The use of generation stochastic models to study an epidemic disease

Abstract

Stochastic models have an important role in modeling and analyzing epidemic diseases for small size population. In this article, we study the generation of stochastic models for epidemic disease susceptible-infective-susceptible model. Here, we use the separation variable method to solve partial differential equation and the new developed modified probability generating function (PGF) of a random process to include a random catastrophe to solve the ordinary differential equations generated from partial differential equation. The results show that the probability function is too sensitive to [mu], [beta] and [gamma] parameters.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

The use of generation stochastic models to study an epidemic disease

S Seddighi Chaharborj1,2,3*, I Fudziah1, MR Abu Bakar1, R Seddighi Chaharborj4, ZA Majid1,5 and AGB Ahmad6

*Correspondence: mailto:[email protected]

Web End [email protected]

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM, Selangor, Malaysia

2Plasma Physics and Nuclear Fusion Research School, Nuclear Science and Technology Research Institute (NSTRI), P.O. Box 14395-836, Tehran, IranFull list of author information is available at the end of the article

Abstract

Stochastic models have an important role in modeling and analyzing epidemic diseases for small size population. In this article, we study the generation of stochastic models for epidemic disease susceptible-infective-susceptible model. Here, we use the separation variable method to solve partial dierential equation and the new developed modied probability generating function (PGF) of a random process to include a random catastrophe to solve the ordinary dierential equations generated from partial dierential equation. The results show that the probability function is too sensitive to , and parameters.

Keywords: epidemic diseases; susceptible-infective-susceptible; deterministic model; stochastic model; probability function

1 Deterministic susceptible-infective-susceptible model

Figure shows a deterministic susceptible-infective-susceptible model for an epidemic disease. In this gure, S is the susceptible population, I is the infective population, > is the natural death rate, > is the removal rate which is a constant. Note that S, I because they represent the number of people. The infection rate, , depends

on the number of partners per individual per unit time (r > ) and the transmission probability per partner ( > ). In this system, the rst susceptible population in class S is going to be infected, then infected population in class I is going to be susceptible again. The following system of ODEs describes this susceptible-infective-susceptible model []

dS(t)dt = I(t) S(t),

dI(t)dt = S(t) .

()

Figure illustrates the system (). This system is nonlinear due to the form of = I.

2 Generation stochastic susceptible-infective-susceptible model

In this section, we present the state of the generation stochastic [] susceptible-infective-susceptible model. The stochastic susceptible-infective-susceptible model is similar to the deterministic susceptible-infective-susceptible model, for the deterministic model we can nd an exact function but for the stochastic model, we cannot obtain

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Figure 1 A schematic of system ().

Figure 2 Stochastic susceptible-infective-susceptible model state diagram.

Table 1 Transition diagram for the stochastic susceptible-infective-susceptible model

Transitions Rates

i i 1 i(m + (n i)) +

i 0

for i = n, n 1, n 2, . . . , 2, 1

In this gure, > 0 is the natural death rate, > 0 is the removal rate which is a constant and > 0 is the transmission probability per partner.

an exact function. Figure shows the state diagram for the stochastic [] susceptible-infective-susceptible model [, ].

At t, if m is infective and n is susceptible, namely S(t) + I(t) = n + m, then Pi(t) = P[S(t) = i|S() = n] is the probability function in the time t and stage i. Here, our goal is to

determine Pi(t). Table shows the transition diagram for this model. To determine Pi(t)s, we should create the Kolmogorov equations. From Figure , we have

P[staying at state i] =

i m + (n i) t + t t

and

t + t t. ()

Now, to produce the forward Kolmogorov equations, we have

Pi(t + t) = P

contact during t|S(t) = i P i(t)

+ P

contact during t|S(t) = i + P i+(t)

= P(staying at state i)Pi(t) + P(moving from state i + to i)Pi+(t)

=

i m + (n i) t + t t P i(t)

+

t + t t

()

P[moving from state i to i ] = i

m + (n i)

(i + )

m + (n i )

Pi+(t).

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So,

Pi(t + t) Pi(t) =

i(m + n i) t + t t

Pi(t)

+

(i + )(m + n i ) t + t

t

Pi+(t), ()

then

Pi(t + t) Pi(t)

t = i(m

+ n i) +

Pi(t)

Pi+(t). ()

Having limited from both sides of Eq. (), when t , we have

P i(t) = lim

t

Pi(t + t) Pi(t)

t = i(m

+

(i + )(m + n i ) +

+ n i) +

Pi(t)

Pi+(t). ()

Therefore, the forward Kolmogorov equations for this model will be as follows:

P i(t) =

i(m + n i) +

+

(i + )(m + n i ) +

Pi(t)

Pi+(t). ()

The probabilities function Pi(t) is found from Eq. (). Also, from probability generating functions (PGFs) and partial dierential functions equations (PDEs), the probabilities function Pi(t) can be obtained. Probability generating functions can be written as

y(x, t) =

n

i=

+

(i + )(m + n i ) +

Pi(t)xi = P(t) + P(t)x + P(t)x + + Pn(t)xn. ()

Now, the partial derivative of y(x, t) with respect to t will be yt = n

i= P i(t)xi, so one can

write the partial derivative as follows:

y

t =

n

i=

i(m + n i) +

Pi(t)

+

(i + )(m + n i ) +

Pi+(t)

xi. ()

Having simplied Eq. (), we can write

y(x, t)

t = (

)

x + (

)y(x, t) + (m + n )( x)

y(x, t)

x

+ x(x )

y(x, t)

x . ()

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The separation variable method is employed to solve Eq. (). If y(x, t) = X(x)T(t), then we have

T (t)

T(t) = (

) + ( )

x

X(x) +

(m + n )( x)X (x) X(x)

+ x(x )X (x)

X(x) . ()

Two sides of Eq. () are equal, so

T (t)

T(t) = c and (

) + ( )

x

X(x) +

(m + n )( x)X (x) X(x)

+ x(x )X (x)

X(x) = c. ()

To include a random catastrophe presented by Gani and Swift in [], we develop the modied probability generating function (PGF) of a random process to solve Eq. () as follows:

G(x, t) =

n

j=

, ()

with , here y(x, t) is the answer of Eq. () when = , = . So, we can put

= , = with m = in Eq. ()

x(x )X (x) n(x )X (x) + (c/)X(x) = . ()

This equation was solved by Bailey in []. After having solved Eq. () by Maple, we have

X(x) = C F /

+ n + n c + ( + n)

,

/

Gj(x, t) =

n

j=

e()ty(x, t) +

t

(

)e()vy(x, v) dv

+ n + n c + ( + n)

; n; x , ()

where F[, ; ; ] is a hypergeometric function.

Notation The standard hypergeometric function F[a, b; c; x] is as follows:

F[a, b; c; x] =

xii! , ()

where (a)i = a(a + )(a + )(a + ) (a + i ) with (a) = is the Pochhammer symbol.

The derivatives of F[a, b; c; x] are given by

dF[a, b, c, x]

dx =

abc F[a + , b + , c + , x], dF[a, b, c, x]

dx =

a(a + )b(b + ) c(c + )

(a)i(b)i

(c)i

F[a + , b + , c + , x],

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dF[a, b, c, x]

dx =

a(a + )(a + )b(b + )(b + ) c(c + )(c + )

F[a + , b + , c + , x],

...

Also, from equation T

(t)

T(t) = c, we have T(t) = kect. So,

y(x, t) = ect

F[/

+n+nc+(+n) , /

+n+nc+(+n) ; n; ]

F

/

+ n + n c + ( + n) ,

/

+ n + n c + ( + n) ; n; x . ()

In Eq. () we can take c = j(N + m j); N = n + , here is so small parameter. Then from Eq. () and Eq. (), one can write

G(x, t) =

n

j=

e()t ej(N+mj)t

F[j, j N , N, ] ()

F[j, j N , N, x] +

t

(

)e()v ej(m+Nj)v ()

F[j, j N , N, ]

F[j, j N , N, v] dv

. ()

Thus,

G(x, t) =

n

j=

j F[j, j N , N, x] (e[()+j(N+mj)]t

+

t

(

)e[()+j(N+mj))]v dv

n

j=

= j F[j, j N , N, x] e[()+j(m+Nj)]t

e(++jNj+j)t

+ + jN j + j

, ()

where

j = ()jn!(N j + )N!

(N + j ) j!(n j)!(N n)!()n+ (n N + j).

Now, to nd P(t), P(t), P(t), . . . , Pk(t), one can calculate G(x, t) as follows:

G(x, t) =

Pk(t)xk = P(t) + P(t)x + P(t)x + . ()

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So,

P(t) = G(, t)

n

j=

= j e[()+j(m+Nj))]t

e(++jNj+j)t + + jN j + j

. ()

Hence, P(t) and P(t) are obtained as follows:

P(t) = dG(x, t)

dx

x= =

n

j=

!

(j)(j N )(N)

j j(t),

P(t) = dG(x, t)

dx

x= =

n

j=

!

(j)(j + )(j N )(j N + ) (N)(N + )

j j(t),

...

Therefore,

Pk(t) = dkG(x, t)

dxk

x= =

n

j=

k!

(j)k(j N )k

(N)k

j j(t), ()

with

j(t) =

ejt(nj)t(m+Nj)t e(++jNj+j)t

+ + jN j + j

. ()

3 Numerical results

Some numerical examples illustrate the behavior of the probability function PK(t, , , )

with the following parameters: t = ; = .; = .; = ..

Figure (a) and (b) show the behavior of the probability functions, P(t, ) and P(t, ) with = . and = . when < t < and < < .

Figure (a) shows that with an increase in t and , the probability function P(t, ) in

creases fast, but in Figure (b) P(t, ) decreases fast.

Figure (a) and (b) show the behavior of the probability functions, P(t, ) and P(t, ) with = . and = . when < t < and < < . Figure (a) shows that as t increases, the probability function P(t, ) increases, but with an increase in , the probability function P(t, ) decreases. Figure (b) shows that as t increases, the probability function P(t, ) decreases, but with an increase in , the probability function P(t, ) increases.

Figure (a) and (b) display the probability functions, P(t, ) and P(t, ) with = . and = .. From Figure (a) when < t < and < < , the probability function P(t, )

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Figure 3 Probability function Pk(t, ) with = 0.3 and = 0.1, (a): k = 0 and (b): k = 8.

Figure 4 Probability function Pk(t, ) with = 0.3 and = 0.1, (a): k = 0 and (b): k = 8.

Figure 5 Probability function Pk(t, ) with = 0.3 and = 0.3, (a): k = 0 and (b): k = 8.

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Figure 6 Probability function Pk(, ) with = 0.1 and t = 1, (a): k = 0 and (b): k = 8.

Figure 7 Probability function Pk(, ) with = 0.3 and t = 1, (a): k = 0 and (b): k = 8.

is nearly zero, but for < t < as t increases, P(t, ) slowly increases, and as increases, P(t, ) sharply increases. In Figure (b), P(t, ) is almost constant for < t < , but with a decrease in from to , P(t, ) increases; also, Figure (b) shows the highest value of P(t, ) when < t < and < < .

Figure (a) and (b) show the probability functions, P(, ) and P(, ) with = . and t = . Figure (a) depicts that for < < , as increases from to , the probability function P(, ) increases. Figure (b) shows that for < < , with an increase in and , the probability function P(, ) increases.

Figure (a) and (b) illustrate the probability functions, P(, ) and P(, ) with = . and t = . Figure (a) shows that the probability function P(, ) decreases with an increase in , but inversely it increases with an increase in . In Figure (a) we observe a separation which means that in the probability function PK(, ) we have = . Figure (b)

depicts that the probability function P(, ) increases when increases, although it decreases with an increase in .

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4 Conclusions

We have presented the generation of a stochastic model for the susceptible-infective-susceptible model. The separation variable method has been applied to solve a partial dierential equation of this generation. So, two ordinary dierential equations have been achieved which relate to the parameter x. To solve this equation, we used the developed modied probability generating function (PGF) of a random process to consider a random catastrophe. Numerical results showed the behavior of the probability function Pk(t, , , ) when < t < and < , , < .

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors carried out the proof and conceived of the study. All authors read and approved the nal manuscript.

Author details

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM, Selangor, Malaysia. 2Plasma Physics and Nuclear Fusion Research School, Nuclear Science and Technology Research Institute (NSTRI), P.O. Box 14395-836, Tehran, Iran. 3Department of Mathematics, Science and Research Branch, Islamic, Azad University, Bushehr Branch, Bushehr, Iran. 4Department of Applied Mathematics and Computer Science, Eastern Mediterranean University, Famagusta, Northern Cyprus. 5Institute of Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang Selangor Darul Ehsan, Selangor, Malaysia. 6School of Mathematical, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia.

Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

Received: 28 October 2012 Accepted: 18 December 2012 Published: 9 January 2013

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doi:10.1186/1687-1847-2013-7Cite this article as: Seddighi Chaharborj et al.: The use of generation stochastic models to study an epidemic disease. Advances in Dierence Equations 2013 2013:7.