Modeling seasonal variation of an SEIR on epidemic diseases

FREDEIC MUHAWENIMANA ([email protected])

African Institute for Mathematical Sciences (AIMS) Rwanda

Supervised by: Dr. Denis NDANGUZA

University of Rwanda, Rwanda

May 2019

Submitted in partial fulfilment of the requirements of a Master of Science in Mathematical

### Sciences at AIMS Rwanda

Abstract

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why they should be interested in your research project but summarises all they need to know if they

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clear and direct. In the rest of the research project, however, you will introduce and use technical terms.

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You may like to repeat the abstract in your mother tongue.

### Declaration

I, the undersigned, hereby declare that the work contained in this research project is my original work, and

that any work done by others or by myself previously has been acknowledged and referenced accordingly.

Firstname Middlename Lastname, May 2019

i

Contents

Abstract i

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 General objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 specific objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.6 Flowchart of model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Mathematical Modeling 4

2.1 Description of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Third Chapter 5

3.1 See? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Numbering in AIMS essays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 This is a section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 The Second Squared Chapter 7

4.1 This is a section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

### References 10

ii

1. Introduction

Mathematical modeling for disease transmission in keeper population is of great practical value in

predicting and controlling the spread of diseases. The fight between infectious diseases and humans

have a long history.

The spread of infectious diseases through human population has been the subject of scientific researchers

for a hundred years. The presence of many researchers on epidemiology diseases has always shown an

important part in human communities. Some of the diseases in any of the countries of the world are

endemic, they always still present in the population like malaria, cholera, etc. Other diseases like AIDS

can spread from the epidemic.

1.1 Background

Many years ago, many researchers have been work on infectious epidemics diseases and strategies

have been increased base on mathematical modeling. The environment change is some of the big

problematic season variations. Daniel Bernoulli, who was born in Groningen, Swiss in 1700, well known

as Mathematician, provided the earliest mathematical model describing the infectious diseases wherein

1760, he modeled the spread of smallpox (Bernoulli, 1760)

Many diseases show seasonal behavior such flu (LONDON and YORKE, 1973)",measles",chickenpox",

mumps by (Earn et al., 2002; AL-AJAM et al., 2006).

In 1942 to 1945, Malaria Control in War Areas(MCWA) was established to control Malaria around

military people where Malaria was a big problem for the soldiers and even the people around those

areas. During this work, many of the bases were established because in those areas the mosquitoes

were absolutely a big problem for people. MCWA pointed to present renew malaria into the civilian

population by mosquitoes that would have fed on malaria-infected soldiers in returning from the endemic

region (Linscott, 2011).

Malaria is one of the infectious diseases that can cause many times with change of climate like rainfall.

Malaria is caused by the parasite called Plasmodium and it can be transmitted by Anopheles’ mosquitoes

in the way they bite human beings and they will be a contact with blood meal for the development of

their eggs. After few weeks, the humans get the contact with Anopheles’ mosquitoes, the symptoms

of malaria may happen due to the rupture of the red blood cells and release the waste of parasites

with the debris of cells into the bloodstream. They are some of the five species that can cause human

Malaria based on the season area such as Plasmodium falciparum, Plasmodium Malaria, Plasmodium

Ovale, Plasmodium vivax, and Plasmodium knowlesi (Smith et al., 1993) . Seasonality, a periodic rise

in disease incidence correspond to season or other calendar periods. The spreading of diseases can be

caused by different factors. The environment factors such as rain fall, temperature, and pollution of

atmosphere can affect the population. The malaria is some of the diseases that can happen many times

during the rain fall and periodic season where many of people suffer with malaria based on different

seasonal variation. How do we incorporate seasonality in our model? What are the effect of seasonality

on the dynamics of infectious diseases?

The epidemic model dynamics, due to its practical and theoretical significance has been studied exten-

sively (Anderson and May, 1979; Hethcote, 2000). In most of the epidemic, models have constants

parameters. This the case of for the contagious diseases spread by mosquitos, where most of the

1

Section 1.2. Problem statement Page 2

mosquitos die out of winter but they reproduce hugely in summer, hence the spread of diseases is

seasonal. Thus under periodic environment, it more realistic to investigate the corresponding epidemic

models with periodic parameters.

The compartment of the model with labels such as S, E, I and R are used in epidemiology diseases and

SEIR is the abbreviation of Susceptible, Exposed, Infectious and Recovered.In1995, Michael Y. Li and

James S. Mulroney studied an SEIR model in epidemiology(M. Y. Li, 1995) . After this study, there

have been researches about epidemic models with latent periods (Herzog and Redheffer, 2004).

1.2 Problem statement

Seasonality, a periodic rise in disease incidence correspond to season or other calendar periods. The

spreading of diseases can be caused by different factors. The environment factors such as rain fall",

temperature, and pollution of atmosphere can affect the population. The malaria is some of the diseases

that can happen many times during the rain fall and periodic season where many of people suffer with

malaria based on different seasonal variation. How do we incorporate seasonality in our model? What

are the effect of seasonality on the dynamics of infectious diseases?

The SEIR model is used to help us to model this disease by including the seasonality of transmission

parameter that will vary depending on the calendar periods of diseases. For examples, we begin by

modeling the population levels as a function of rainfall depending on rainy seasons in a year.

1.3 General objective

To formulate model and analyze malaria with seasonal variation

1.4 specific objective

• Formulation of SEIR model with seasonality transmission rate.

• Estimation of model parameter.

• Numerical solution.

1.5 Problem statement

Seasonality, a periodic rise in disease incidence correspond to season or other calendar periods. The

spreading of diseases can be caused by different factors. The environment factors such as rain fall",

temperature, and pollution of atmosphere can affect the population. The malaria is some of the diseases

that can happen many times during the rain fall and periodic season where many of people suffer with

malaria based on different seasonal variation. How do we incorporate seasonality in our model? What

are the effect of seasonality on the dynamics of infectious diseases?

Section 1.6. Flowchart of model Page 3

1.6 Flowchart of model

Let’s demonstrate a figure by looking at Fig. 1.1.

Figure 1.1: Epidemic diseases flowchart.

Remember how to include code with verbatim and to fix the tabs in python in a verbatim environment?

It may be best to have an ‘include’ command for code, not to have to re-edit it all the time.

# This program prints hello

import Scipy as S

if __name__ == "__main__":

print "hello"

2. Mathematical Modeling

2.1 Description of Model

The model starts with the assumption saying that the total population N is constant at any time “t” and

the individuals are assuming to be homogeneous and mix uniformly. The basic assumption is saying that

the population N can be subdivided into 4 groups depends on the level of diseases. Our model classifier

the individuals as susceptible, Exposed, infectious and recovered and it is called the SEIR mathematical

model. The individual born into susceptible group S(t) and the susceptible individuals are people who

have never come into contact with the disease and who are able for getting disease except not able

to spread to other people can be called exposed group E(t). The exposed group can stay in the same

group up to (1� ). In time the exposed individuals start to spread the disease, they are said to move into

the infectious group I(t). The infected individual can spread the disease to susceptible and can stay in

the infectious group for a certain period of time( 1γ ) before moving into the recovered group. Lastly, the

recovered individual’s group are assumed to be immune for life. And the whole population is given as

N= S(t)+ E(t)+I(t)+ R(t)

dS

dt = α− rβ(t)S

I

N − µS

dE

dt = rβ(t)S

I

N − �E

dI

dt = �E − γI − µI

dR

dt = γI − µR

• β(t) = β0(1 + β1 cos(ωt))

The model will focus in seasonal transmission rate

2.2 Problem statement

2.2.1 Theorem (Jeff’s Washing Theorem). If an item of clothing is too big, then washing it makes it

bigger; but if it is too small, washing it makes it smaller.

Proof. Stated without proof. But a proof would look like this.

4

3. Third Chapter

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3.0.1 Theorem (My Theorem2). This is my theorem2.

Proof. And it has no proof2.

Lorum ipsum.

3.1 See?

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3.1.1 Theorem (My Theorem2). This is my theorem2.

Proof. And it has no proof2.

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orci sit amet orci dignissim rutrum.

x = y + y (3.1.1)

= 2y (3.1.2)

5

Section 3.2. Numbering in AIMS essays Page 6

Equations (3.1.1) and (3.1.2) are trivial.

3.2 Numbering in AIMS essays

Here is a conjecture:

3.2.1 Conjecture. The washing operation has fixed points.

And here is an example:

3.2.2 Example. 5 Rand coin.

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3.3 This is a section

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4. The Second Squared Chapter

An average research project may contain five chapters, but I didn’t plan my work properly and then ran

out of time. I spent too much time positioning my figures and worrying about my preferred typographic

style, rather than just using what was provided. I wasted days bolding section headings and using

double slash line endings, and had to remove them all again. I spent sleepless nights configuring

manually numbered lists to use the LATEX environments because I didn’t use them from the start or

understand how to search and replace easily with texmaker.

Everyone has to take some shortcuts at some point to meet deadlines. Time did not allow to test model

B as well. So I’ll skip right ahead and put that under my Future Work section.

4.1 This is a section

Some research projects may have 3, 5 or 6 chapters. This is just an example. More importantly, do you

have at close to 30 pages? Luck has nothing to do with it. Use the techniques suggested for writing

your research project.

Now you’re demonstrating pure talent and newly acquired skills. Perhaps some persistence. Definitely

some inspiration. What was that about perspiration? Some team work helps, so every now and then

why not browse your friends’ research project and provide some constructive feedback?

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7

Appendix

Some text

8

### Acknowledgements

This is optional and should be at most half a page. Thanks Ma, Thanks Pa. One paragraph in normal

language is the most respectful.

Do not use too much bold, any figures, or sign at the bottom.

9

### References

AL-AJAM, M. R., BIZRI, A. R., MOKHBAT, J., WEEDON, J., and LUTWICK, L. Mucormycosis in

the eastern mediterranean: a seasonal disease. Epidemiology and Infection, 134(2):341–346, 2006.

doi: 10.1017/S0950268805004930.

Anderson, R. M. and May, R. M. Population biology of infectious diseases : Part i. Nature 280.5721",

(7), 1979.

Bernoulli, D. Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages

de l’inoculation pour la prévenir. Histoire de l’Acad., Roy. Sci.(Paris) avec Mem, pages 1–45, 1760.

Earn, D., Dushoff, J., and Levin, S. Ecology and evolution of the flu. Trends in Ecology and Evolution",

17(7):334–340, 7 2002. ISSN 0169-5347. doi: https://doi.org/10.1016/S0169-5347(02)02502-8.

Herzog, G. and Redheffer, R. Nonautonomous seirs and thron models for epidemiology and cell biology.

Nonlinear Analysis-real World Applications - NONLINEAR ANAL-REAL WORLD APP, 5:33–44, 02

2004. doi: 10.1016/S1468-1218(02)00075-5.

Hethcote, H. W. The mathematics of infectious diseases. SIAM Review, 42(4):599–653, 2000.

Linscott, A. J. Malaria in the united states–past and present. Clinical Microbiology Newsletter, 33(7):

49–52, 2011.

LONDON, W. P. and YORKE, J. A. RECURRENT OUTBREAKS OF MEASLES, CHICKENPOX AND

MUMPS: I. SEASONAL VARIATION IN CONTACT RATES1. American Journal of Epidemiology",

98(6):453–468, 12 1973. ISSN 0002-9262. doi: 10.1093/oxfordjournals.aje.a121575. URL https:

//doi.org/10.1093/oxfordjournals.aje.a121575.

M. Y. Li, J. S. M. Global stability for the seir model in epidemiology. math. biosci. 5:155–164, 1995.

doi: 125.

Smith, T., Charlwood, J., Kihonda, J., Mwankusye, S., Billingsley, P., Meuwissen, J., Lyimo, E., Takken",

W., Teuscher, T., and Tanner, M. Absence of seasonal variation in malaria parasitaemia in an area

of intense seasonal transmission. Acta tropica, 54(1):55–72, 1993.

Williams, B. and Dye, C. Infectious disease persistence when transmission varies seasonally. Mathemat-

ical biosciences, 145(1):77–88, 1997.

10

https://doi.org/10.1093/oxfordjournals.aje.a121575

https://doi.org/10.1093/oxfordjournals.aje.a121575

Abstract

### Introduction

Background

### Problem statement

### General objective

specific objective

### Problem statement

### Flowchart of model

### Mathematical Modeling

### Description of Model

### Problem statement

### Third Chapter

See?