (BC) [1]. Recent outbreaks of these diseases is not only hampering the production of tea, but also stupendously hampering our national economy. In this paper, a simple SEIR model has been used to analyze the dynamics of Branch Canker in the tea garden. The stability of the system is analyzed for the existence of the disease free equilibrium. We established that there exists a disease free equilibrium point that is locally asymptotically stable when the reproduction number Ro< 1 and unstable when Ro> 1. Theoretically, we analyze the BC model. Finally, we numerically tested the theoretical results in MATLAB.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
A Numerical Study of the Spread of Malaria Disease with Self and Cross-Diffus...IOSRJM
: A study of the SIS model of malaria disease with a view to observing the effects of self and crossdiffusion on spatial dynamics is undertaken. Three different cases based on self-diffusion and cross-diffusion are chosen for the investigation. Two cases of cross-diffusion without self-diffusion are also considered in order to see the effects of diffusion on the transmission of malaria. Basic reproductive numbers and bifurcation values are calculated for each case. A series of numerical simulations based on self and cross-diffusion is performed. It is observed that with positive cross-diffusion and self-diffusion in the system, there is a significant increase in the proportion of both infected human and mosquito populations. The proportion of infected humans increases markedly with cross diffusion in the system. This also gives rise to some oscillations across the domain.
Fibonacci and Lucas Identities with the Coefficients in Arithmetic ProgressionIOSRJM
We define an analogous pair of recurrence relations that yield some Fibonacci, Lucas and generalized Fibonacci identities with the coefficients in arithmetic progression. One relation yields same sign identities and the other alternating signs identities. We also show some new results for negative indexed Fibonacci and Lucas Sequences.
A Note on Hessen berg of Trapezoidal Fuzzy Number MatricesIOSRJM
The fuzzy set theory has been applied in many fields such as management, engineering, theory of matrices and so on. in this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNS) are defined. We also have been defined some operations on trapezoidal fuzzy matrices(TrFMs). The notion of Hessenberg fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.
Enumeration Class of Polyominoes Defined by Two ColumnIOSRJM
Abacus diagram is a graphical representation for any partition µ of a positive integer t. This study presents the bead positions as a unite square in the graph and de ne a special type of e-abacus called nested chain abacus 픑 which is represented by the connected partition. Furthermore, we redefined the polyominoes as a special type of e-abacus diagram. Also, this study reveals new method of enumerating polyominoes special design when e=2
A New Method for Solving Transportation Problems Considering Average PenaltyIOSRJM
Vogel’s Approximation Method (VAM) is one of the conventional methods that gives better Initial Basic Feasible Solution (IBFS) of a Transportation Problem (TP). This method considers the row penalty and column penalty of a Transportation Table (TT) which are the differences between the lowest and next lowest cost of each row and each column of the TT respectively. In a little bit different way, the current method consider the Average Row Penalty (ARP) and Average Column Penalty (ACP) which are the averages of the differences of cell values of each row and each column respectively from the lowest cell value of the corresponding row and column of the TT. Allocations of costs are started in the cell along the row or column which has the highest ARP or ACP. These cells are called basic cells. The details of the developed algorithm with some numerical illustrations are discussed in this article to show that it gives better solution than VAM and some other familiar methods in some cases.
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Devaney Chaos Induced by Turbulent and Erratic FunctionsIOSRJM
Let I be a compact interval and f be a continuous function defined from I to I. We study the relationship between tubulent function, erratic function and Devaney Chaos.
Bayesian Hierarchical Model of Width of Keratinized GingivaIOSRJM
The purpose of this paper is to offer a method for studying the treatment result of gingival recession. A parameter showing the success of the surgical treatment of gingival recessions is the keratinized gingival width. It was measured four times: at baseline, after 1 month, after 3 months and after 6 months. Every patient has data that can be described by an individual trend. Bayesian hierarchical model of the keratinized gingiva width’s increase rate is built.
A Novel Approach for the Solution of a Love’s Integral Equations Using Bernst...IOSRJM
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The Probability distribution of a Simple Stochastic Infection and Recovery Processes
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. III (Jan. - Feb. 2017), PP 20-29
www.iosrjournals.org
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 20 | Page
The Probability distribution of a Simple Stochastic Infection and
Recovery Processes
Olabisi O. Ugbebor1
, Joshua O. Okoro 2
1
(Department of Mathematics, University of Ibadan, Ibadan, Nigeria.)
2
(Industrial Mathematics, Physical Science Department College of Science and Engineering Landmark
University, Omu-Aran, Kwara, Nigeria)
Abstract: It is true that when infections occur there might not be recovery. In this case either infections will
occur or will do not occur. When measures are put in place to reduce the rate of infection, there might be a
tendency for this rate to have an effect on the growth of the infection. When this rate is not checked the whole
population might get infected with time. In this work we use the birth-death process to describe a simple but
classical infection processes, recovery processes, infection and recovery processes and finally, infection and
recovery with immigration processes. For each of these processes explicit formulas are derived for their
probability distributions and moment generating functions. We formulate a general infection and recovery
process. The conditions for existence of a unique stationary probability for this general infection and recovery
process is stated. A density-dependent infection and recovery process is formulated. A quasi-stationary
probability distribution is defined, where the process is conditioned on non-extinction.We note here that the
infection process has Negative Binomial Distribution and recovery process has a Binomial Distribution.
Keywords: Generating Functions, Kolmogorov Differential Equations, Markov Property, Method of
Characteristics
Subject Classification: MSC 2010 60G05, 60G20, 62H10.
I. Introduction
When considering a random experiment with sample space, say , we define a random variable
as a single valued function that maps or assigns a real number called the value of to each sample
point of . A probability space consist of three part , where is the sample space, is the event
space and maps events to the interval . Distribution functions and probability distribution function is a
rule that helps you describe the spread of random variable. Example of such maps, assignment or rule is the
Probability distribution function or the cumulative distribution function (CDF) of a real-valued variable
at . The probability distribution function may be design for discrete and continuous random variable
respectively. A well-known probability mass function for a discrete random variable are for Binomial and
Negative Binomial random variable which are given below;
,
Where and
(1)
,
Where
(2)
respectfully.
The mean for a binomial random variable is given by , while that of negative binomial is given by , with
variance and respectively, while the moment generating function for the binomial and
negative binomial distribution is given as and respectfully. If i.e,
.
A collection of random variable , where is some index set and is the
common sample space of the random variable is called a stochastic process. Now, for each fixed ,
denotes a single random variable defined on , also for each fixed , corresponds to a function
defined on that is called a sample path or a stochastic realization of the process. However for the collection
random variable , where for all , in this case is some index set.
One of the classification of random processes is Markov Processes. When
, whenever ,
such that is a random variable, such a random variable possesses a Markov Process.
2. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 21 | Page
Our interest in this work is to show the probability distribution function of Infection and recovery
processes. we formulate a general infection and recovery process. Then the conditions for existence of a unique
stationary probability for this general infection and recovery process is stated. It is shown that if the process is
nonexplosive, then the general infection and recovering process converges to this stationary probability
distribution. Simple but classical infection and recovery processes are presented such as: infection, recovery,
infection and recovery, and infection and recovery with immigration processes. Explicit formulas are derived for
the moment generating functions. In addition, for the simple infection and simple recovery processes, explicit
formulas are derived for their probability distributions. Queueing processes are taken as important application of
infection and recovery processes, where infection and recovery are arrivals and departures in the system. A
positive stationary probability distribution may not exist for many infection and recovery processes in biology
because the zero state (extinction) is absorbing. For such types of processes, the probability of extinction and the
expected time until population extinction are investigated. Example of an infection and recovery process with an
absorbing state at zero is logistic growth of infection. This density-dependent infection and recovery process is
formulated. A quasi-stationary probability distribution is defined, where the process is conditioned on non-
extinction. two types of processes that have not been considered previously, an explosive infection process and a
nonhomogeneous infection and recovery process.
A good book for stochastic processes with applications to biology is the classic textbook by Bailey,
which he title; The Elements of Stochastic Processes with Applications to the Natural Sciences, which has been
referenced frequently since its initial publication in 1964.
In his book he said suppose that the random variable has a m.g.f. , it can be shown that the . of
is and that gives the moment of
about its mean [1]. A good book for probability theory is the book by [2]. Another one for probability theory
and probability models is the book by [3]. Other references on the basic theory of probability and statistics
include [4]. For a more general proof based on characteristic functions, see [5] or [6]. [7], and [8]. The history of
the development of the theory of stochastic process began with the study of biological as well as physical
problems. According to [9]. One of the first occurrences of a Markov chain may have been in explaining
rainfall patterns in Brussels by Quetelet in 1852. The simple branching process was invented by Bienaym´e in
1845 to compute the probability of extinction of a family surname. In 1910, Rutherford and Geiger and the
mathematician Bateman described the disintegration of radioactive substances using a Poisson process. In 1905,
Einstein described Brownian motion of gold particles in solution, and in 1900, Bachelier used this same process
to describe bond prices [9]. The simple birth and death process was introduced by McKendrick in 1914 to
describe epidemics. Gibbs in 1902 used nearest-neighbor models to describe the interactions among large
systems of molecules [9]. Stochastic processes are now used to model many different types of phenomena from
a variety of different areas, including biology, physics, chemistry, finance, economics, and engineering. [10],
simulate the simple birth process show that that the random variable for the inter-event time is exponentially
distributed. He simulates the time , by mapping it into . The function subroutine RAND in the
FORTRAN [11] program is a pseudo–random number generator and is based on the recursion relation
, where and the modulus is a Mersenne prime.
The term “rand” in the MATLAB program is a built-in MATLAB function for a uniform random number
generator on . A birth and death process with immigration was developed by [12] based on a spatially
implicit patch model. in the stochastic formulation by [12], it is assumed that is the random variable for the
number of patches occupied at time t. If , then the birth and death rates will be denoted as birth
rate and death rate (see, e.g., [1]; [13]; [14]; [15]; [16]. For a more thorough but elementary introduction to
queueing systems, please consult [17], [8], or [16] the distribution can be assumed to be approximately normal
or lognormal, referred to as moment closure assumptions (e.g., [18]; [19]).
II. General infection and recovery Process
Let denote the change in state of the stochastic process from to . That is,
. If , then the infection and recovery rates will be denoted as infection rate
and recovery rate. For the general birth and death process see [1], [13], [14], [16]. The continuous-time
infection and recovery Markov chain may have either a finite or infinite state space
or . Assume the infinitesimal transition probabilities for this process are;
(3)
for sufficiently small, where for and . It is often the case that ,
except when there is immigration. The initial conditions are where is the identity
matrix . In a small interval of time , at most one change in state can occur, either a infection,
3. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 22 | Page
or a recovering, . From (3), we can derive the forward Kolmogorov differential
equations for . Now consider the transition probability , for sufficiently small, we have;
(4)
which holds for all and in the state space with the exception of and (if the population size is
finite). If , then . If is the maximum
population size, then
(5)
If is the maximum population size, then
(6)
where and for . Subtracting , , and from equation (4), (5), and
(6) respectively, dividing by ∆t and taking the limit as ∆t → 0, yields the forward Kolmogorov differential
equations for the general birth and death process,
for and . For a finite state space, the differential equation for is
.
which can be written in matrix form to give the forward Kolmogorov differential equations which are
, where is the generator matrix.
Theorem 0.1
Suppose the CTMC is a general continuous-time infection and recovery Markov chain has
infection and recovery rates, and , If the state space is infinite then a unique positive stationary
probability distribution exists iff and which is given by
for
Theorem 0.2
Suppose a continuous-time infection and recovery Markov chain has infection and recovery rates ,
and , where . The embedded Markov chain is a semi-infinite
random walk model with reflecting boundary conditions at zero. The chain has a unique stationary probability
distribution iff
Iff .
The stationary probability distribution is a geometric probability distribution,
–
and
for
Theorem 0.3
Let be a nonexplosive, positive recurrent, and irreducible CTMC with transition matrix
and generator matrix . Then there exists a unique positive stationary probability
distribution , , such that It follows that the mean recurrence time
can be computed from the stationary distribution:
(7)
4. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 23 | Page
III. The Simple Infection
In the simple infection process, the only event is infection. Let represent the population size at time and
, so that . Since the only event here is infection, the population size of infection can only
increase in size. For sufficiently small, the transition probabilities are
The probabilities are solutions of the forward Kolmogorov differential equations,
, where and for
– . Then so that the solution is , for
3.1 Generating Functions for the Simple Infection Process
To derive the partial differential equation for the ., multiply the differential equations by and sum over
, we have; . The initial condition is . The partial differential equation for
the . can be derived by a change of variable. Let Then and . so
that the is a solution of the following partial differential equation: with initial
condition Rewriting the differential equation for we have;
. The method of characteristics is applied to find the solution . Hence we
have;
. Along characteristic curves, and , , , and are solutions of
, , , with initial conditions , , and .
Separating variables and simplifying leads to . Integrating yields the relations,
, and . Applying the initial condition , the second expression can be
written as therefore we have .
Finally, the solution must be expressed in terms of the original variables and Using the preceding
formulas, can be expressed in terms of and , . Since , the
. for the simple infection process is
(8)
The can be found directly from the . by making the change of variable,
(9)
Letting the is
(10)
The p.g.f. and m.g.f. for the simple infection process correspond to a negative binomial distribution. The
probabilities in (10) can be written as
Let and replace and by and , respectively,
Now for , the initial infectious population, we have;
for
5. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 24 | Page
IV. The Simple Recovery Process
In the simple recovery process, the only event is a recovery. Let . The infinitesimal transition
probabilities are
Since the process begins in state N, the state space is . The probabilities
are solutions of the forward Kolmogorov differential equations, where by the forward Kolmogorov
equations we have
with initial conditions . It can be easily seen that zero is an absorbing state
and the unique stationary probability distribution is .
The generating function technique and the method of characteristics are used to find the probability distribution.
Multiplying the forward Kolmogorov equations by and summing over the partial differential equation for
the p.g.f. is
The initial condition is
The partial differential equation for the . can be derived by a change of variable. Let , hence,
, = , then and
so that the is a solution of the following partial differential equation:
with initial condition Hence we can have;
Along characteristic curves, and , , , and are solutions of , ,
. Rewriting the differential equation for . The method of characteristics
is applied to find the solution with initial conditions , , and .
Separating variables and simplifying leads to or
Integrating yields the relations at , and .
Applying the initial condition , the second expression can be written as
6. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 25 | Page
Finally, the solution must be expressed in terms of the original variables and Using the preceding
formulas, can be expressed in terms of and , . Since
, the . for the simple recovery process is
(11)
The can be found directly from the by making the change of
variable, ,
(12)
Let and . Then the has the form , corresponding to a
binomial distribution, . The probabilities;
for .
V. The Simple Infection And Recovering
In the simple birth and death process, an event can be a birth or a death. Let . The infinitesimal
transition probabilities are
The forward Kolmogorov differential equations are
for with initial conditions . As in the simple death process, , so that zero is an
absorbing state and is the unique stationary probability distribution. Applying the generating
function technique and the method of characteristics from the Appendix for Chapter 6 of [20], the moment
generating (m.g.f.) function and probability generating function (p.g.f.) is ;
and respectfully if the rate of
infection is not equal to the rate of recovery.
and if the rate of infection is equal to the rate of
recovery
The is not easy as it was for the simple infection and recovery processes, this is made more, since
functions cannot be associated with a well-known probability distribution, but we know that;
and . Using computer software, we can find the terms of the series
expansion. is the first term of the series expansion. For ,
and for ; . As , the probability of extinction
will be;
(13)
The mean and variance of the simple infection and recovery process can be derived from the generating
functions.
(14)
And
(15)
From equation (14) and (15), we see that correspond to exponential growth, when and exponential decay
when .
7. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 26 | Page
VI. The Simple Birth and Death with Immigration
When there an immigration in the compartment of infected, the rate of immigration will added to the rate of
general infection; whatever it is. Suppose the immigration rate is assume constant rate . We let the initial
population of infected to be . Transition probabilities for this process is given below;
The forward Kolmogorov differential equations are
for with initial conditions . the m.g.f. is a solution of
with as initial condition. [1] gave the solution as
The moments of the probability distribution can be found by differentiating with respect to and
evaluating at . The mean is
When , the mean increases exponentially in time and linearly when . However, in the case ,
the mean approaches a constant:
Thus, for the case , the process is nonexplosive and irreducible.
From
Theorem 0.1 and
Theorem 0.2, it can be shown that the process has a unique positive stationary distribution. [14] shows that the
conditions of being nonexplosive, irreducible, and having a positive stationary distribution imply the process is
positive recurrent. Then,
Theorem 0.3 implies the limiting distribution exists and equals the stationary distribution.
VII. Result and Conclusion
The mean and variance for the simple infection process are and
. The moments can be calculated directly from one of the generating functions. It is interesting to note that
the mean of the simple infection process corresponds to exponential growth with . The variance also
increases exponentially with time.
8. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 27 | Page
Fig 0.1: Probability distributions of for the simple infection process for
Fig 0.2: Ten sample paths when and . The mean and variance are
The mean and variance of a binomial distribution are and . and
.
The mean corresponds to exponential decay. Also, the variance decreases exponentially with time.
9. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 28 | Page
Fig 0.3: Probability distributions of I(t) for the simple death process for t = 1, 2, 3, 4
Fig 0.4: three sample paths when and . The mean and variance are
and .
We simulates the 'Recovery process. Starting with a given initial individuals, each having probability of
recovering in each time step, we continue until all have recover. The infected population size is plotted over
time and compared to the deterministic model for expected infected population size.
For ,
and for ; . As , the probability of extinction
will be;
(16)
The mean and variance of the simple infection and recovery process can be derived from the generating
functions.
(17)
And
(18)
10. The Probability distribution of a Simple Stochastic Infection and Recovery Processes
DOI: 10.9790/5728-1301032029 www.iosrjournals.org 29 | Page
From equation (14) and (15), we see that correspond to exponential growth, when and exponential decay
when .
Fig 0.5: Three sample paths for the simple infection and recovery process when λ = 1 = µ and .
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