Partial Differential Equation Optimation for Infectious Disease Model

Partial Differential Equation Optimation for Infectious Disease Model

Lifeng Zhang School of Science Dalian Ocean University

Dalian, China e-mail:zlfa@dlou.edu.cn

Peng Lian

College of Marine and Environmental Dalian Ocean University

Dalian, China

Jianing Zou School of Science Dalian Ocean University

Dalian,China

Huakai Deng School of Science Dalian Ocean University

Dalian,China

Abstract—The most basic infectious disease model ,SIR model is widely used and upgraded to evaluate the affect of viruses now, because of the outbreak of Ebola. In this paper, we give a brief introduction to SIRS model, and adopt the SEIRS (Suspectable-Exposed-Infective-Recovered Suspectable), which apply partial differential equations to optimize the SIRS. By solving four Partial Differential Equations, we set a description of the relationship between the four kinds of people, time and space. The advantage of this method is that we consider the incubation period of viruses. Meanwhile, we show the numerical solution through diffusion figure of Ebola by using matlab.

Keywords-differential equation; model ;SIRS;SEIRS.

I. INTRODUCTION

The world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems (sending the medicine to where it is needed), (geographical) locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain.

This paper main model a equation set of partial differential equations. We based on practical data from the official website from WHO, and we explain the importance of prophylaxis and treatment according to the basic reproduction number.

II. THE SIRS MODEL

SIR model is one of the most classics model in the aspect of infectious disease (founded by Daniel Bernoulli in

1760). There are 3 kinds of people basically. The Susceptible means people who is non-ill, but is compromised. The Infective means people who is ill, and can spread the disease to Susceptible. And the Removal means people who is segregated and immune to the disease.

Because they are easy and usual, we don’t describe this part but the more advanced model-SIRS model in followed passage. And there is a important number named basic reproduction number. It means a number of people who are infected by an Infective in his or her whole life, when the other people are non-immunity and there is no other factors.

A. Assumption of Model

a. This model covers 4,100,000 people, exampled from Sierra Leone.

b. The initial infective quantity is 10.

c. On the border population movements is zero in SEIRS model.

d. The virus variation can not happen during this period.

B. Symbols definition

S:The people who can’t be infected.

I: The people who have been infected.

R: The people who are vulnerable to be infected.

Ka: The rate of infection.

Kb: The rate of antibody disappear.

La: The percent of the people who can bevaccination per day.

Lb: The percent of the people who can be curedper day.

T:Time.

C. The SIRS Model

The main differential equations function:

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DOI 10.1109/ICICTA.2015.160

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dS Ka

SI KbR LaS

dt n

dI Ka SI LbI dt n

dR LaS LbI dt

∃ % ∃

!!

! ∃

∀!

! %

!#

(1)

Where dS dI dR, , dt dt dt

means the increasing rate of each kind of people along the time variable t . The S is proportionate to the decreased volume of S and I . But Iis proportionate to the increased volume of S and I . Then S and I will slowly change into R at the rate of

La

and

Lb

. When medicine is introduced in the system, S and Iwill decrease at the rate of

La

and

Lb

.

The system cycles as descriptions before. And when the number of I decreased to 1 or below , the cycle will stop and all the Swill be turned toR ^{[ 2]} .

D. The Results of SIRS Model

This figure shows the the people who can’t be infection(S ) varies along the variable time t . We can see that after Ebola exploded 5 ~ 6 weeks , S decreased immediately.

0 1 2 3 4 5 6 7 8 9 101112131415161718 0

0.5 1 1.5 2 2.5 3 3.5 4

4.5x 10⁶ S

t(week) n(people)

Figure 1. The people who can’t be infected

500 and about 5 or 6 weeks later , Iincreased.

0 1 2 3 4 5 6 7 8 9 1 0 1

1 1 2 1

3 1 4 1

5 1 6 1

7 1 8 0

0.5 1 1.5

2 2.5

3 3.5

4 x⁶ I

t(week)

Figure 2.The people who have been infected

This figure shows that when S decreased , Rincreased nearly synchronously .

0 1 2 3 4 5 6 7 8 9 1 0 1

1 1 2 1

3 1 4 1

5 1 6 1

7 1 8 0

5 10 15

x R

t(wee k)

Figure 3. The people who are vulnerable to be infected

However , we found that viruses havetheir incubation period andthey spreads not only along time , but also space^[3] . We considered some space spreading models of some kinds of viruses and finally chose one of them . We decide to introduce a new variable volumeEto describe the the people who are in the incubation period , and a constant volumed to describe the space spreading speed of Ebola . Thus we came up with a new model as SEIRS MODEL and the main partial differential equationsfunction of our own .

III. THE SEIRS MODEL

A. Symbols Definition

E: The people who are in the incubation period.

d: The diffusion rate of the Ebola virus.

r: Location variable.

B. The SEIRS Model

Dynamic transmission model,the main partial differential equationsfunction:

t

t

t t

S d S KaSI KbR LaS n

E d E KaSI LcE n

I d I LcE LbI R d R LaS LbI KbR ∃ & ∃ % ∃

!!

! ∃ & ∃

∀!

∃ & ∃

!! ∃ & % ∃

#

(2)

Boundary condition:

(0, ) (0, ) (0, ) (0, ) 0 (1, ) (1, ) (1, ) (1, ) 0

r r r r

r r r r

S t E t I t R t

S t E t I t R t

∀

#

(3)

Which means the speed at the boundaries will be 0.

Initial condition:

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2

( , 0) 0 ( , 0) 0

1 ( 0)

( , 0) 25 0 exp

0.5 ( , 0) 0

S r S

E r E

I r I x

R r R

!

!! ∋ ∃ ∗

∀ (∃ +

! ) ,

!!

#

(4)

Which means the initial status of every parameters.

Whered S d E d I d R& , & , & , & means the transmission rate of each kind of people along the position variable r^[4] . The S is proportionate to the decreased volume of S and I. But Ebola has its incubation period,Swill change into Eat first.

Then Ewill slowly change into I at the rate of _Lc. When medicine is introduced in the system, Iand Swill decrease at the rate of La and Lc. But this is a SEIRS model,some R will lose their immunity in 10 or 15years ,which will cause the reverse of R to S , of course at a very low speed .Thus the system cycles. In the equations:

d=2.5,ka 0.029, Kb 0.0000027, La 0.00007, Lb 0.0005, Lc 0.05, n 4100000;

0 4099990, 0 0, 0 10, 0 0

S E I R

The basic reproduction number

R0will decide whether the cycle continues. Based on our equations,

0 (La )

Ka Kb

R Lb %Kb .^[5] If

R

₀>1 ,the cycle occurs. If

R

₀<1, the cycle will stop.^[6] The R₀ of our model will be 2.15. We solved the equations through MATLAB, and made some figures about each kind of people to show how they change along the position r and time t as follows:

The figure shows the suspects along position r and time t.

When time goes on, Sdecreases Linearly.

0

1

2 0

1 4.042 4.05 4.06 4.07 4.08 4.09 4.1

x 10⁶

r(km) S

t(week)

n(people)

Figure4. The people who can’t be infected in SEIRS

This figure shows a non-liner relationship between the exposed people along t and r. It shows that in 2 weeks, people increased fast.

0 1 2

0 1 20 50 100 150 200 250

r(km) E

t(week)

n(people)

Figure5. The people who are in the incubation period in SEIRS

This figure shows the infected people change alongt and r. At first it exploded as the medicine is not awake.

0 1 2

0 1 20 100 200 300 400 500 600

r(km) I

t(week)

n(people)

Figure6. The people who have been infected in SEIRS

This figure shows the recovered people change along t and r. It almost steady increase with t.

0

1

2

0 1 20 1 2 3 4 5 6

x 10⁴

r(km) R

t(week)

n(people)

Figure7. The people who are vulnerable to be infected in SEIRS

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IV. DISCUSSION AND CONCLUSION According to The World Health Organization(WHO), the current Ebola epidemic is likely to be controlled in the coming months with their new medication. However, evidence has been accumulating that a diverse group of enveloped viruses is gradually to unstoppable growth if there is no effective measures. To balance the contradiction between affected areas and medical structures, we dedicated to the proposition that scientific decisions are created rebirth of Africa. we can conclude:

Our suggested solution, which is easy to implement, includes a detailed timetable and the arrangement of pipes.

Since our model is based on SIR model, it can be applied to other domain.

We find that SEIRS model has work very well. It’s can describe the outbreak reasonable. And we find the

R

₀ is 2.15, which means the viruses will break out if there is no action be taken.

ACKNOWLEDGMENT

This work was supported by the project of Dalian Ocean UniversityMathematical modeling of excellent resource sharing constructionAnd when we have a problem in model’s equations, the doctor Mr. Kang comes from Dalian

University of Technology give some suggests to us, who main direction of studying is differential equation. We are grateful to this gentlemen.

REFERENCES

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http://maps.who.int/MapJournal/?appid=5b29178821dd4584b78dcb18d280 326a

[2] Zhang Qunying. Study on some nonlinear problems related to free boundary,2012.

[3] Christian L. Althaus.Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa,2014.

[4] Abhishek Pandey, Katherine E.Atkins.Strategies for containing Ebola in West Africa. Science,2014,PP. 346- 991.

[5] Ni Shunjiang.Research on Modeling of Infectious Disease Spreading Based on Complex,Network Theory,2009.

[6] Zhang Yun-hui,Wang Shu,Chen Yu-qi,Li Jun.Research advances in Ebola virus disease and disease update in the 2014 Ebola outbreak.China.

[7]Niu Yinwu,. Operations Research. Xi'an Jiao Tong University Press,1994.

[8] Wei Guohua,He Jialiang,Zhou Zhongliang. Application of operations research.Shanghai:Fudan University Press,1987.

[9] Xiong Qicai, Methods and Application of Mathematics Modeling, Chong Qing, Chongqing University Press,2005.

[10] Zhang Zhubao. Mathematics Modeling and Mathematics Experiment, Beijing, High Education Press ,2005.

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