Controlling Ebola Spread with Optimal Medication Delivery

Team #34235 Page 1 of 39
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2015 Mathematical Contest in Modeling (MCM) Summary Sheet
We establish an Ebola infection model based on the newly invented medication to
analyze not only the medication deliver system but also the medication production
rate in order to control and eradicate Ebola as soon as possible.
On the one hand, we extend the classical Compartment Model (SIR) to Extending
Multiple Compartment Model (EMCM) that is more scientifically applied to Ebola.
Furthermore, we introduce Genetic Algorithm (GA) to train the parameters in EMCM
with the early-stage data of disease condition in Sierra Leone and then implement
fitting process on the late-stage data, which finally exhibits an excellent fitting effect.
Simultaneously, we utilize EMCM to predict the total quantities of medications
needed to eradicate Ebola and nonlinear programming to solve the producing rate at
the minimum total cost within different transportation methods taken into
consideration. We conclude that the producer will produce at the maximum velocity in
the early stage while it’s reasonable to decrease the velocity to achieve the best of the
economical benefits in the late stage with the capacity to make timely supplements.
On the other hand, we adopt the Cellular Automaton Method(CAM) to simulate
the effective contact pattern among individuals from a realistic network perspective.
In addition, we extend the Center Method to Extending Center Method to determine
the optimal medical stations in the plates. Finally, we take the Kenema of Sierra
Leone as the targeted region, using simulation program to imitate the spread and
control process of Ebola. The results show that it’s effective enough to control or
even eradicate Ebola with the establishment of medical stations which can both
deliver medications and publicize useful information.

How to Eradicate Ebola?
Summary........................................................................................................................3
1 Introduction.................................................................................................................3
2 Restatement of the Problem........................................................................................4
3 Assumptions & Symbol Description...........................................................................4
3.1 Assumptions of Compartment Model..............................................................4
3.1.1 Additional Assumptions of Compartment Model based on Classical
Population Dynamics.....................................................................................5
3.1.2 Additional Assumptions of Extending Multiple Compartment Model.5
3.2 Assumptions of Micro Individuals Model Based on Human Activity.............7
3.2.1 Assumptions of Simple Human Activity Model...................................7
3.2.2 Assumptions of Micro Individuals Model ............................................8
4 Model Development....................................................................................................8
4.1 Extending Multiple Compartment Model........................................................8
4.1.1 Compartment Model Based on Classical Population Dynamics ..........8
4.1.2 Extending Multiple Compartment Model.............................................9
4.2 Micro Individuals Simulation Model.............................................................11
4.2.1 Simple Human Activity Model ...........................................................12
4.2.2Micro Individuals Simulation Model Based on Human Activity ........14
4.3 The Analysis of Medication Delivery & Distribution Rules .........................15
4.3.1 From Producing area to Infected Countries to Infected Cities ...........15
4.3.2 From Cities to Hospitals .....................................................................16
4.3.3 The Center Method & The Extending Center Method .......................18
4.4 Quantity & Production Rate of Medication...................................................19
4.4.1 Medication Quantity ...........................................................................19
4.4.2 Production Rate of Medication ...........................................................20
5 Instance Analysis ......................................................................................................21
5.1 Construct the Extending Multiple Compartment Model ...............................23
5.1.1Genetic Algorithm................................................................................23
5.1.2Application Results..............................................................................24
5.1.3 Model Advancement...........................................................................26
5.2 Application of Micro Individuals Model .......................................................28
5.2.1 Medication Delivery System In Low Rank Plates..............................30
5.2.2 Medication Delivery System In Medium Rank Plates........................31
5.2.3 Medication Delivery System In High Rank Plates .............................32
5.2.4 Conclusion ..........................................................................................34
5.3 Analysis of Medication Quantities & Production Rate..................................35
6 Conclusion ................................................................................................................36
7 Strengths & Weaknesses ...........................................................................................37
8 Reference ..................................................................................................................38
The non-technical letter ...............................................................................................39

Summary
We establish an Ebola infection model based on the newly invented medication to
analyze not only the medication deliver system but also the medication production
rate in order to control and eradicate Ebola as soon as possible.
On the one hand, we extend the classical Compartment Model (SIR) to Extending
Multiple Compartment Model (EMCM) that is more scientifically applied to Ebola.
Furthermore, we introduce Genetic Algorithm (GA) to train the parameters in EMCM
with the early-stage data of disease condition in Sierra Leone and then implement
fitting process on the late-stage data, which finally exhibits an excellent fitting effect.
Simultaneously, we utilize EMCM to predict the total quantities of medications
needed to eradicate Ebola and nonlinear programming to solve the producing rate at
the minimum total cost within different transportation methods taken into
consideration. We conclude that the producer will produce at the maximum velocity in
the early stage while it’s reasonable to decrease the velocity to achieve the best of the
economical benefits in the late stage with the capacity to make timely supplements.
On the other hand, we adopt the Cellular Automaton Method(CAM) to simulate
the effective contact pattern among individuals from a realistic network perspective.
In addition, we extend the Center Method to Extending Center Method to determine
the optimal medical stations in the plates. Finally, we take the Kenema of Sierra
Leone as the targeted region, using simulation program to imitate the spread and
control process of Ebola. The results show that it’s effective enough to control or
even eradicate Ebola with the establishment of medical stations which can both
deliver medications and publicize useful information.
1 Introduction
Ebola virus disease, a disease with a high fatality between 50 to 90 percent, is
gradually endangering people’s life. The latent period of Ebola is 5 to 10 days in
general. The symptoms it results in can be divided into 3 stages. Early stage presents
fever, vomit, diarrhea, muscular stiffness symptoms. Medium stage involves
hemorrhage. Late stage concerns hypovolemic shock and multiple system organ
failure to death. Furthermore, the infection route of Ebola is due to effective contact
infection with blood, sweat, vomitus, excreta, urine, saliva and other infected body
fluids.
The outbreak of Ebola initially occurred in 1976 in Congo and South Sudan and
became intermittently popular in Sub-Saharan Africa later on. The large scale

outbreak of Ebola nowadays was derived from December 2013 in Sierra Leone and
then spread to Liberia and Guinea. It is the most serious one in history on account that
it has caused significant mortality with reported case fatality rates of up to 71 percent
and specifically 57 to 59 percent among hospitalized patients. Although a small
outbreak of several cases occurred in Nigeria, Senegal, Mali, the United Kingdom, the
United States and Spain, but none of them experienced further diffusion and all now
declare disease-free. As of 3rd February 2015, the WHO and respective governments
in different countries had reported a total of 22,560 suspected cases and 9,019 deaths,
though the WHO believed that this substantially understates the magnitude of the
outbreak. Extreme poverty, a dysfunctional healthcare system, a mistrust of
government officials after years of armed conflict and the delay in responding to the
outbreak for several months have all contributed to the failure to control the epidemic.
Other factors include local burial customs that include washing of the body after death,
the spread to densely populated cities, and international indifference.
2 Restatement of the Problem
With the gradual havoc of Ebola, the medications that can cure the late stage
infected have been invented. However, how to effectively prevent Ebola from
spreading is the issue we need to focus at present. In order to determine the number of
medications, we need to initially build an Ebola spread model to determine the
number of the infected. Furthermore, so as to get the infected treated as soon as
possible, we need to design a relative efficient and scientific delivery system and
locate the medical stations in the map with the consideration of both economical
benefits and social demand. And it’s also imperative for us to propose some advice for
the producers concerning the production rate of the medications.
3 Assumptions & Symbol Description
3.1 Assumptions of Compartment Model
1. We divide people into several types and assume that everyone is completely
identical except types.
2. All the people mix uniformly.
3. The contact among people is instantaneous and the variation in the contact
moment does nothing with its historical state.
4. We don’t define the number of each type.
5. Both of the infection rate and recover rate are constants and they’re only different

to different type of people. Namely, people of the same type all possess the same
infection rate and recover rate.
3.1.1 Additional Assumptions of Compartment Model based
on Classical Population Dynamics
1. Patients can generate antibodies after medication treatment and they can be
immune to the viruses perpetually.
2. People are divided into 3 types including susceptible individuals S, Infected
individuals I and ones that become perpetually immune after medication treatment
R.
3. Birth rate is equal to mortality rate in the region.
4. We assume the parameters below are constants:
Symbol Explanation
𝛼
the average times of one individual contacting effectively with others
per unit time
(effective contact means the contact that can spread viruses)
𝛽 the number of individuals who are effectively cured per unit time
𝜇 birth rate and mortality rate in the region
3.1.2 Additional Assumptions of Extending Multiple
Compartment Model
1. People are divided into 8 types as follows:
Type Symbol Explanation
Average individuals A
Individuals who’re not so
susceptible to viruses
Susceptible individuals S
Individuals who’re susceptible
to viruses
Latent individuals L
Individuals who are infectious
and possess viruses latent in the
bodies without symptoms. They
can be cured with medication
treatment.
Early-stage infected EI
Individuals who’re infectious
and posses only slight

symptoms similar to influenza,
leading to the possibility of
misdiagnosis. They can be
cured with medication
treatment.
Late-stage infected LI
Individuals who’re more
infectious and possess typical
Ebola symptoms. They can be
treatment.
Variant V
Individuals who are infectious
and possess LI symptoms.
However, they cannot be
treatment as viruses mutate.
Disease-induced dead D
The corpses of the dead are
strongly infectious and they
will be cleared.
Recovered R
Individuals who have been
cured after medication
treatment.
2. We assume parameters are constants as follows:
Parameter
symbol
Explanation
𝜆𝑖 the probability of Average individuals A being infected
𝜇𝑖 the probability of Susceptible individuals S being infected
𝜔
the average times of Latent individuals L contacting effectively with
A and S per unit time
𝜎
the average times of Early-stage infected EI contacting effectively
with A and S per unit time
𝜈
the average times of Late-stage infected LI and Variant V contacting
effectively with A and S per unit time
𝜏
the average times of Late-stage infected D contacting effectively with
A and S per unit time
𝜑 the possibility factor of A being infected

𝜓 The possibility factor of S being infected
𝛼
the average number of individuals who turn into EI from L per unit
time
𝑝 the average times of effective treatment that EI obtain per unit time
𝛽
the average number of individuals who turn into LI from EI per unit
time
𝜂
1/𝜂 represents the period that EI need to recover from the moment
they obtain effective medication treatment
𝑞 average times of effective treatment that LI obtain per unit time
γ average number of individuals who turn into V from LI per unit time
𝜉
1/𝜉 represents the period that LI need to recover from the moment
they obtain effective medication treatment
𝜅
the average number of individuals who turn into A from R per unit
time
𝛿 average mortality rate of V
Φ average number of patients’ corpses that are cleared per unit time
3.2 Assumptions of Micro Individuals Model Based on
Human Activity
3.2.1 Assumptions of Simple Human Activity Model
1. Every citizen either possesses fixed residence or being homeless and the homeless
will wander about at night.
2. The citizens who are office workers will have different number of working days
with the consideration of different jobs but they will have specific clock in and out
time in spite of varying with different job requirements.
3. The citizens who are office workers will stay at home after work without going
out anymore on that day.
4. The citizens without office jobs will wander about or search the nearest park to
play or the nearest store to shop or the nearest hospital to get medical
examination.
5. The citizens with fixed residence have fixed get-out-of home time and
go-back-home time in spite of varying with different citizens.
6. The city region is marked with 5 main parts including residence, office building,
hospital, park, commercial center and the other parts are streets or green fields.

7. During the period in which Ebola viruses are active, we divide citizens into 8
types according to the different infected sates. As is the 8 types listed in
3.1( including Average individuals A, Susceptible individuals S, Latent individuals
L, Early-stage effected EI, Late-stage infected LI, Variant V, Disease-induced
dead D and Recovered R), we redefine the definition of Recovered R and the
definition XI, that is using R to substitute XI.
As is shown in the table below:
Type Symbol Explanation
X times infected
individuals
XI
The individuals who have been
infected X times but they are
cured after medical treatment
every time.
3.2.2 Assumptions of Micro Individuals Model
1. The uninfected will get infected at a certain probability after coexisting with the
infected for a period of time and the degree of probability is concerned with
different regions.
2. The infected in different stages have different infectious capacity and the
uninfected (including A, S, R,XI) hold different infected capacity.
3. The probability of XI being infected is inversely proportional to 𝑿.
4 Model Development
4.1 Extending Multiple Compartment Model
Compartment Model is a “Single Group Model” which constructs model from a
macro perspective. It applies partial differential equation to describe the evolution law
and the evolution process of total population state.
4.1.1 Compartment Model Based on Classical Population
Dynamics
The Classical Compartment Model is also called the SIR Model. However, the
SIR Model neglects the process of population dynamics. When some diseases have
long epidemic period or total population varies obviously, the SIR Model will lead to
a great error. Therefore, we construct the SIR Model based on population dynamics.
We let total population be 𝑁 and it is divided into 3 types including susceptible

individuals S, Infected individuals I and ones that become perpetually immune after
medication treatment R. According to the assumptions and parameter definitions
above, we draw the state transition diagram as follows to describe the relationships
among different types.
Figure 1: state transition diagram(the direction of arrow is the direction of
population transition and each formula above each arrow represents the proportion
that the transferred population account for total population )
We construct the SIR Model based on population dynamics as the following left
four formulas and if we let 𝜇 = 0, then it’ll degrade into the SIR Model as is shown
in the following right four formulas:
𝑁 = 𝑆 + 𝐼 + 𝑅
𝑑𝑆
𝑑𝑡
= 𝜇𝑁 − 𝜇𝑆 − 𝛼
𝐼𝑆
𝑁
𝑑𝐼
𝑑𝑡
= 𝛼
𝐼𝑆
𝑁
− 𝛽𝐼 − 𝜇𝐼
𝑑𝑅
𝑑𝑡
= 𝛽𝐼 − 𝜇𝑅
let 𝜇=0 then get SIR
⇒
𝑁 = 𝑆 + 𝐼 + 𝑅
𝑑𝑆
𝑑𝑡
= −𝛼
𝐼𝑆
𝑁
𝑑𝐼
𝑑𝑡
= 𝛼
𝐼𝑆
𝑁
− 𝛽𝐼
𝑑𝑅
𝑑𝑡
= 𝛽𝐼
The model points out a significant fact that we can introduce parameter
𝛿 = 𝛼/(𝛽 + 𝜇) whose definition is the average number of population that an infected
effectively contacts in the infection period within the consideration of mortality.
If 𝛿 < 1 or 𝐼(0) = 0, then lim 𝑛→∞ 𝐼(𝑡) = 0 represents infectious diseases disappear
and if 𝛿 > 1, then lim 𝑛→∞ 𝐼(𝑡) = 𝜇(𝛿 − 1)/𝛼 < ∞ represents the velocity of being
infected is equal to the velocity of being cured, namely, the infected maintain a
constant number in the region.
4.1.2 Extending Multiple Compartment Model
The Classical SIR Model is suitable for the diseases that once an infected are
cured, he will get perpetual immunization. Nevertheless, the medication for Ebola
cannot guarantee the point on account that although Ebola medication can cure the
𝑆 𝐼 𝑅
𝛽𝐼𝛼
𝐼
𝑁
𝜇 dead 𝜇 dead 𝜇 dead
𝜇 𝑁 = 𝑆 + 𝐼 + 𝑅

infected, the infected still can get immunization or not and even if they get
immunization, the immunity may degrade in a certain period. Simultaneously, Ebola
viruses possess a latent period in bodies, in the period of which the infected is difficult
to be recognized without typical Ebola symptoms.
As discussed above, these problems cannot be solved by the Classical SIR Model.
As a consequence, we introduce multiple compartment model to further analysis the
spread of Ebola viruses.
We pay attention to the symptoms of Ebola as follows:
1. The latent period is usually from 5 to 10 days.
2. Early-stage symptoms are similar to influenza, thus, there exists a high
possibility of misdiagnosis and then lose the best treatment time.
3. Late-stage infected show typical Ebola symptoms and late stage is also
considered as medicable period.
4. Ebola viruses will mutate in the late-stage of infection and medication can’t
work on the mutated Ebola viruses.
5. The corpses of the dead are strongly infectious before being cleared.
Let total population be 𝑁 and it is divided by 8 types including Average
individuals A, Susceptible individuals S, Latent individuals L, Early-stage effected EI,
Late-stage infected LI, Variant V, Disease-induced dead D and Recovered R.
we draw the state transition diagram as follows to describe the relationships
among different types.
Figure 2: transition state diagram(the direction of arrow is the direction of
population transition and each formula above each arrow represents the proportion
that the transferred population account for total population )
From the above diagram, we know
𝜆 =
𝜑
𝑁
[𝜔𝐿 + 𝜎𝐸𝐼 + 𝜈(𝐿𝐼 + 𝑉) + 𝜏𝐷]
𝜇 =
𝜓
𝑁
𝐴 𝑆
𝐿 𝐸𝐼 𝐿𝐼 𝑉
𝐷𝑅
(1 − 𝑝)𝛽𝛼𝑖
(1 − 𝑞)𝛾
𝛿𝜆 𝜇
𝑞𝜉𝑝𝜂
Φ
Be cleared
𝜅

We construct extending multiple compartment model as follows:
𝑁 = 𝐴 + 𝑆 + 𝐿 + 𝐸𝐼 + 𝐿𝐼 + 𝑉 + 𝑅
𝑑𝐴
𝑑𝑡
= 𝜅𝑅 − 𝜆𝐴 = 𝜅𝑅 − 𝐴
𝜑
𝑁
[𝜔𝐿 + 𝜎𝐸𝐼 + 𝜈(𝐿𝐼 + 𝑉)]
𝑑𝑆
𝑑𝑡
= −𝜇𝑆 = −𝑆
𝜓
𝑁
[𝜔𝐿 + 𝜎𝐸𝐼 + 𝜈(𝐿𝐼 + 𝑉)]
𝑑𝐿
𝑑𝑡
= 𝜆𝐴 + 𝜇𝑆 − 𝛼𝐿
𝑑𝐸𝐼
𝑑𝑡
= 𝛼𝐿 − [ 𝑝 𝜂 + (1 − 𝑝) 𝛽]𝐸𝐼
𝑑𝐿𝐼
𝑑𝑡
= (1 − 𝑝) 𝛽 𝐸𝐼 − [ 𝑞 𝜉 + (1 − 𝑞) 𝛾]𝐿𝐼
𝑑𝑉
𝑑𝑡
= (1 − 𝑞) 𝛾 𝐿𝐼 − 𝛿𝑉
𝑑𝐷
𝑑𝑡
= 𝛿𝑉 − Φ
𝑑𝑅
𝑑𝑡
= 𝑝𝜂𝐸𝐼 + 𝑞𝜉𝐿𝐼 − 𝜅𝑅
4.2 Micro Individuals Simulation Model
Composite Group Model is the extension of Single Group Model and it
subdivides every group into subgroups which are dealt with different compartment
models. They both assume individuals are not only homogenous but also mixed
uniformly. However, every individual can only contact with limited individuals and
the contact pattern have great differences.
The Network Structural Model is beneficial to construct a more realistic spread
model and Micro Individuals Model based on network has gained momentum to
develop in recent years. There are 2 aspects in the Network Structural Model. One is
the Realistic Network Model based on actual surveys, which aims to construct more
realistic contact network from a statistical perspective. The other is the Ideal Network
Model regardless of exactly accurate social contact data, which establishes model
from a perspective concerning both network characteristics and viruses spread pattern.
This paper intends to initially establish a human activity concerned simulation
platform on the basis of the ideal network in which a mathematical model describing
human activity rules has been introduced and eventually construct a micro individuals
model similar to realistic social contact pattern. The targeted model not only possesses
simplicity of the Ideal Network Model but also the reality of the Realistic Network
Model.

4.2.1 Simple Human Activity Model
Urban citizens always live a regular life. With the above 3.2.1 assumptions, we
mark the city region with 5 main types including residence, office building, hospital,
park and commercial center and we divide citizens into 8 types including Average
individuals A, Susceptible individuals S, Latent individuals L, Early-stage effected EI,
Late-stage infected LI, Variant V, Disease-induced dead D and X times infected
individuals XI. Furthermore, we assume citizen activity will be influenced as soon as
Ebola viruses begin spreading.
we draw the state transition diagram as follows to describe the relationships
among the 8 types.
Figure 3: the state transition diagram
Citizens will have simple activities rules in the period of Ebola spreading as
follows:
1. The infected won’t represent any symptoms during the latent period so they won’t
go to the hospitals deliberately except for the condition that they go there for
physical examinations on the non-workday.
2. The symptoms of the early-stage infected are similar to influenza, thus, there
exists a high possibility of misdiagnosis and they are only in a low possibility to
go to the hospitals.
3. The infected of Late-stage and Variant stage will show typical Ebola symptoms
and they are in a high possibility to go to the hospitals.
4. The infected of recovery stage will stay in the hospital until they completely
recover and then the successfully recovered infected will belong to X times
infected individuals XI.
𝐴
𝑆
𝐿 𝐸𝐼 𝐿𝐼 𝑉
𝐷
symptoms
aggravates
latent
finishes
virus
mutates
dead
medication treatment
be
cleared
𝑋𝐼
get infected
get
infected

We develop an algorithm design to match the above activity rules and virus
transition process and then we use an algorithm flowchart to illustrate the state
renewal for every individual as below:
Figure 4: the flowchart of the human activity algorithm
Every individual can specify his behavioral objective, that is everyone has the
capacity to subjectively select his destination and successfully go to the targeted
destination. We adopt the method of data structure of priority queue to store the
information concerning destinations and prior information includes the departure time
to go to the destination and whether a boolean value marks the special information or
not. We rank the destination information according to the departure time and rule out
non-special and outdated information.
Moreover, we utilize the cellular automaton method to implement the above
model and algorithm on the computer. To start with, we construct the whole region
with grid pattern, specifically, every individual can only move from one vertex of a
grid to another, that is to say the individuals can only exist on the vertexes. And we
define the regional shape for all of the 5 types including residence, office building,
Start
Is EI or LI or
V ?
Transform
destination to
hospital at a certain
possibility
Disease aggravates
(controlled by timer )
A-star algorithm
to search for the
shortest path
Have next
destination?
Determine a new
destination
Wander around
randomly with
destination region
End
YES
YES YES
NO
NO
NO
Have reached
destination?

hospital, park and commercial center as rectangles on account that when this
simulation model is applied to disease research, the size of the whole selected region
will be large enough relative to the size of the single unit (referring to the connected
region of the same type) of the region for each type. Therefore, the shape of each type
represents such little difference that we define all of them as rectangles.
So far, the Single Human Activity Model has been successfully established.
4.2.2Micro Individuals Simulation Model Based on Human
Activity
Since Ebola viruses are mainly spread through body fluids and contact with
others, we can suppose that an uninfected will get infected at a certain possibility after
coexisting with an infected for a period of time 𝑻 and the possibility of spread varies
as the region changes because different region has different degree of effective contact
with different population density.
We suppose a certain vertex be in the type Q(Q=residence, office building,
hospital, park, commercial center，street) . If we let the probability of type
𝒊(𝒊 = 𝑨, 𝑺, 𝑳, 𝑬𝑰, 𝑳𝑰, 𝑽, 𝑫 , 𝑹 ) being infected at the vertex be 𝑃𝑄(𝑖), then
𝑃𝑄(𝑖) =
1
𝑇
∗ 𝑓(𝑄, 𝑖) ∗ ∑ [𝑔(𝑖, 𝑗) ∗
𝑛𝑢𝑚(𝑗)
𝑛
]
𝑗∈{𝐿,𝐸𝐼,𝐿𝐼,𝑉,𝐷}
𝑖𝑓 𝑖 = 𝑋𝐼, 𝑃𝑄(𝑖) =
the formula above
𝑋
In the above formula,
1. 𝑛𝑢𝑚(𝑖) represents the number of type 𝑖 at the vertex
2. 𝒏 represents the number of individuals at the vertex(the dead are included), so
𝑛 = ∑ 𝑛𝑢𝑚(𝑖)𝑖∈{𝐴,𝑆,𝐿,𝐸𝐼,𝐿𝐼,𝑉,𝐷,𝑅,𝑋𝐼}
3. 𝑋 represents the times that 𝑋𝐼 have been infected
4. 𝑓(𝑄, 𝑖) is the regional factor and it symbols the probability of type 𝑖 effectively
contacting with the viruses in 𝑄. And we need to mention that the viruses can not
only directly originate from the infected but also indirectly from the remaining
viruses that the infected leave behind.
5. 𝑔(𝑖, 𝑗) is the crowd type factor and it symbols the probability of crowd type 𝑗
resulting in aggravation of crowd type 𝑖, for example, 𝑔(𝐸𝐼, 𝐿𝐼) ≠ 0 symbols
EL can make LI aggravated to V.

4.3 The Analysis of Medication Delivery &
Distribution Rules
From the problems, we know that the newly-invented medication can cure the
infected whose condition haven’t developed to the variant stage. Given that at present
the seriously infected regions are located to the south of the Sahara desert in west
Africa and the their development levels are generally low, therefore, they don’t have
the capacity to develop and massively produce the medication and we assume the
producing area of the medication are other countries. Considering that the medication
delivery is designed to make the medication reach the hospitals of the infected regions
and the infected ones as soon as possible to control and eventually eradicate Ebola
rapidly, we develop the delivery system with the following processes included:
Figure 5: the flowchart of medication delivery system
We will discuss the delivery system through the above 4 processes.
4.3.1 From Producing area to Infected Countries to Infected
Cities
When the medications are successfully developed, our focused regions has been
seriously infected. Thus, it’s imperative for us to deliver the first batch of medications
at the fastest speed to prevent the further aggravation. As a consequence, air
transportation is the best choice. However, the air transportation requires high cost
and considering the current seriously infected countries are all near the Atlantic Ocean,
it’s economical to deliver subsequent medications by shipping.
As the national economic development in all of the seriously infected countries
are in a low level with only 1 or 2 small international airports, it’s difficult to deliver
large quantities of the medication from the producing area to the targeted countries by
Air transportation
or shaping
Seriously infected
countries
Medication
producing area
Seriously infected
countries
Highway
transportation
Hospital or
medical stations
the infected

air transportation. Therefore, we only use air transportation initially to control the
spread velocity. Subsequently, large quantities of medications will be delivered to the
medication stores which are set in harbors of infected countries y shipping. And then,
the medication will be delivered to the targeted infected places by highway
transportation.
4.3.2 From Cities to Hospitals
When both the size of the infected city and the infected coverage rate is large,
putting the medications only in one hospital aren’t timely enough to deliver
medications to the infected. In addition, with other factors such as geological factors
of hospitals into consideration, putting the medications only in hospitals to wait the
infected to come is also ineffective.
In order to solve the problem, to start with, we divide the cities into different
plates.( Note that the plates are different from the stated regions above.) Then, we
rate each plate with low, medium and high ranks according to the serious degree of
infection. At last, we design different delivery systems to the regions with different
ratings.
4.3.2.1 City Plate Division
If we let the city be covered with the grid whose spacing is 𝑲, then the city can
be divided into 𝑴 plates.
4.3.2.2 City Plate Rating
We let total number of the infected be 𝑵.
For a certain plate 𝜴𝒊, if we set the number of the infected as 𝒏𝒊, then the
proportion that the infected in the plate accounts for the total infected will be
𝑝(𝛺𝑖) =
𝑛𝑖
𝑁⁄ , i = 1,2, … , M. Let the variance of sequence {𝑝(𝛺𝑖)}𝑖=1,2,…,𝑀 be 𝝈.
Regulations: If 𝑝(𝛺𝑖) < 1
𝑀⁄ − 𝜎, then it is a low rank plate, if 𝑝(𝛺𝑖) > 1
𝑀⁄ + 𝜎,
then it is a high rank plate and if 1
𝑀⁄ − 𝜎 < 𝑝(𝛺𝑖) < 1
𝑀⁄ + 𝜎, then it is a medium
rank plate.
4.3.2.3 Delivery System of Different Rank Plate
We define the medical stations with following characteristics. To start with, they
are set in some certain plates in the cities. Moreover, they not only provide the
infected condition diagnosis and medication for the patients but also send the infected
of the variant stage to the nearest hospital for isolation. More specifically, the

condition diagnosis can figure out whether the individuals has been infected and
whether the infected are in the variant stage. In addition, it’s impossible for the
medication to cure the infected of variant stage.
For a certain plate 𝜴, we set the number of the hospitals as 𝒏𝒖𝒎(𝑯) and we
let the number of the intended medical stations be 𝒙.
We hope the hospitals and the intended medical stations in this plate will serve
the infected effectively, that is setting the medications in a reasonable position which
is convenient for the infected. The next chapter 4.3.3 will describe the specific method
to calculate the number of the medical stations 𝒙 and to select the positions for
medical stations construction. It includes Center Method and Extending Center
Method.
As the infected cities’ economic conditions are all in a low level without many
medical facilities, so there may be no hospitals in some plates, that is 𝒏𝒖𝒎(𝑯) = 𝟎.
Therefore, we attach more attention to the collective services provided by hospitals
and the medical stations, namely, we emphasize the control of the disease condition
based on 𝒏𝒖𝒎(𝑯) + 𝒙 medical facilities.
Let 𝚲 = 𝒏𝒖𝒎(𝑯) + 𝒙.
In different plates with different ranking, we follow the principle of utilizing the
hospitals in priority and deliver the medication as according to the procedure
below.( The Center Method and the Extending Center Method will be shown in 4.3.3)
1. For the low rank plates: Λ ≥ 1
a) If num(H) > 0, then we will select a hospital under the Center Method
and we will deliver the number of medications according to the number
of the infected to the selected hospital(at this time, 𝑥 = 0）
b) If num(H) = 0，then we will select a position in the plate and construct
the medical station.（at this time 𝑥 = 1）
2. For the medium rank plates：Λ ≥ 2
a) If 𝑛𝑢𝑚(𝐻) > 1, we will select a hospital as the main hospital in the
plate and then deliver a certain number of medications to other hospitals
as inventories according to the infected in the main hospital.
Additionally, when the number of inventories are consumed below a
certain threshold, The main hospital will deliver additional medications
as supplements.（at this time 𝑥 = 0）
b) If 𝑛𝑢𝑚(𝐻) ≤ 1，we will select（2 − 𝑛𝑢𝑚(𝐻)）points in the plate
under the Center Method to construct medical stations.（at this time

𝑥 = 2 − 𝑛𝑢𝑚(𝐻)）
3. For the high rank plates：Λ ≥ 4 and 𝑥 ≥ 2
a) If 𝑛𝑢𝑚(𝐻) > 2, we will initially deal with it under the same method of
medium rank plates, then we will select 2 points in the plate following
the Extending Center Method to construct medical stations.（at this time
𝑥 = 2）；
b) If 𝑛𝑢𝑚(𝐻) ≤ 2，we will select（4 − 𝑛𝑢𝑚(𝐻)）points in the plate
under the Center Method to construct medical stations.（at this time
𝑥 = 4 − 𝑛𝑢𝑚(𝐻) > 2）；
4.3.3 The Center Method & The Extending Center Method
In this problem, the demand points are the regions with dense crowd activities
including residence, office building, hospitals, green field and malls and the station
points are the positions where the intended medical stations locate in.
As the crowd activity density varies with the change of demand points, we will
recalculate after endowing the different demand point with different weight.
According to the definition of region division in the Micro Individuals Model, the city
is divided into 5 types, they are residence R, office building O, hospitals H, green
field G and malls M. And the weights are 𝝎 𝑹, 𝝎 𝑶, 𝝎 𝑯, 𝝎 𝑮, 𝝎 𝑴 respectively.
Hospitals belong to the station points, so set 𝝎 𝑯 = 0.
4.3.3.1 The Center Method
We randomly select a point 𝑷 in the plate 𝛺. If we set ||𝑷 − 𝑿|| represents the
longest liner distance from 𝑷 to the region 𝑿，then 𝑚𝑎𝑥 𝑿||𝑷 − 𝑿|| represents the
maximum distance from 𝑷 to all of the regions.
The maximum weighted distance from 𝑷 to all the regions can be shown as
𝐷𝑖𝑠(𝑷) = 𝜔 𝑿 ∗ 𝑚𝑎𝑥
𝑿
||𝑷 − 𝑿||
The center point in the plate 𝜴 is the point 𝑷 which makes 𝐷𝑖𝑠(𝑷) minimum
that is
𝐶𝑒𝑛𝑡𝑒𝑟(𝛺) = arg 𝑚𝑖𝑛
𝑷∈𝛺
𝐷𝑖𝑠(𝑷)
The significance of this selecting method is to make the necessary time of the
individual from each region reaching the intended station point controlled to an upper
limit, which represents the least time and the point is the optimal position to construct
the medical station.
4.3.3.2 The Extending Center Method

In most cases, we need to select more than one point in the plate 𝜴 and we hope
the maximum distance of any region 𝑿 reaching the selected points to be minimum
enough.
More commonly, we assume that there has existed 𝒎 points in the plate 𝜴 ,
that is {𝑃𝑖}𝑖=1,2,…,𝑚 and we select 𝒏 more intended station points, that
is {𝑃𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛。
Set the minimum weighted distance from region 𝑿 to {𝑃𝑖}𝑖=1,2,…,𝑚+𝑛 as
𝑚𝑖𝑛
𝑖=1,…,𝑚+𝑛
𝜔 𝑿 ∗ ||𝑷𝒊 − 𝑿||
For all the region 𝑿, the above formula has the maximum as follows
𝑚𝑎𝑥
𝑿
𝑚𝑖𝑛
𝑖=1,…,𝑚+𝑛
𝜔 𝑿 ∗ ||𝑷𝒊 − 𝑿||
We hope the additional selected points {𝑃𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛 can make the
formula above reach the minimum.
We call {𝑃𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛 as 𝑛 − center point in 𝛺 under {𝑃𝑖}𝑖=1,2,…,𝑚 ,
which can be called 𝑛 − center point in 𝛺 for abbreviation. And we set it as
𝑛𝐶𝑒𝑛𝑡𝑒𝑟(𝛺|{𝑃𝑖}𝑖=1,2,…,𝑚).
We describe the selecting method of the above {𝑃𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛 by the
formulas below:
{𝑃𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛 = 𝑛𝐶𝑒𝑛𝑡𝑒𝑟(𝛺, {𝑃𝑖}𝑖=1,2,…,𝑚)
= arg 𝑚𝑖𝑛
{𝑃 𝑖}𝑖=𝑚+1,𝑚+2,…,𝑚+𝑛
𝑚𝑎𝑥
𝑿
𝑚𝑖𝑛
𝑖=1,2,…,𝑚+𝑛
𝜔 𝑿 ∗ ||𝑷𝒊 − 𝑿||
The significance of this selecting method is to make the necessary time of the
individual from each region reaching the nearest station point controlled to an upper
limit, which represents the least time and the points are 𝑛 optimal positions to
construct the medical stations.
4.4 Quantity & Production Rate of Medication
After having determined the delivery systems and medical stations, it’s
imperative for us to obtain the information concerning the total medication quantities
that the infected regions need and negotiate the appropriate production rate with the
manufacturers. Therefore, we will analyze the quantity and the publication rate of the
medications in the following parts.
4.4.1 Medication Quantity
According to the above assumptions, the infected of latent period don’t need to
take medication. (As the former cannot realize being infected and the latter cannot be

cured by the medications) And here we assume that both 𝑬𝑰 and 𝑳𝑰 will recover
after having taken only one dose of the medications. Therefore, we can know that the
total number of medications needed every day are equal to the total number of 𝑬𝑰
and 𝑳𝑰 on that day.
We set day 1 as the day the medications being delivered to the infected at the
first time and let everyday’s required quantities of medications be 𝑩𝒕 during the
infected period, namely, 𝑩𝒕 = 𝑬𝑰(𝒕) + 𝑳𝑰(𝒕). Then let the total required quantities of
medications be 𝑩.
The Ebola viruses spread model established in 4.2.1 can predict the total number
of 𝑬𝑰 and 𝑳𝑰 on day 𝒕.
If we assume the supply of everyday’s medication quantities extend the demand
ones, then the number of the infected are on the decline. Additionally, when the
number of the infected are zero continuously for two weeks, we consider the Ebola to
be eradicated completely. This is because there are possibilities that when the number
of the infected declines to zero at the first time, patients of the latent period won’t
have been discovered concerning the 5 to 10 days long latent period.
Set the total time needed to eradicate Ebola as 𝑇, then we get the total quantities
needed as below:
𝐵 = ∑ 𝐵𝑡
𝑇
𝑡=1
4.4.2 Production Rate of Medication
According to the calculations above, the total quantities of medications we need
is 𝑩.
The maximum velocity for the producers to produce the targeted medication is
𝒗 𝒎𝒂𝒙 dose per day.
In the early stage of production, in order to deliver enough quantities to the
targeted regions, we not only choose the most rapid delivery pattern but also produce
the medications with the maximum velocity. We set the maximum quantities the cargo
airplane can carry be 𝐻 𝑎𝑖𝑟 and it can meet consumption for 𝑇𝑎𝑖𝑟 days.
That is
∑ 𝐵𝑡
𝑇 𝑎𝑖𝑟
𝑡=1
≤ 𝐻 𝑎𝑖𝑟 < ∑ 𝐵𝑡
𝑇 𝑎𝑖𝑟+1
𝑡=1
We assume that the total cost related to medication production 𝑦 shows an
exponential relationship with the production velocity per day, that is 𝑦 = 𝑎 𝑣
∗ 𝑐, in
which 𝑐 represents the essential cost of daily production. Besides, we set the time of

shipping be 𝑇𝑏𝑜𝑎𝑡 and 𝐻 𝑏𝑜𝑎𝑡 additional medications will be transported by shipping.
The time needed to produce 𝐻 𝑏𝑜𝑎𝑡 is 𝑇𝑒𝑙𝑠𝑒 and the total cost of producing them are
𝑌.
The relations among the variables are listed below:
{
𝐵 = 𝐻 𝑎𝑖𝑟 + 𝐻 𝑏𝑜𝑎𝑡
0 < 𝑇𝑒𝑙𝑠𝑒 ≤ 𝑇𝑎𝑖𝑟 − 𝑇𝑏
0 < 𝑣 ≤ 𝑣 𝑚𝑎𝑥
𝐻 𝑏𝑜𝑎𝑡 = 𝑣 ∗ 𝑇𝑒𝑙𝑠𝑒
𝑦 = 𝑎 𝑣
∗ 𝑐
𝑌 = 𝑦 ∗ 𝑇𝑒𝑙𝑠𝑒
For further simplify:
Inequality constraint conditions:
{
0 < 𝑇𝑒𝑙𝑠𝑒 ≤ 𝑇𝑎𝑖𝑟 − 𝑇𝑏
0 < 𝑣 ≤ 𝑣max
Nonlinearity constraint condition:
𝑣𝑇𝑒𝑙𝑠𝑒 = 𝐵 − 𝐻 𝑎𝑖𝑟
The objective function:
𝑌 = 𝑐𝑇𝑒𝑙𝑠𝑒 𝑎 𝑣
According to the nonlinearity programming method, we can get (
𝑇𝑒𝑙𝑠𝑒
𝑣
), which
enables 𝑌 to achieve the minimum and the corresponding production rate 𝑣 is the
desirable production rate of the late stage.
5 Instance Analysis
The World Health Organization (WHO) initially announced the spread of Ebola
viruses in Sierra Leone on May 27th
2014. However, it’s not until around August 10th
2014 that the WHO and the local authorities began to take on a more scientific and
effective measure and the crowds obviously enhanced vigilance against the viruses
from then on. Therefore, it’s reasonable to assume that the data before August 10th
2014 is without the intervention of medication.
Sierra Leone, Liberia and Guinea, lying in the southwest of Africa, are 3 most
seriously infected nations nowadays. The northeast of Sierra Leone is surrounded by
Guinea and its southeast borders on Liberia. As not only Kenema is the third largest
city in Sierra Leone but also its infected velocity is the most rapid, therefore, we
consider Kenema typical enough for us to implement simulation and then we carry on
analysis on the real map for Kenema.

Figure 6: the selected region Kenema
Figure 7: a plate of Kenema

5.1 Construct the Extending Multiple Compartment
Model
According to the characteristics of Ebola viruses, its latent period lasts for about
5 to 10 days, the early stage period lasts for about 5 days, the late stage period lasts
for about 3 days and the variation period lasts for about 10 days, more specifically, the
mortality rate of the variation period reaches above 90 percent. We define the
parameters of the EMCM and then construct the viruses spread model without the
intervention of medication.
Parameter
symbol
Assumed value Assumption basis
𝛼 1/8
the statistical mean for the latent period is
about 8 days
𝛽 1/5
the statistical mean for the early stage period
is about 5 days
γ 1/3
the statistical mean for the late stage period
is about 3 days
𝛿 1/10
the statistical mean for the variation period is
about 10 days
𝜂 0
None of the infected can be treated or
recovered without medication
𝑝 0
𝑞 0
𝜉 0
𝜅 0
Φ 5
We assume utmost 5 corpses will be cleared
every day
In addition, there are 6 extra parameters that cannot be obtained form the
empirical data, they are 𝜔, 𝜎, 𝜈, 𝜏, 𝜑, 𝜓.
The paper adopts GA(Genetic Algorithm) to train the model and we select the
data that Ebola spread in Sierra Leone from May 27th
,2014 to July 15th
,2014 as our
training data. The specific GA is narrated in the following part.
5.1.1Genetic Algorithm
The aim of GA is to utilize actual data to optimize the parameters of the existing
model and the model based on the newly obtained data can fit the actual data

perfectly.
Set model parameters as vector 𝑾 = (𝜔 𝜎 𝜈 𝜏 𝜑 𝜓) 𝑇
and set the
model as 𝒚 = 𝑭(𝒙 𝟎, 𝑾, 𝑡), in which 𝒙 𝟎 represents the initial value, 𝑡 represents for
days, 𝒚 represents the model output of day 𝑡. And we also set 𝒀 as the actual data
of day 𝑡.
The coding process of 𝑾:
We use 8-bit binary series {𝑎𝑖}𝑖=1,2,…,8 to represent a decimal 𝑝 ∈ [0,1]: 𝑝 =
∑ (𝑎𝑖2−𝑖
)8
𝑖=1
As all of the parameters are decimals in the 𝑾 , we expand
(𝜔 𝜎 𝜈 𝜏 𝜑 𝜓) into 8-bit binary series in sequence, then we get 48-bit
binary series and this binary series is the coding chromosome that 𝑾 corresponds
with.
The fitness calculation of 𝑾:
We define the fitness calculation of 𝑾 with a set of parameters as
𝐺(𝑾) = 1 −
||𝒀 − 𝑭(𝒙 𝟎, 𝑾, 𝑡)||
||𝒀||
∈ [0,1]
And we know that when the model fitting effect is better, so is the fitness.
The selection process of 𝑾:
We use the roulette method to select the survival individuals and they will enter
into the next breeding process.
As the roulette method has random process, there exists the probability of ruling
out the good individuals. Therefore, we adopt elite selection method, that is before
each roulette, we send the most excellent ones directly into the next breeding process
without implementing roulette.
The crossover and mutation of 𝑾:
According to the crossover probability and mutation probability, we carry on
cross breeding among the survivals’ chromosomes(that is binary series hybridize
under hybrid rules ) and few individuals’ chromosomes will experience mutation(that
is the position value in the binary series has been inversely selected).
The filial generations after crossover and mutation will repeat the above process
consecutively and after multiple generations we can obtain 𝐺(𝑾 𝟎), the maximum of
𝐺(𝑾) and the corresponding parameter setting 𝑾 𝟎.
5.1.2Application Results
We select the data that Ebola spread in Sierra Leone from May 27th
,2014 to July
15th
,2014 as our training data and the data from July 15th
,2014 to August 9th
,2014 as

our inspection data.
We use GA based on the training data to find out the optimal parameter setting
𝑾 𝟎 and apply 𝑾 𝟎 to the model to simulate the variation of the number of
subsequent individuals. Then we implement goodness of fit test with inspection data
and determine the accuracy of the model.
GA defines the number of individuals as 100, iterative breeding generations as
200, chromosome crossover probability as 0.6 and mutation probability as 0.01. The
optimal individual fitness degree for each generation is shown in the figure below.
Figure 8 : The Variation Of Optimal Individual Fitness Degree in GA
The specific value results are shown on the table below.
Optimal individual fitness
degree
Optimal individual parameters
𝑾 = (𝜔 𝜎 𝜈 𝜏 𝜑 𝜓) 𝑇
0.9368
Optimal individual
generations (0.3730 0.4668 0.2617 0 0.2090 0.4727) 𝑇
130

Figure 9 : The Fitting Effect Of Extending Multiple Compartment Model
In general, we can see that GA has a good effect. The obtained model fits well
with the training data and the inspection data of dead tolls. However, as for the
inspection data of infected tolls, it only represents good fitting with the early stage
data instead of the late stage data.
5.1.3 Model Advancement
The reason why it represents poor fitting with the late stage data is that when it
comes to the late stage of Ebola viruses spreading, the WHO and the local authorities
begin to take on a more scientific and effective measure and the crowds obviously
enhance vigilance against the viruses, however, the model doesn’t take them into
consideration. (The model only considers medical treatment, nevertheless, it’s obvious
that the weakening growth rate of the infected does nothing with the medical
treatment and there is no medication to treat Ebola at this time.)
Based on this reason, we change the original infected probability of average
individuals A and susceptible individuals S to advanced ones. The process is
illustrated by the formulas below.
The original ones:

𝜆 =
𝜑
𝑁
𝜇 =
𝜓
𝑁
The advancedd ones:
𝜆 =
𝐴 + 𝑆 + 𝐿 + 𝑋𝐼
𝑁
∗
𝜑
𝑁
𝜇 =
𝐴 + 𝑆 + 𝐿 + 𝑋𝐼
𝑁
∗
𝜓
𝑁
We use GA to find out the optimal parameter setting once more. And GA also
defines the number of individuals as 100, iterative breeding generations as 200,
chromosome crossover probability as 0.6 and mutation probability as 0.01. The
specific value results are shown on the table below.
Optimal individual
fitness degree
Optimal individual parameters
𝑾 = (𝜔 𝜎 𝜈 𝜏 𝜑 𝜓) 𝑇
0.9616
Optimal individual
generations (0.4941 0.3125 0.0977 0.0156 0.2734 0.4434) 𝑇
181
Figure 10 : The Fitting Effect Of Advanced Extending Multiple Compartment Model

In general, we can see that GA has a good effect with both training data and
inspection data.
5.2 Application of Micro Individuals Model
To start with, we compile the Java program based on the Micro Individuals
Model concerning human activity. Then we implement simulation on the computer.
The parameter values in the model are listed below:
𝑓(𝑄, 𝑖) 𝑓(𝑄, 𝐴) 𝑓(𝑄, 𝑆) 𝑓(𝑄, 𝐿) 𝑓(𝑄, 𝐸𝐼) 𝑓(𝑄, 𝐿𝐼) 𝑓(𝑄, V)
𝑓(𝑅, 𝑖) 0.54 0.50 0.46 0.47 0.44 0.52
𝑓(𝑂, 𝑖) 0.52 0.47 0.44 0.42 0.44 0.48
𝑓(𝐻, 𝑖) 0.001 0.004 0 0 0.01 0.004
𝑓(𝐺, 𝑖) 0.47 0.49 0.48 0.45 0.52 0.40
𝑓(𝑀, 𝑖) 0.83 0.89 0.85 0.88 0.82 0.81
𝑓(𝑆𝑡, 𝑖) 0.56 0.55 0.58 0.59 0.52 0.55
𝑔(𝑖, 𝑗) 𝑔(𝐿, 𝑗) 𝑔(𝐸𝐼, 𝑗) 𝑔(𝐿𝐼, 𝑗) 𝑔(𝑉, 𝑗) 𝑔(𝐷, 𝑗)
𝑔(𝑖, 𝐴) 0.046 0.080 0.084 0.185 0.240
𝑔(𝑖, 𝑆) 0.080 0.145 0.149 0.155 0.440
𝑔(𝑖, 𝐿) 0 0.057 0.071 0.072 0.140
𝑔(𝑖, 𝐸𝐼) 0 0 0.57 0.58 0.140
𝑔(𝑖, 𝐿𝐼) 0.83 0.89 0.85 0.88 0.140
𝑔(𝑖, 𝑉) 0 0 0 0 0.005
𝑔(𝑖, 𝑋𝐼) 0.064 0.064 0.085 0.100 0.130
We need to mention that 𝑓(𝑄, 𝐷) = 𝑓(𝑄, 𝑋𝐼) = 0 ， 𝑔(𝑖, 𝐷) = 𝑔(𝐴, 𝑗) =
𝑔(𝑆, 𝑗) = 𝑔(𝑋𝐼, 𝑗) = 0 and 𝑔(𝑖, 𝑋𝐼) won’t function until being divided by X, which
is the infected times of XI.
We obtain the satellite map of Kenema in Sierra Leone from the internet and
intercept a part of its 1.381×0.802 square kilometers as a plate. Then we utilize
image recognition to identify buildings, greenbelts, wildernesses and streets and make
simple classification of buildings to recognize different regions including
𝑹, 𝑶, 𝑯, 𝑮, 𝑴 and 𝑺𝒕 in the targeted plate.
We draw the regions above in the display panel of the program according to the
colors in the table below:
region 𝑅 𝑂 𝐻 𝐺 𝑀
color dark blue blue yellow dark green light green

Figure 11: The region identification picture in the Kenema plate
Now we add citizens of different type 𝑨, 𝑺, 𝑳, 𝑬𝑰, 𝑳𝑰, 𝑽 and the specific numbers
are listed below:
type 𝐴 𝑆 𝐿 𝐸𝐼 𝐿I 𝑉
number 1122 848 14 9 5 2
We set the refreshed step as one minute in the simulation environment and then
begin simulating.
Figure 12: The simulation picture of 3 o’clock a.m.

In the picture, the red dots represent the infected ones 𝑬𝑰, 𝑳𝑰 and 𝑽. the purple
dots represent the infected of latent period 𝑳. We further enlarge the dots of these two
types of individuals, letting them be identified clearly from the picture.
5.2.1 Medication Delivery System In Low Rank Plates
We assume that the plate shown below are determined to be low rank plate with
slight infection condition and select the hospitals whose coordinates are (262,228)
and (301,254) as the main hospitals in the plate according to the Center Method and
they are marked with brownish red in the picture. In order to identify the patients’
specific infected states, we label the individuals with letter 𝑳, 𝑬𝑰, 𝑳𝑰, 𝑽, 𝑫.
The picture below shows the first day condition of the main hospitals delivering
medications.
Figure 13: The picture of the delivery system in the low rank plate
As of the 21st
day, the number of the infected of latent period, early stage, late
stage and variation period all decrease into zero which means the Ebola viruses have
been eradicated completely. The number of the infected(excluding the ones of latent
period) , dead tolls and the medicable infected are shown in the following picture.

Figure 14: the line graph based on simulation data in the low rank plate
5.2.2 Medication Delivery System In Medium Rank Plates
We assume that the plate shown below are determined to be medium rank plate
with medium infection condition and select the hospitals whose coordinates are
(262,228), (301,254) and (460,93) , (481,112) as the main hospitals in the plate
according to the Extending Center Method and they are marked with brownish red in
the picture.
Figure 15: The picture of the delivery system in the medium rank plate

As of the 16th
Figure 16: the line graph based on simulation data in the medium rank plate
5.2.3 Medication Delivery System In High Rank Plates
We assume that the plate shown below are determined to be medium rank plate
with medium infection condition and select the hospitals whose coordinates are
(262,228), (301,254) and (353,40), (362,48)as the main hospitals in the plate
according to the Extending Center Method and they are marked with brownish red in
the picture.(As the latter area is too small to be seen clearly, we mark it with the red
ring.) Additionally, we construct medical stations at the coordinate (493,99) and
(453,239), which are marked with orange below. Since he medical stations will
publicize the relevant Ebola viruses knowledge to the nearby citizens, the citizens
with Ebola symptoms( the crowds of EI and LI ) near the medical stations will have a
greater probability to go there whether to diagnose or fetch the medications.

Figure 17: The picture of the delivery system in the high rank plate
As of the 17th
day, the infected around the medical stations are few, which
represents it’s significant for the medical stations to make publicity.
Figure 18: The mid-stage effect picture in the high rank plate
As of the 28th

Figure 19: the line graph based on simulation data in the high rank plate
5.2.4 Conclusion
The medication delivery system adopted by the medium rank plates exhibits
excellent effect as it saves not only human resources and material resources but also
the financial resources in contrast with the high rank plates. In addition, achieving the
excellent effects which transcend far more than the low rank ones and it exhibits
strong practice value.
Although there’s no denying that high rank plates are seriously infected and it
will take relatively longer tome to eradicate the viruses, however, the medication
delivery system adopted by the high rank plates are the most functional, especially at
the setting of the medical stations which can prevent the viruses from spreading
effectively. While it puts on relative heavy burden on the human and financial
resources, it still exhibits great function at the seriously infected regions or during the
special periods.

5.3 Analysis of Medication Quantities & Production
Rate
According to EMCM, we add some medication factors that is if we set
𝑝, 𝑞 > 0, 𝑡ℎ𝑒𝑛 𝜂 = 𝜉 = 𝜅 = 1, more parameters will be shown as below:
parameter 𝜔 𝜎 𝜈 𝜏 𝜑 𝜓 𝑝 𝑞
value 0.3340 0.4551 0.2910 0.1191 0.3398 0.4102 0.0801 0.0781
The variation of the infected including EI and LI is shown as the figure below.
And we need to mention that at the point that 𝑡 = 219, the value decreases below1.
Figure 20: The variation of the infected including EI and LI
We solve the total quantities of medications 𝐵 = 27088.
We assume that Ebola has spread for 2 months about 61 days and the first batch
of medications have reached the targeted region, as is shown from the marked points
in the dashed line. At the point 𝑡 = 61, the region has accumulated ∑ 𝐵𝑡
61
𝑡=1 = 8897
EI and LI waiting the medications.
We also assume that the maximum quantities the cargo airplane can carry

𝐻 𝑎𝑖𝑟 =9000, according to 𝐵𝑡 , we can solve 𝑇𝑎𝑖𝑟 that meets ∑ 𝐵𝑡
𝑇 𝑎𝑖𝑟
𝑡=1 ≤ 𝐻 𝑎𝑖𝑟 <
∑ 𝐵𝑡
𝑇 𝑎𝑖𝑟+1
𝑡=1 , that is 𝑇𝑎𝑖𝑟 = 2.
We assume𝑣 𝑚𝑎𝑥 = 30000, 𝑇𝑏 = 1 and 𝑎 = 1.0001(1.000130000
≈ 20.1，it
represents that the total cost with the maximum velocity production is 20 times bigger
than that without production)
Then the feasible region will be 0 < 𝑇𝑒𝑙𝑠𝑒 ≤ 1, 0 < 𝑣 ≤ 30000, the constraints
condition will be 𝑇𝑒𝑙𝑠𝑒 ∗ 𝑣 = 𝐵 − 𝐻 𝑎𝑖𝑟 = 18088 and the objective function will be
𝑌 = 𝑐𝑇𝑒𝑙𝑠𝑒 ∗ 1.0001 𝑣
.
Note that the feasible region and the constraint condition exactly construct a
limited segment of the inverse proportional function. Therefore, we explore on it with
the 0.00001 step length and them we will find out the optimal solution of the above
nonlinear programming, that is
𝑇𝑒𝑙𝑠𝑒 = 1, 𝑣 = 18088
In conclusion, the velocity should be controlled with 18088 doses per day.
6 Conclusion
1. We introduce the parameters from GA into the model and it shows good
fitthing with the infected data of the late stage.
2. It’s effective to take different delivery systems in different rank plates.The
medication delivery system adopted by the high rank plates are the most
functional, especially at the setting of the medical stations which can prevent
the viruses from spreading effectively. While it puts on relative heavy burden
on the human and financial resources, it still exhibits great function at the
seriously infected regions or during the special periods.
3. The total quantities and the producing rate of the medications can be
calculated rationally.

7 Strengths & Weaknesses
Model Strengths Weaknesses
Extending Multiple
Compartment Model
It divides the crowd type
specifically and can simulate
different types of diseases.
In addition, it has a strong
poplularity.
It has the collective
disadvantages that all the
comparment model have
and it’s not reasonable
enough to assume
individuals mix uniformly
and fails to describe the
actual contact relationships
among people.
Additionally, it requires
too many parameters
setting.
The Micro Individuals
Simulation Model
It can simulate scientifically
from the individual level and
describe actual contact
relationship among people
rationally, which represents
strong actuality and
popularity.
There are too many
parameters and they all
have high sensitivity, that
is a change of a single
parameter may result in
the change of the model
results.
The model of the total
quantities and the
producing rate of
medications
It has simple patterns and
is easy to comprehend.
Furthermore, it is
constructed following the
EMCM, therefore, it’s
effective and actual.
Because it is constructed
on the EMCM, there are
too many parameters to be
set.

8 Reference
[1] Nuno M, Chowell G, Gumel A B. Assessing the role of basic control measures,
antivirals and vaccine in curtailing pandemic influenza: scenarios for the US, UK and
the Netherlands[J]. Journal of the Royal Society Interface, 2007, 4(14): 505-521.
[2] http://en.wikipedia.org/wiki/Ebola_virus_epidemic_in_West_Africa
[3] Kermack W O, McKendrick A G. A contribution to the mathematical theory of
epidemics[C] Proceedings of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences. The Royal Society, 1927, 115(772): 700-721.
[4] Wallinga J, Edmunds W J, Kretzschmar M. Perspective: human contact patterns
and the spread of airborne infectious diseases[J]. Trends in microbiology, 1999, 7(9):
372-377.
[5] Liljeros F, Edling C R, Amaral L A N. Sexual networks: Implications for the
transmission of sexually transmitted infections [J]. Microbes and Infection, 2003, 5(2):
189-196.
[6] Hassin R, Levin A, Morad D. Lexicographic local search and the p-center
problem[J]. European Journal of Operational Research, 2003, 151(2): 265-279.

The non-technical letter
Since May 2014 Ebola outbreak, the terrible situation worldwide, which are
rapid growth in the number of cases and the treatment situation of no medicine can
cure, caused great panic.
But now treatment drug has been successful developed. In order to better use of
this drug to curb Ebola outbreaks, we establish a model to simulate the propagation
mechanism of Ebola. Simulation results show that the transmission speed is high if
the outbreak uncontrolled. Under the government and department of health promotion,
people's conscious to take preventive measures, then the outbreak propagation speed
will be slow but not zero. Based on propagation model, we simulated the existing
drug treatment after the outbreak of development. The simulation results show that
under the condition of drugs can be allocated to each patient, the outbreak will be
faster under control and can completely eliminate the outbreak in a certain time. And
combined with the current epidemic situation report, our model fitting effect is good
and estimate the outbreak and the actual situation is similar.
In distribution system, we set up a distribution system that distributes the drug
from the factory to country to city to blocks to hospitals (clinics) and end at the
patient. The system will distribute task layer upon layer, and extended to the various
blocks, which set up distribution station. Its service cover families. And it can be a
system which truly distribute each medicine to every man who need this medicine.
And fine-tune the system can be combined with reality, and less cost.
And the simulation propagation model we build in the preparation, because of
the good fitting effect, can be used to estimate the required number of drugs in the
fight against the disease process. As estimate more accurately, it can provide a more
reliable reference to pharmaceutical manufacturer drugs, so that the drug maker can
effectively avoid the abundance production caused by the waste and the drug shortage
caused by insufficient production.

Controlling Ebola Spread with Optimal Medication Delivery

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