IT

1. Viktors Gopejenko
Department of Natural Sciences and Computer Technologies
Professor, Dr. sc. ing.
Mathematic Simulation
Course Paper 1
Development of an Epidemic
ISMA University of Applied Sciences
Riga, Latvia

2. Development of an Epidemic
• Let us imagine a population that is initially healthy.
• In this population a number people infected with a
contagious disease appear.
• An individual could transmit or catch the illness from other
individuals. The transmission of the illness is due to the
physical proximity.
2

3. Development of an Epidemic
• During the infectious process the individuals can pass
through some or all of the following states:
• Susceptible (S), the state in which the individual can
catch the illness from another infected person.
• Infected (I), the state in which the individual finds
himself infected and can infect others.
• Recuperated (R), or cured, the state during which the
individual can not infect or be infected because he will
have acquired an immunity (temporary or permanent)
nor can he infect others as he has recuperated or has
passed through the contagious stage of the illness.
3

4. Development of an Epidemic
• In the various infectious diseases, we can find two
principle groups:
• Those that produce immunity (temporary or permanent)
in individuals who have been infected and have since
recuperated. The majority of these illnesses are of viral
origin (sarampion, varicela, poliomyelitis).
• Those that, once recuperated, may turn again to
susceptibility. These illnesses are mainly caused by
bacterial agents (venereal disease, pest and some
forms of meningitis) or protozoos (malaria).
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5. Development of an Epidemic
• Bearing in mind the different states in the infectious
process, the epidemiological models can be divided into
three big groups:
• SIR: The model susceptible-infected-recuperated,
related to illnesses that produce permanent immunity
and a typical cycle that includes the three states. This
does not mean that all the individuals of the population
must pass through these stages, some will not be
infected and remain healthy, in other words, they will
remain in state (S). Others will be immunised artificially
by vaccination or another method and will pass directly
to state (R) without having been infected.
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6. Development of an Epidemic
• SIRS: The model susceptible-infected-recuperated-
susceptible, the same as the previous model but
applicable in cases where the immunity is not
permanent and the individual is again susceptible
after a certain amount of time, such as the ‘flu.
• SIS: The model susceptible-infected-susceptible is
used in cases where the illness does not produce
immunity, the individual can pass from being
infected to being susceptible again skipping the
recuperation stage completely.
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7. Development of an Epidemic
• A model can bear in mind the vital dynamics of the
population (births, deaths, migratory movements)
depending on the temporal horizon studied, the
characteristics of the illness and the population being
studied.
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8. THE MODEL
Development of an Epidemic
• We are going to use one of the most well known
epidemiological models in biology as a reference for
our model. It is Kermack and McKendrick that is
expressed as:
𝑑𝑆
𝑑𝑡
= −𝛽 ∙ 𝑆 ∙ 𝐼
𝑑𝐼
𝑑𝑡
= 𝛽 ∙ 𝑆 ∙ 𝐼 − 𝛾 ∙ 𝐼
𝑑𝑅
𝑑𝑡
= 𝛾 ∙ 𝐼
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9. Development of an Epidemic
Mathematical MODEL
𝑑𝑆
𝑑𝑡
= −𝛽 ∙ 𝑆 ∙ 𝐼
𝑑𝐼
𝑑𝑡
= 𝛽 ∙ 𝑆 ∙ 𝐼 − 𝛾 ∙ 𝐼
𝑑𝑅
𝑑𝑡
= 𝛾 ∙ 𝐼
S is population who is susceptible to catching illness, I
the population infected and R the population that has
passed through the illness and has recuperated.
There are two constants, the rate of infection β and the
rate γ of recuperation.
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10. Development of an Epidemic
Mathematical MODEL
10
(01) beta= 0.001
(02) "dI/dt"= beta*S*I-gamma*I
(03) "dR/dt"= gamma*I
(04) "dS/dt"= -beta*I*S
(05) FINAL TIME = 20
(06) gamma= 0.4
(07) I= INTEG ("dI/dt", 100)
(08) INITIAL TIME = 0
(09) R= INTEG ("dR/dt", 0)
(10) S= INTEG ("dS/dt", 900)
(11) SAVEPER = TIME STEP
(12) TIME STEP = 1
Vensim Model and Equation list

11. Development of an Epidemic
11
Epidemic Model
1000
500
0
0 3 6 9 12 15 18
Time (Month)
S : Current.vdfx
I : Current.vdfx
R : Current.vdfx

12. Model 2
The mental Model
• Let’s start from creation 3 stocks:
• Add the flow of the vaccinated individuals
• Short cut: Press and hold down the SHIFT key, click inside the initial Level, move
the mouse and click again in each corner of the flow, click inside the final Level.
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13. Model 2
The mental Model
• We draw the flow of the exposed and repeat the
process obtaining a set of flows that feed back each
other:
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14. Model 2
The mental Model
• Complete the model with the flows of make ill, cure,
and death:
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15. Model 2
The mental Model
• We can add shadow variables.
• Pressing the key
A menu opens and we can choose the variable which we
want to copy.
The usefulness of these copies is that we can avoid
producing excessive crossing of arrows in the model.
In our case, we will create the variable Total Population
as the sum of the three possible states:
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16. Model 2
The mental Model
16

17. Model 2
The mental Model
• We can add the remaining elements of the system,
introduce the equations and simulate.
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18. Model 2
The mental Model
• The software pre-writes the equations of the stocks
based on how we have drawn the flows.
• The equations are simple as the stocks vary depending
on the entries and exits that we have assigned to them
with their corresponding sign and the flows are, in
general, a product of the value of a stock by the value
of a rate.
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19. Model 2
The mental Model
• There are formulas that are slightly more complicated -
those of the infection:
‘make ill’
We can apply the equation formula from the model by
Kermack and McKendrick. According to this model, the
number of people who infect can be calculated as the
product of the number of people susceptible by the
number of people infected by the rate of contagion.
•
𝑑𝐼
𝑑𝑡
= 𝛽 ∙ 𝑆 ∙ 𝐼
19

20. Model 2
The mental Model
‘stress’
• We create a concept of stress to gather the fact that
depending on the number of people that infect in
relation to the people who are susceptible to infect, a
higher quantity of vaccinations is produced.
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21. Model 2
Equations The mental Model
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(01) Cure= Infected*rate of cure
(02) Dead= Infected*rate of mortality
(03) Exposed= rate of exposition*Recuperated
(04) FINAL TIME = 20
(05) Infected= INTEG ( Make ill-Cure-Dead, 100)
(06) INITIAL TIME = 0
(07) Make ill= Infected*rate of contagion*Susceptible
(08) rate of contagion= 0.001
(09) rate of cure= 0.4
(10) rate of exposition= 0.05
(11) rate of mortality= 0.1
(12) rate of vaccination= 0.5
(13) Recuperated= INTEG (Cure+Vaccinated-Exposed, 0)
(14) SAVEPER = TIME STEP
(15) stress= (Make ill/Susceptible)*rate of vaccination
(16) Susceptible= INTEG (Exposed-Make ill-Vaccinated, 900)
(17) TIME STEP = 1
(18) Total Population= Infected+Recuperated+Susceptible
(19) Vaccinated= stress*Susceptible

22. Model 2
The mental Model
• Pressing the automatic simulation icon
(SyntheSim) we can observe that in a simultaneous
form the evolution of each variable in the model, and
also simulate the effect of changes on the variables in
the rates that are constant by displacing the cursor to
the cursor to the left and right.
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23. Model 2
The mental Model
23

24. Model 2
The mental Model
• The behaviour of specific variables:
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Epidemic Dymanics
900
450
0
0 40 80 120 160 200
Time (Month)
Susceptible : 2.vdfx
Infected : 2.vdfx
Recuperated : 2.vdfx