STOCHASTIC EPIDEMIC MODELS WITH POISSON INFECTIVES AND CARRIERS

S.Revathi, Assistant Professor1, C.Nithya, Assistant Professor2

Department Of Statistics,

Marudhar Kesari Jain College For Women, Vaniyambadi1

Department Of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi2

Mail id: revathiramesh06@gmail.com1

Mail Id: nithimadhavan287@gmail.com2

Abstract

In this Paper, we have derived stochastic epidemic models Poisson infection, carrier and removal rates. Stochastic modelling of epidemic is importent when the number of infectious individuals is small or when the variability in transmission, recovery, births, deaths or the environment impacts the epidemic outcome.

Keywords: Epidemics, Susceptible, infection rate, carrier, removal rate.

Introduction

An Epidemic is the rapid spread of infections disease to a large number of people in a given population within a short period of time, usually two weeks or less. An Epidemic may be restricted to one location, however if it spreads to other countries or continents and affects a substantial number of people, it may be termed a pandemic. The occurrence of more cases of a disease than would be expected in a community or region during a given time period is called an epidemic. We have derived stochastic epidemic models carrier rates with Poisson infection.

1. Definition

An Epidemic is an usually large short term outbreak of a disease, such as cholera and Plague etc….

The spread of disease depends on

(i) The mode of transmission

(ii) Susceptibility

(iii) Infections period

(iv) Resistance and many other facts.

1.1 Definition

The population in an epidemiological model can be classified into three types.

(i) Susceptible – S(t)

(ii) Infected – I(t)

(iii) Removal – R(t)

Where S(t) = number of individuals in the population who can be infected

I(t) = number of infected individuals in the population

R(t) = number of individuals removed from population by recovery, death, immunization.

1.2Definition

Consider the discrete random variable which taking the non negative values

(i.e) π‘π‘˜=𝑃(𝑋=π‘˜) where k = 0,1,2,3……

The probability generating function X is defined as

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𝐺𝑋(𝑆)=Ξ£π‘ƒπ‘˜π‘†π‘˜=𝐸(𝑆𝑋)∞π‘˜=0

Note

Consider 𝐴1𝐴2….𝐴𝑛 be a partition of Ξ©, for any event B

Pr(𝐡)= Ξ£Pr(𝐴𝑗)Pr (𝐡/𝐴𝑗)𝑛𝑗=1

2. Stochastic Epidemic Model with Infectives and Carries

Let m, n, p denote the number of susceptible, infectives and carriers. A susceptible can become infective by contact with either an infected or a carrier. The result is represented in the following model.

Let π‘ƒπ‘š,𝑛,𝑝(t) be the probability that there are m susceptible, n infectives and p carriers in the population at time t. If N is the total size of the population, then the number of persons in the removed category is N-m-n-p.

Let the probability of a susceptible being infected in the time interval (t, t + Δ𝑑) be 𝛽mnΔ𝑑 + O(Δ𝑑), and let the probability of a susceptible being infected due to contact with a carrier in the time interval (t,t + Δ𝑑) be 𝛾mpΔ𝑑 + O(Δ𝑑), and also let the corresponding probability of one carrier being removed in the same time interval be 𝛿𝑝Δ𝑑 + O(Δ𝑑).

In this case, we assume that 𝛽mn is the Poisson rate of susceptible becomes infected with parameter πœ†πΌ , 𝛾mp is the Poisson rate of susceptible becomes infected due to contact with a carrier with parameter πœ†πΆ , and 𝛿P is the Poisson rate of carrier becomes removed with parameter πœ†π‘….

i.e, 𝛽mn = = 𝑒−πœ†πΌ ( πœ†πΌ)π‘š π‘š!, m≥0 ( 1)

𝛾mp = 𝑒−πœ†πΆ ( πœ†πΆ)π‘š π‘š! , m≥0 (2)

𝛿𝑝= 𝑒−πœ†π‘… ( πœ†π‘…)π‘š 𝑝! , p≥0 ( 3)

Here, πœ†πΌ be the rate of infection from susceptible to infected, πœ†πΆ be the rate of carrier and πœ†π‘… be the rate of removal carrier to removed.

The probability that there is no change in the time interval (t, t + Δ𝑑) is then given by

1- 𝛽mn Δ𝑑 - 𝛿𝑝 Δ𝑑 + O (Δ𝑑) ( 4)

Now there can be m susceptible , n infectives and p carriers at the time t + Δ𝑑 if there are

a) m+1 susceptibles , n-1 infectives and p carriers at the time t and if one person has become infected in time Δ𝑑, (or)

b) m+1 susceptibles , n-1 infectives and p carriers at the time t and if one person has become infective due to contact with a carrier in time Δ𝑑, (or)

c) m susceptibles , n infectives and p +1 carriers at the time t and if one carrier has been removed in time Δ𝑑, (or)

d) m susceptibles , n infectives and p carriers at the time t and if there is no change in time Δ𝑑.

We assume that the probability of more than one change in time Δ𝑑 is 𝑂(Δ𝑑).

Then, using the theorems of total and completed probability , we get

π‘ƒπ‘š,𝑛,𝑝(t + Δ𝑑) = π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛽 (m+1) (n-1) Δ𝑑 + π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛾 (m+1) p Δ𝑑

+ π‘ƒπ‘š,𝑛,𝑝+1(t) 𝛿 (p+1) Δ𝑑 + π‘ƒπ‘š,𝑛,𝑝 (t) (1- 𝛽mn Δ𝑑 - 𝛿𝑝 Δ𝑑) + O (Δ𝑑)

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So that,

π‘ƒπ‘š,𝑛,𝑝(t + Δ𝑑) = π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛽 (m+1) (n-1) Δ𝑑 + π‘ƒπ‘š−1,𝑛−1,𝑝 (t) 𝛾 (m+1) p Δ𝑑

+ π‘ƒπ‘š,𝑛,𝑝+1(t)𝛿 (p+1) Δ𝑑 + π‘ƒπ‘š,𝑛,𝑝 (t) - π‘ƒπ‘š,𝑛,𝑝 (t)𝛽mn Δ𝑑

- π‘ƒπ‘š,𝑛,𝑝 (t) 𝛿𝑝 Δ𝑑) + O (Δ𝑑)

π‘ƒπ‘š,𝑛,𝑝(t + Δ𝑑) - π‘ƒπ‘š,𝑛,𝑝 (t) = π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛽 (m+1) (n-1) Δ𝑑 + π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛾 (m+1) p Δ𝑑

+ π‘ƒπ‘š,𝑛,𝑝+1(t)𝛿 (p+1) Δ𝑑 - π‘ƒπ‘š,𝑛,𝑝 (t)𝛽mn Δ𝑑

- π‘ƒπ‘š,𝑛,𝑝 (t) 𝛿𝑝 Δ𝑑 + O (Δ𝑑) (5)

Dividing on the both sides by Δ𝑑 , and proceeding to the limit as Δ𝑑 →0 in ( 5 ) we get

limΔ𝑑→0 π‘ƒπ‘š,𝑛,𝑝(t + Δ𝑑) − π‘ƒπ‘š,𝑛,𝑝 (t)Δ𝑑 = π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛽 (m+1) (n-1) + π‘ƒπ‘š+1,𝑛−1,𝑝 (t) 𝛾(m+1) p

+ π‘ƒπ‘š,𝑛,𝑝+1(t)𝛿 (p+1) - π‘ƒπ‘š,𝑛,𝑝 (t) 𝛽mn

- π‘ƒπ‘š,𝑛,𝑝 (t) 𝛿𝑝 + O

Therefore,

𝑑𝑑𝑑 (π‘ƒπ‘š,𝑛,𝑝(t)) = 𝛽 (m+1) (n-1) π‘ƒπ‘š+1,𝑛−1,𝑝 (t) - 𝛽mn π‘ƒπ‘š,𝑛,𝑝 (t) + 𝛾(m+1) p π‘ƒπ‘š+1,𝑛−1,𝑝 (t)

+ 𝛿 (p+1) π‘ƒπ‘š,𝑛,𝑝+1(t) – 𝛿𝑝 π‘ƒπ‘š,𝑛,𝑝 (t)

Initially, let there be a susceptible, b infective and c carriers.Then we define the probability generating function

πœ•πœ•π‘‘(Ξ£Ξ£Ξ£π‘ƒπ‘š,𝑛,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0) =

Ξ£Ξ£Ξ£π‘‘π‘‘π‘‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 (π‘ƒπ‘š,𝑛,𝑝 (t)) π‘₯π‘šπ‘¦π‘›π‘§π‘ ( 6 )

Multiplying ( 6) by π‘₯π‘šπ‘¦π‘›π‘§π‘ and summing over p from 0 to π‘Ž+𝑏+𝑐−π‘š−𝑛;𝑛 π‘“π‘Ÿπ‘œπ‘š 0 π‘‘π‘œ π‘Ž+𝑏−π‘š π‘Žπ‘›π‘‘ π‘š π‘“π‘Ÿπ‘œπ‘š 0 π‘‘π‘œ π‘Ž, we get

πœ•πœ•π‘‘(Ξ£Ξ£Ξ£π‘ƒπ‘š,𝑛,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0) =

Ξ£Ξ£Ξ£π‘‘π‘‘π‘‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 (π‘ƒπ‘š,𝑛,𝑝 (t)) π‘₯π‘šπ‘¦π‘›π‘§π‘

So that,

πœ•πœ‘πœ•π‘‘ = ΣΣΣ{ π›½π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 (m+1) (n-1) π‘ƒπ‘š+1,𝑛−1,𝑝(𝑑)-𝛽mnπ‘ƒπ‘š,𝑛,𝑝(t)

+𝛾(π‘š+1)𝑝 π‘ƒπ‘š+1,𝑛−1,𝑝(𝑑)+ 𝛿 (𝑝+1)π‘ƒπ‘š,𝑛,𝑝+1(𝑑) –

𝛿𝑝 π‘ƒπ‘š,𝑛,𝑝(𝑑) } π‘₯π‘šπ‘¦π‘›π‘§π‘

= 𝛽 { ΣΣΣ(π‘š+1)(𝑛−1)π‘ƒπ‘š+1,𝑛−1,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛 𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 –

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Ξ£Ξ£Ξ£π‘šπ‘›π‘ƒπ‘š,𝑛,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 }

+𝛾 {ΣΣΣ(π‘š+1)π‘π‘ƒπ‘š+1,𝑛−1,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0

+𝛿 {ΣΣΣ(𝑝+1)π‘ƒπ‘š,𝑛,𝑝+1π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘−π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0

Ξ£Ξ£Ξ£π‘π‘ƒπ‘š,𝑛,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 }

Now , the definition of probability generating function , we have

π‘₯ πœ•πœ‘πœ•π‘₯ = Ξ£Ξ£Ξ£π‘š π‘π‘›π‘šπ‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘šπ‘¦π‘›π‘§π‘ ;

𝑦 πœ•πœ‘πœ•π‘¦ = ΣΣΣ𝑛 π‘π‘›π‘šπ‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘šπ‘¦π‘›π‘§π‘ ;

𝑧 πœ•πœ‘πœ•π‘§ = ΣΣΣ𝑝 π‘π‘›π‘šπ‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘šπ‘¦π‘›π‘§π‘ ;

π‘₯𝑦 πœ•2πœ‘πœ•π‘₯πœ•π‘¦ = Ξ£Ξ£Ξ£π‘šπ‘› π‘π‘›π‘šπ‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘šπ‘¦π‘›π‘§π‘ ; and π‘₯𝑧 πœ•2πœ‘πœ•π‘₯πœ•π‘§ = Ξ£Ξ£Ξ£π‘šπ‘ π‘π‘›π‘šπ‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘šπ‘¦π‘›π‘§π‘ .

∴πœ•πœ‘πœ•π‘‘ = 𝛽{ 𝑦2ΣΣΣ(π‘š+1)(𝑛−1)π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 π‘ƒπ‘š+1,𝑛−1,𝑝 (t)

π‘₯π‘šπ‘¦π‘›−2𝑧𝑝 – π‘₯𝑦 Ξ£Ξ£Ξ£π‘šπ‘›π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 π‘ƒπ‘š,𝑛,𝑝 (t) π‘₯π‘š−1𝑦𝑛−1𝑧𝑝 }

+ 𝛾 {𝑧ΣΣΣ(π‘š+1)π‘π‘ƒπ‘š+1,𝑛−1,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘−1π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 }

+ 𝛿 {ΣΣΣ(𝑝+1)π‘ƒπ‘š,𝑛,𝑝+1π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘−π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0

𝑧 Ξ£Ξ£Ξ£π‘π‘ƒπ‘š,𝑛,π‘π‘Ž+𝑏+𝑐−π‘š−𝑛𝑝=0(𝑑)π‘₯π‘šπ‘¦π‘›π‘§π‘−1π‘Ž+𝑏−π‘šπ‘›=0π‘Žπ‘š=0 } (7)

By using the above relation ,we get

πœ•πœ‘πœ•π‘‘ = 𝛽 ( 𝑦2 πœ•2πœ‘πœ•π‘₯πœ•π‘¦− π‘₯𝑦 πœ•2πœ‘πœ•π‘₯πœ•π‘¦ )+ 𝛾 (𝑧 πœ•2πœ‘πœ•π‘₯πœ•π‘§ )+ 𝛿 (πœ•πœ‘πœ•π‘§−𝑧 πœ•πœ‘πœ•π‘§ )

that is, πœ•πœ‘πœ•π‘‘ = 𝛽 (𝑦2−π‘₯𝑦 ) πœ•2πœ‘πœ•π‘₯πœ•π‘¦+ 𝛾𝑧 πœ•2πœ‘πœ•π‘₯πœ•π‘§+ 𝛿 (1−𝑧)πœ•πœ‘πœ•π‘§ ( 8)

3. Classification of the solutions

We find the classification of the solution of the partial differential equation in equation (8).

Now, find a linear partial differential equation of the second order in four independent variables π‘₯,𝑦,𝑧 and t.

Equation (8) can be written as

0.πœ‘π‘₯π‘₯ + 𝛽2 (𝑦2− π‘₯𝑦) πœ‘π‘₯𝑦 +𝛾2 π‘§πœ‘π‘₯𝑧+0. πœ‘π‘₯𝑑 + 𝛽2 (𝑦2− π‘₯𝑦) πœ‘π‘¦π‘₯

+ 0.πœ‘π‘¦π‘¦ + 0.πœ‘π‘¦π‘§ + 0.πœ‘π‘¦π‘‘+ 𝛾2 π‘§πœ‘π‘§π‘₯+ 0.πœ‘π‘§π‘¦ +0.πœ‘π‘§π‘§+0.πœ‘π‘§π‘‘+0.πœ‘π‘‘π‘₯

+0.πœ‘π‘‘π‘¦+ 0.πœ‘π‘‘π‘§ +0.πœ‘π‘‘π‘‘ +0.πœ‘π‘₯+0.πœ‘π‘¦+Ξ΄ (1−z)Ο†z−1.Ο†t+0.Ο†=0 (9)

Let us define the matrix A, is given by

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A= [ π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒π‘₯π‘₯π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑦π‘₯π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑧π‘₯π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑑π‘₯ π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒π‘₯π‘¦π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘¦π‘¦π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘§π‘¦π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑑𝑦 π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒π‘₯π‘§π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘¦π‘§π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘§π‘§π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑑𝑧 π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒π‘₯π‘‘π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘¦π‘‘π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ π‘’π‘§π‘‘π‘π‘œπ‘’π‘“π‘“.π‘œπ‘“ 𝑒𝑑𝑑 ] (10)

Therefore, A = [ 0𝛽2 (𝑦2− π‘₯𝑦)𝛾2 𝑧0 𝛽2 (𝑦2− π‘₯𝑦)000 𝛾2 𝑧000 0000]

Since, π’‚π’Šπ’‹= π’‚π’‹π’Š , 𝑨= [π’‚π’Šπ’‹]πŸ’π‘ΏπŸ’ is a symmetric matrix of order 4 x 4.

So that,

|𝐴| = ||0𝛽2 (𝑦2− π‘₯𝑦)𝛾2 𝑧0 𝛽2 (𝑦2− π‘₯𝑦)000 𝛾2 𝑧000 0000|| = 0

Also, the Eigen values of A is given by the form |𝐴−πœ†πΌ|=0

||−πœ†π›½2 (𝑦2− π‘₯𝑦)𝛾2 𝑧0 𝛽2 (𝑦2− π‘₯𝑦)−πœ†00 𝛾2 𝑧0−πœ†0 000−πœ†|| = 0

That is,

πœ†4− πœ†2 𝛽22 (𝑦2− π‘₯𝑦)2− πœ†2 𝛾24 (𝑧)2 =0

πœ†2 (πœ†2− 𝛽22 (𝑦2− π‘₯𝑦)2− 𝛾24 (𝑧)2 =0)

Ξ» = 0, 0; πœ†2= 𝛽22 (𝑦2− π‘₯𝑦)2+ 𝛾24 (𝑧)2

Therefore,

Ξ» = 0, 0; Ξ» = 12 √𝛽2(𝑦2− π‘₯𝑦)2+ 𝛾2 𝑧2

in above equations, the determinant value of A is zero and one of the Eigen values of A is also zero. the solution of the given partial differential equation (8) is of parabolic type.

Conclusion

Hence we have classified the stochastic epidemic models by considering infectives, carriers and removal rates. There are three types of removal recovery, immunization and death this can be elaborately studied by considering each of the removals separately.

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ISSN NO: 0972-1347

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INTERNATIONAL JOURNAL OF INFORMATION AND COMPUTING SCIENCE

Volume 9, Issue 1, January 2022

References:

[1] Bailey NT. The mathematical theory of infectious diseases, Second Edition, Griffin, London, 1975.

[2] Burghes DN, Borrie M.S. Modeling with differential equations, Ellis Horwood Ltd, 1981.

[3] Diekmann O, Heesterbeek JA. Mathematical epidemics of infectious diseases: Model

building, analysis and interpretation, John Wiley, New York, 2000.

[4] Hethcote HW. Three basic epidemiological models. In S.A. Levin, editor, Lect. Notes

in Biomathematics, Springer-Verlag Heidelberg. 1994; 100:119-144.

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