• No results found

Deterministic and Stochastic Schistosomiasis Models with General Incidence

N/A
N/A
Protected

Academic year: 2020

Share "Deterministic and Stochastic Schistosomiasis Models with General Incidence"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

Applied Mathematics,2013, 4, 1682-1693

Published Online December 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.412229

Deterministic and Stochastic Schistosomiasis Models with

General Incidence

Stanislas Ouaro, Ali Traoré

Laboratoire d’Analyse Mathématique des Equations (LAME), Unité de Formation et de Recherche en Sciences Exactes et Appliquées, Département de Mathématiques, Ouagadougou, Burkina Faso

Email: souaro@univ-ouaga.bf, ouaro@yahoo.fr, traoreali.univ@yahoo.fr Received September 11, 2013; revised October 11, 2013; accepted October 18, 2013

Copyright © 2013 Stanislas Ouaro, Ali Traoré. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, deterministic and stochastic models for schistosomiasis involving four sub-populations are developed. Conditions are given under which system exhibits thresholds behavior. The disease-free equilibrium is globally asymp- totically stable if and the unique endemic equilibrium is globally asymptotically stable when . The populations are computationally simulated under various conditions. Comparisons are made between the deterministic and the stochastic model.

01

R R01

Keywords: Computational Simulation; General Incidence; Reproduction Number; Schistosomiasis Model; Stochastic

Differential Equation

1. Introduction

Schistosomiasis (Bilharzia or snail fever) remains a se- rious public health problem in the tropics and subtropics. There are about 240 million people worldwide infected and more than 700 million people live in endemic areas [1]. It probably originated around the Great Lakes of Central Africa and possibly spread to other parts of Africa, the west Indies and South America [2]. Human schistosomiasis is a parasitic disease caused by five species of the genus Schistosoma or flatworms: Schisto- soma mansoni, Schistosoma intercalatum, Schistosoma japonicum, Schistosoma mekongi and Schistosoma hae- matobium. The three most widespread are Schistosoma japonicum, Schistosoma haematobium and Schistosoma mansoni. The large number of parasite eggs retained in the host tissues rather than shed in stools can cause the inflammatory reactions. Indeed, a study conducted in Sudan indicated that schistosomiasis depleted the blood hemoglobin levels to the extent that oxygen flowed to the muscles and the brain and impaired physical activity [3].

Several mathematical models for schistosomiasis dis- ease have been done ([4-15] and the references therein). Stochastic differential equation (SDE) model is a natural generalization of ordinary differential equation (ODE) model. SDE models became increasingly more popular

in mathematical biology ([4,16,17] and the references therein). In [4], a SDE model for transmission of schi- stosomiasis was analyzed. That model assumes that hu- man births and deaths are neglected. So, the compu- tational work is involved in a computation of B that one requires other schemes in which we solve an initial value problem.

In this paper, we derive an equivalent stochastic model for schistosomiasis infection which involves four sub- populations. This model allows the recruitments and the natural deaths of human. Additionally, the incidence forms are taken as general as possible. The paper is structured as follows: In Section 2, we present the schi- stosomiasis model while Section 3 provides the basic properties of the model. Section 4 provides an analysis of the full model. We construct a stochastic differential schistosomiasis model in Section 5. In Section 6, we derive an equivalent stochastic model. We describe a numerical method to solve the equivalent stochastic mo- del in Section 7. Computational simulations are per- formed in Section 8 and finally, in the last section, we end with a conclusion.

2. Mathematical Model

(2)

suppose that all newborns are susceptible and the infec- tion does not imply death or isolation of the different hosts.

Then, according to Figure 1, we obtain the following system of four differential equations

d

, ,

d d

, ,

d d

, ,

d d

, ,

d

s

h h s i s i

i

i s h i i

s

s s s i s

i

i s s i

H

b d H f S H H t

H

f S H d H H t

S

b d S g H S t

S

g H S d S t

 

 

 

 



(1)

where

 

s

H t is the size of the susceptible human population;

 

i

H t

 

S t

is the size of the infected human population;

s

 

S t

is the size of the susceptible snails;

i

b

is the size of the infected snails;

h is the recruitment rate of human hosts; s

b is the recruitment rate of snails;

h

osts

d is the per capita natural death rate of human h ;

s

d is the per capita natural death rate of snails;

 is the treatment rate of human hosts;

f is the infection function of susceptible human due to infected snails;

g is the infection function of susceptible snails due to in

nctions fected human.

We assume that the fu f and g satisfies the following assum tions. p

H1 f and g are nonnegative functions on the

no r t.

H For all ,

1 C

nnegative o

,Si

than

2 f

4

, ,

s i s

H H S

, 0

0,Hs

f S

i

0 and g

0,Ss

g H

i, 0

0.

Remark 2.1 f and g are two incidence functions

ich explain the contact between two species. Therefore,

is no infected in the human and snail populations, then the incidence functions are equal to zero. The incidence functions are also equal to zero when there is no sus- ceptible in the human and snail populations.

3. Model Basic Properties

are nneg ve. Note also that when there

no ati

In this section, we study the basic properties of the solu- tions of system (1).

Theorem 3.1 [18] Let : be a dif-

ferentiable function and let

L

n

X x be a vector field. Let

a and let

n:

 

x a

Gx L be such that

 

L x 0

  for all

 

 

1

:

n

xLax L xa . If X x

 

L x

 

0 for all 1

 

xLa then, the set G is positively in- variant.

Lemma 3.2 The system Equation (1) preserve the

positivity of solutions.

proof. We start by proving that

4

, , , ; 0

s i s i i

H H S S  H  is positively invariant. For that, let x

H H Ss, i, s,Si

and L x

 

 Hi.

 

L x is differentiable and for all

 

0

L x

 

0, 1, 0, 0

 

4

 

: x 0

1

0

xL  x L  . The vector field on the set

4

, , , ; 0

s i s i i

H H S S  H  is

 

, ,

.

h i s h s

i s

s s s

s i b f S H d H

f S H X x

b d S d S

   

 

 

  

 

 

 

(2)

We obtain X x

 

L x

 

 f S H

i, s

0. Then,

using Theorem 3.1, we conclude that

4

, , , ; 0

s i s i i

H H S S  H  is positively invariant. Similarly we prove that

H H S Ss, i, s, i

4;Hs 0

,

4

, , , ; 0

s i s i s

H H S S  S  and

4

, , , ; 0

s i s i i

H H S S  S  are positively invariant 

Lemma 3.3 Each nonnegative solution of model

system (1) is bounded. wh

[image:2.595.111.487.577.725.2]

f and g

(3)

Proof. Let H t

 

Hs

 

tH ti

 

 

s S ti . Then, adding t

second two Equations and

he first two equa- pectively, we get

 

 

S tS t

tions and the of (1) res

and .

h h s s

Hbd H Sbd S (3) According to [19], it follows that

 

 

 

 

0 e

and 0 e ,

h

s d t

h h

d t

s s

s s

b b

S t S

d d

 

 

 

(4)

where 0 and

h h

b b

H t H

d d

 

 

 

 

0 s

 

0 i

 

HHH

 

 

0 s 0 i 0

SSS . ,

 

Thus

 

 

lim

t and lim

h s

t

h s

b b

H t S t

d d

     (5)

Corollary 3.4 The set

 

   

 

 

 

 

 

4

, ,

s i s

K  H t H t S

 , ;

0 , 0

i

h s

s i s i

h s

t S t

b b

H t H t S t S t

d d

 

      

is invariant and attracting for system (1).

Theorem 3.5 For every nonzero, nonnegative initial

ist for all time

dard uments since the right-hand side of system (1) is lo

nd analyze the eventual equilibria. For that, let us denote value, solutions of model system (1) ex

0.

Proof. Local existence of solutions follows from stan-

arg

t

cally Lipschitz. Global existence follows from the a priori bounds. 

4. Reproduction Number and Equilibria

Analysis

In this section, we define the reproduction number a

1 i, s , 2 i, s .

i s

g g

1 i, s , 2 i, s ,

i s

f f

f S H f S H

S H

g H S g H S

H S

 

 

 

System (1) has a disease-free equilibrium (DFE) given by

 

 

 

0, 0, 0, 0

bh, 0,bs, 0 .

EDFEHs Ss  

h s

d d

 

 

Proposition 4.1 The reproduction number f r system

(1) is

o

 

0 0

0, 0,

1 1

0 .

s s

s h

f H g S

dd

R

Proof. We use the method in [20] to compute the

reproduction number Let

0

R .

1, 2

XX X , where X1 

H Ss, s

= i, i

is the healthy population and X2 H S , the infected

population.

1

, h i i

i s

i f S H d H H

H X

S

  

   

 

1 2

,

.

s i i s

i g H S d S

X

 

   

X X1, 2

X ,

 

Then,

  

 

1 1

1 2

1 2

0 0,

, 0

0, 0

0

and , 0 .

0

s s

h

s

f H

F X

g S

X

d

V X

d X

 

 

 

 

 

Hence,

1

1 1

1

0, 1

0 0

and .

1 0,

0 0

s

h s

s

s h

f H

d d

V FV

g S

d d

 

 

 

 

 

 

 

 

 

 

 

 

The basic reproduction number is defined as the dominant eigenvalue of matrix FV1 [20].

Therefore,

 

0 0

1 1

0

0, s 0, s

s h

d d

The basic reproduction num valuate

f H g S

 

R

ber e the average nu nfections generated by a single infected individual in a completely susceptible population.

Using Theorem 2 in [20], the following result is estab- lished.

mber of new i

Lemma 4.2 If R01 , then EDFE is locally

asymptotically stable.

Proof. Using the assumption H2, it follows that

0,

0

f2 Hs  and g2

0,Ss

0 for all Hs and Ss.

Then, the linearization of system (1) at EDFE gives the

following equation



 



 

1 0 1

s h

d d

f , 0 0, 0 0.

s s

H g S

h s

d d

     

 (6)

We can see that all solution  of Equation (6) have negative real part. Indeed, the Equation (6) has negative roots  dh,  ds and other roots are given by



0

 

1 0, 1 0,

s h s s

d d f H g S

    0

. (7)

Now, we suppose that there ex ts at least one root is 1

(4)

0

 

1 0 s 1 0

f H g 0

2

0

, ,SsR ds dh . Since R0< 1, we obtain

0

 

0

1 0,Hs g1 0,Ss ds dh 1 ds 1 dh .

f        

This show that the Equation (7) cannot have roots with positive real part. Hence, EDFE

ing to T is locally

asymptotically stable accord heorem 2 in [20]  et us a bal behavior of the DFE. The global stability of the DFE will be studied using the basic reproduction number . We first make the fo

Now, l nalyze the glo

0

R llowing additional assumption.

H3 For all

H H S Ss, i, s,

4,

0

1

, 0,

i s s i

i

f S Hf H S and

0

1

, 0,

i s s i

g H Sg S H .

Theorem 4.3 The disease-free equilibrium is globally

asymptotically stable in K whenever R0< 1.

Proof. The proof is based on using a comparison

theorem [21]. Note that the equations of the infected components in system (1) can be expressed as

,

d

i i i

F V S

S  

   

   

 

(8) d

d

i

H

H t

 

   

dt

 

where F and V , are defined in the proof of Pro- position 1.

Using the fact that all the eigenvalues of the matrix

FV

whenev eigenvalues

have negative real parts, it follows that the linearized differential inequality syste

er . Indeed, we can see that all

m (8) is stable

0< 1

R

 of the matrix FV satisfy Equation th

(7). By e same asoning as in the proof of Lemma 4.2 we deduce that, the eigenvalues of the matrix

re

FV

have negative real parts since R0< 1.

Thus,

H ti

   

,S ti

 

0, 0 as t  for the system (8). Consequently, by a standard comparison theorem [21

i

   

t ,S ti

 

0, 0 as t  and substituting HiSi0 into system (1) gives

0

], H

s s

HH and 0

s s

SS as t .

Thus,

 

     

0 0

, , , , 0, ,

s i s i s

H t H t S t S tH as t  for R0< 1. The

0

s S refore, EDFE is globally

lly stable if R0< 1. 

asymptotic

Lemma 4.4 If R01 then,

a

DFE

E is unstable and there exists a unique endemic equilibrium

s, i, ,

EH H S

Proof Let

s S . i

. E

H Hs, i,Ss,Si

. It follows that E is

a positive equilibrium if and only if

, 0,

h h s i s

b f S H H

H

   

, 0,

0,

i h i

s

f S H H

 

 (9) ,

, 0.

i

s i

s s i s

i s s i

d H

d

b d S g H S

H S d S

  

 

 

wo equations of sys- tem (9) respectively, we get

g

 

Adding the first two and the last t

0 and 0.

h h s h i s s s s i

bd Hd Hbd Sd S  Then,

and .

h h i s s i

s s

h s

b d H b d S

H S

d d

 

 

Let

, ,

, , .

h h i

i i i

h

s s i

h i i s i

s b d H h S H f S

d

b d S

d H g H d S

d

   

   

  

  

    

 

on is continuous and so sy (9) hold whenever

We can see that the functi stem

h

,

0

h S  . Therefore any zero of h in the set

i Hi

0, s 0, h

s h

b b

d d

  

  

  

 

 where Hi and

i

S are positives corresponds to an endemic e .

According to H2,

quilibrium

 

0, 0 0

h  and bs ,bh 0

h 

s h

d d

 

olution of equ

 

Then, in order to have a s ation h 0 .

 in

0, s 0, h

s h

dd

  

  , h 0. This lea to

b b

   

 must increase at ds

 

2

d

0, 0 0,

h

(10) dX

where X2 is defined in the proof of Proposition 4.1.

We can see that

0

0

0, s h , 0, s s .

1 1

2

d d

h

f Hd  g Sd X

It follows from inequality (10) that

0 1

1 0 1

0 1

0, 0

or 0, 0

and 0, 0.

s h

s s

s h

s s

f H d

g S d

f H d

g S d

  

 

  

 

0

0, 0

and

 

0

0,

f H g

Which implies that

0

1 1 0,

1

s s

s h

S

d d   , that is

To study the global behaviour of this endemic equilibrium, we second make this additional assumption.

H4 For all

01

R .

4

, , ,

s i s i

(5)

,

1

,

,

and 1 .

,

i s

s i

i

i s

s i s

s i

i i s

s f S H

H S

S f S H

H g H S

S H

H g H S

S

 

  

     

  

and

 

 

 

 

 

 

 

, ,

, ,

, ,

, .

s

hs i s s

s

i

hi i s i

i

s

ss i s s

s i

si i s i

i H t V t g H S H h

H

H t V t g H S H h

H

S t V t f S H S h

S

S t V t f S H S h

S

 

  

 

 

 

 

 

  

 

  

(11)

*

(14)

Theorem 4.5 When  0 and R01, the endemic

equilibrium E is globally asymptotically stable in K. when R 1,

We can see that h: *   

  has the strict global minimun h

 

1 0. Thus,

hs with equality if and only

if 0,

VVhi 0,Vss0,Vsi 0

, ,

s s i i s s

HH HH SS and i i . We will

study the behaviour of the Lyapunov function

Proof. If w em

ere exists a

e consider the syst (1) th unique endemic equilibrium

0

E. We now ic establish the global asymptotic stability of this endem equilibrium. Evaluating both sides of (1) at E with

0

 

SS

gives

 

.

hs hi ss si

V tVVVV (15)

, ,

, ,

, ,

, .

h h s i s

h i i s

s s s i s

s i i s

b d H f S H

d H f S H

b d S g H S

d S g H S

  

 

 

 

 

 

We can see that V t

 

0 with equality if and only if

 

 

 

 

1, 1, 1 and 1.

s i s i

s i s i

H t H t S t S t

HHSS

(12)

(1

The derivatives of Vhs,V Vhi, ss and si will be

calculated separately and then combined to get the V

Let

 

1 ln

h x   x x 3) desired quantity d

d V

t .

,

1 s

,

.

i s h h s i s

s H

S b d H f S H

t H

 

  

 

 

Usi Equation of (12) to replace gives

d d

, 1

d d

hs s s

i s

s

V H H

g H S g H

t H

 

 

ng the first bh



2

 



2

 

d

, 1 , ,

d

,

, , , 1 1

,

, ,

, , , 1 .

, ,

hs s

i s h s s i s i s

s

s s i s s

h i s i s i s

s i s s

s s s i s i s s

h i s i s

s i s i s s

V H

g H S d H H f S H f S H

t H

H H f S H H

d g H S g H S f S H

H f S H H

H H H f S H f S H H

d g H S g H S f S H

H H f S H f S H H

 

  

 

 

 

    



 

 

 

     

 

 

By adding and substracting the quantity

i s

s

,

1 ln ,

i s s

s

i s

f S H H

H f S H

 , we obtain



 



 

2

2

d

, 1 ln

, , , ,

1 ln 1 ln

, , , ,

,

, , ,

s s s s

i s

s s

i s i s i s s i s s

s s

i s i s i s i s

s s s i

h i s i s i s

s s

H H

V H H

g S H

H H

f S H f S H f S H H f S H H

H H

f S H f S H f S H f S H

H H H f S H

d g H S g H S f S H h h

H H



  

 

 

   

   

      

   

   

  

   

 

, ,

d

hs

h i s i s

s

d g H S H S f

t   H

,

.

, ,

s i s s

s

i s i s

f S H H h

H f S H f S H

    

    

   



(16)

(6)

d d

hi V

t . Next, we calculate

d d

, 1 ( , ) 1 ,

d

,

, 1 , .

,

hi i i i

i s i s i s h i

i i

i s

i i

i s i s h i

i i s i

V H H H

g H S g H S f S H d H

t H dt H

f S H

H H

g H S f S H d H

H f S H H

   

   

 

 





h i d H

Using the second Equation of (12) to replace gives

 

,

 

,

d

, , 1 , , 1

d , ,

i s i s i s

hi i i i i

i s i s i s i s

i i s i i i s i i s

f S H f S H f S H

V H H H H

f S H g H S f S H g H S

t H f S H H H f S H H f S H

  

 

     

 



, . ,

   

By adding and substracting the quantity

,

1 ln ,

i s i

i

i s

f S H H H f S H

 , we obtain

 

 

, ,

d

, , 1 ln 1 ln

d , ,

, ,

n , ,

, ,

i s i s

hi i i i i

i s i s

i i i s i i s i

i s i i s i i s

i s i s

i i

i s i s i s i s

f S H f S H

V H H H H

f S H g H S

t H H f S H H f S H H

S H H f S H H f S H

f S H g H S h h h

H H

S H f S H f S H

 

 

       

 

    

   

    

  

    

.

 

, ,

1 l

, ,

i s

f S H f

f S H f

 

   

(17)

Now, we calculate d d

ss V

t .

d d

, 1 , 1 ,

d d

ss s s s

i s i s s s s i s

s s

V S S S

f S H f S H b d S g H S

t S t S

   

 

    .

Using the third Equation of (12) to replace bs gives

 



 



 

2

2

d

, 1 , ,

d

,

, , , 1 1

, ,

, , , 1 .

, ,

ss s

i s s s s i s i s

s

s s i s s

i s i

s

s s s i s i s s

s i s i s i s

s s i s i s s

V S

f S H d S S g H S g H S

t S

S S g H S S

d f S H f S H g H S

S S

S S S g H S g H S S

d f S H f S H g H S

S S g H S g H S S

 

      

 

 

 

     

  

 

 

     

 

 

By adding and substracting the quantity

,

s i s s

s i s

S g H

,

1 ln ,

i s s

s

i s

g H S S

S g H S

 , we obtain



 



 

2

d

, 1 ln

, , , ,

1 ln 1 ln

, , , ,

,

, , ,

s s s s

s i s i s i s

s s s

i s i s i s s i s s

s s

i s i s i s i s

i s

s i s i s i s

s s

S S

V S S

H f S H g H S

S S S

g H S g H S g H S S g H S S

S S

g H S g H S g H S g H S

g H S S

d f S H f S H g H S h h

S S



  

 

   

   

      

   

   

 

   

 

, ,

d

ss

d f S t  

2

s s

SS

,

.

, ,

s i s s

s

i s i s

g H S S h

S g H S g H S

    

    

   



(18)

(7)

After that, we evaluate d d

si V

t .

d d

, 1

d d

, 1 ,

,

, 1 , .

si i i

i s

i

i

i s i s s i

i

i s

i i

i s i s s i

,

i

V S S

f S H

t S t

S

f S H g H S d S S

g H S

S S

f S H g H S d S

i

S

i s

S

 

 

 

    

 

 

 

g H S

 

Using the last Equation of (12) to replace



 

 

, d

, , 1

d ,

, ,

, ,

1 .

, ,

i s

si i

i s i s

i i s i

i s i s

i s i s

i i

i i s i i s

g H S

V S

g H S f S H

t S g H S

f S H g H S

g H S g H S

S S

S g H S S g H S

 

 





 

 

   

 

 

i

S S

By adding and substracting the quantity

,

1 ln ,

i

g H

g H

s i d S gives

s i

i

i s

S S

S

S , we obtain

 

 

d

, , 1 ln

d

, , ,

1 ln 1 ln

, , ,

, ,

, ,

, ,

si i i

i s i s

i i

i s i i s i i s i s

i i

i s i s i s i s

i s i s

i i

i s i s

i i s i i s

V S S

f S H g H S

t S S

g H S S g H S S g H S g H S

S S

g H S g H S g H S g H S

g H S g H S

S S

f S H g H S h h h

S g H S S g H S

 

  

 

  

  

      

   

   

  

 

   

 

  .

 

 

 

 

,

,    

6)-(1 , we obtain

(19)

Combining equations (1 9)





 

2 2

, d

, , , ,

d ,

, , ,

, , ,

s s

s s i i s s

h i s s i s i s i s

s s i i s

i s i s i s

i s i s s i

i i s s i s i s s i s i

S S

H H S f S H H

V

d g H S d f S H g H S f S H h h

t H S S f S H

g H S f S H g H S

H S H S H S

h h h h h h

H g H S S f S H H S H g H S S

  

 

      

   

 

s

H

  

   

  

   

      .

 

 

 

 

(20)

Since the function is monotone on each side of 1 and is minimized a es 

t 1, H4 impli

h

,

and

,

.

, ,

i s s i i s s

s i s

i s i s

f S H H S g H S S H

h h h

H S S H

f S H g H S

   

   

   

   

i i

h

Since h0, then

 

t 2.

d 0, d

V

t  (21) for all

H H S Ss, i, s, i

K with equality only for

, ,

s s i i s s

HH HH SS and SiSi.

rium

Hence, the endemic equilib E is the only positively invariant set of the system (0.1) contained in

4

, , , ; , , ,

s i s i s s i i s s i i

H H S S  HH HH SS SS

. Then, it follows that E is globally asymptotically stable on [22]. 

5. Stochastic Differential Equation Model

To derive a stochastic model, we apply a similar procedure to that described in Allen [23]. Here, we

neglect the possibility of multiple events of order

K

The possible changes in the populations over a short time

t

 , concern individual births, deaths and transform se changes are produced in Table 1, togeth

corresponding probability. Let’s denote th

ation. er with is change The

their

by  

Hs,Hi,Ss,Si

T.

Neglecting terms of the order , the m system (0.1) is given by

t

 

2 t

 ean of

 

10 1

, ,

. ,

,

h h s i s i

i s h i i

i i

i s s s i s

i s s i

b d H f S H H f S H d H H

E P t

b d S g H S g H S d S

 

    

 

 

 

 

 

 

  

(8)
[image:8.595.56.287.100.289.2]

Table 1. Possible changes in the population.

Change Probability

 T

11,0,0,0

P1 b th

 T

2 1,0,0,0

P2d Hh st

 T

3 1,1,0,0

P3f S Hi, st

 T

0, 1,0,0  

4 4 h i

 T

1, 1,0,0  

PH t

Pd H t

5 5 i

 T

6 0,0,1,0

P6 b ts

 T

7 0,0, 1,0

P7d Ss st

 T

8 0,0, 1,1

P8g H Si, st

0,0, 

 

T T 9 0, 1

P9d S ts i

 

10 0,0,0,0

 9

10 1 i1i

P  

P

, the covariance ma by

23)

Further trix of system (1) is given

 

11 12

1 33 34

0 0

0

i i i i B B B t B B         10 12 22 T T 34 44 0 0 , 0 0 0 B B

E P t

B B          

   ( where

11 22 12 33 , , , , , , , , ,

h h s i s i

i s h i i

i s i

s s s i s

B b d H f S H H

B f S H d H H

B f S H H

B b d S g S H

B g S H

                 

44

andBg S Hi, sd Ss

Note that it has been

34

.

i s

i

proved in [23] that the changes

 are normally distributed. Then,

t  t

 

t  

t  t

 

t   t B t ,

Y YY Y  

where for

Furt converges strongly

to the solution of th m

 

0,1

i N

 

hermore, as

1, 2, , 4

i  . 0

 t , Y

 

t

e stochastic syste

 

 

 

 

dY t dW t

,

d  Y tB Y t d (24)

where and is the four-

dimensi 3].

The computational work of Equation (24) involves the calculation of the qu tity

t t

 

T

, , ,

s i s i

tH H S S

Y

onal Wiener process [2

 

t

W

an B

Y

 

t

ften cumb

at each time step [24,25]. This procedure is o ersome.

In the next section, we derive an equivalent ochastic model which seem to e easier to implement.

6. Equivalent Stochastic Differential

Equation Model

In this section, we develop a stochastic model to e the changes occured on each vector individually. Unl we say otherwise, we adopt the vectors defined in the previous section but here the Poisson processes

st b

xamine ess

 

P

ean ti are used to establish the different probabilities as m mes before occurence. Then, we have

1 2 3 4

3 4 5

6 7 8

8 9 , , , , s i s i

H u u u u

H u u u

S u u u

S u u

                    (25) where

1 2 3 4

5 6 7

8 9

, ,

, , ,

,

, , .

h h s

i s i

h i s s s

i s s i

u P b t u P d H t

u P f S H t u P H t

u P d H t u P b t u P d S t

u P g H S t u P d S t

        

, ,          

We now normalize the Poisson process to get

1 2 3 4 3 5 4 6 7 8 8 , , , , , , , , , , ,

s h h h s h s

i s i s i i

i i s i s

h i h i i i

s s s s s s s

i s i s

i i s i s

s i

t b t d H t d H t H b

f S H t f S H t H t H t

H f S H t f S H t

d H t d H t H t H t

S b t b t d S t d S t

S t g H S t

S g H S t g H S t

d S               g H                                                   9, s i t d S t

               (26) where    

 

0,1

i N

  for i1, 2,, 9. Then, as  t 0, chastic

the system erge llowing Itô sto

differential equation [24]

(26) conv s to the fo

1 2

d , d d

, d

s h h s i s i h

h s i s

H b d H f S H H t b W

d H dW f S H

       

3 4 3 6 7 8 8 9 d ,

d , d , d

,

d , d ,

d , d , d d .

i

i i s h i i i s

h

s s i s

i i s s i i s s i

W H W

H f S H d H H t f S H W

H W

d S W g H S W

S g H S d S t g H S W d S W

                  llows. 4 5 d d

d , d d

i i

s s s s i s s

H W

S b d S g H S t b W

         d   (27)

Figure

Figure 1. Transfer diagram.
Table 1. Possible changes in the population.
Table 2. Mean and standard deviation for the ODE epi- demic model (1) and the SDE epidemic model (28) at t = 300  years where R = 0.970
Figure 2. Deterministic and Equivalent stochastic models for R = 0.97. 0
+2

References

Related documents

atmosphere: Terra-forma and Atom-sphere Inc New, designed life forms to solve our energy and climate crisis... (example of

Face shield and safety glasses Use equipment for eye protection tested and approved under appropriate government standards such as NIOSH (US) or EN 166(EU).

Our observation that calcium sensitivity of ten- sion in failing preparations decreases to a greater extent after in vitro treatment with the catalytic subunit of protein kinase

The measured electron doses can be quantitatively explained by the inelastic scattering of the incident high- energy electrons by the MgH 2 plasmon.. The obtained insights are

cytomegalovirus infections following nonmyeloablative compared with myeloablative allogeneic stem cell transplantation, a matched control study. Immunologic recovery

Use Image print to print your images using the USB data cable supplied with your device, Bluetooth connectivity, or a compatible memory card (if available).. You can only

of professional staff Contact person Phone Email address Website Fund name Previous fund(s) name Date of Inception Duration Total asset value (m EUR) Dedicated asset value to CEE

Subject: Approval of the Project Scope and Budget, Appropriation for Construction Services, and Approval for Construction for the Student Recreation Center Project, Prairie