Deterministic and Stochastic Schistosomiasis Models with General Incidence

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 asymptotically 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 0 1  R


Introduction
Schistosomiasis (Bilharzia or snail fever) remains a serious 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: Schistosoma mansoni, Schistosoma intercalatum, Schistosoma japonicum, Schistosoma mekongi and Schistosoma haematobium.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 disease have been done ( [4][5][6][7][8][9][10][11][12][13][14][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 schistosomiasis was analyzed.That model assumes that human births and deaths are neglected.So, the computational 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 subpopulations.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 schistosomiasis 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 model in Section 7. Computational simulations are performed in Section 8 and finally, in the last section, we end with a conclusion.

Mathematical Model
Here we consider a relatively isolated community where there are no immigration or emigration.Additionaly, we suppose that all newborns are susceptible and the infection does not imply death or isolation of the different hosts.
Then, according to Figure 1, we obtain the following system of four differential equations is the size of the susceptible human population; is the size of the infected human population; , , 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 susceptible in the human and snail populations.

Model Basic Properties
are nneg ve.Note also that when there

 no ati
In this section, we study the basic properties of the solutions of system (1).Theorem 3.1 [18] Let : be a differentiable function and let then, the set is positively invariant.
The vector field on the set We obtain According to [19], it follows that and 0 e , where 0 and Corollary 3. 4 The set

Reproduction Number and Equilibria Analysis
In this section, we define the reproduction number a , , , .
Proof.We use the method in [20] to compute the reproduction number Let , where is the health and y population 2 X H S , the infected population.
The basic reproduction number is defined as the dominant eigenvalue of matrix 1 FV  [20].Therefore, The basic reproduction num valuate the average nu nfections generated by a single infected individual in a completely susceptible population.
Using Theorem 2 in [20], the following result is established.
is locally asymptotically stable.
Proof.Using the assumption H2, it follows that H and s S .Then, the linearization of system (1) at DFE E gives the following equation We can see that all solution  of Equation ( 6) have negative real part.Indeed, the Equation (6) has negative roots Now, we suppose that there ex ts at least one root 1 is  of the Equation ( 7) which have positive real part.
According to Proposition 4.1, it follows that This show that the Equation ( 7 , , , 3 The disease-free equilibrium is globally asymptotically stable in K whenever 0 < 1 R .Proof.The proof is based on using a comparison theorem [21].Note that the equations of the infected co system (1) can be e ssed as mponents in xpre where F and V , are defined in the proof of Proposition 1.
Using the fact that all the eigenvalues of the matrix F V  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 satisfy Equation th (7).By e same asoning as in the proof of Lemma 4.2 we deduce that, the eigenvalues of the matrix H t S t  as t   for the system (8).Consequently, by a standard comparison theorem [21 , is unstable and there exists a uni m q endemic equilibri ue u   , , , wo equations of system (9) respectively, we get Adding the first two and the last t 0 and 0.
and .
on is continuous and so sy (9) hold whenever


We can see that the functi stem where i H and i S are positives corresponds to an endemic e .
According to H2, X is defined in the proof of Proposition 4.1.We can see that , that is

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

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
(1


The derivatives of and , , hs hi ss V V V si will be calculated separately and then combined to get the , .

Usi
Equation of (12) to replace gives , By adding and substracting the quantity , d H Using the second Equation of (12) to replace gives , By adding and substracting the quantity Using the third Equation of (12) to replace s b es giv

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 g H S g H S S S d f S H f S H g H S S S S g H S g H S
By adding and substracting the quantity , we obtain


After that, we evaluate d d .
Using the last Equation of ( 12) to replace By adding and substracting the quantity , we obtain

g H S g H S g H S g H S S 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
Combining equations (1 9) Since the function is monotone on each side of 1 and is minimized a es , for all   , , , with equality only for , , rium Hence, the endemic equilib E is the only po ntained in sitively invariant set of the system (0.1) co . Then, it follows that E is globally asymptotically stable on [22].

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 Neglecting terms of the order , the m system (0.1) is given by , the covariance ma by Note that it has been
i s i proved in [23] that the changes  are normally distributed.Then, where for Furt converges strongly to the solution of th m where and is the fourdimensi 3].The computational work of Equation (24) involves the calculation of the qu tity 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.

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 where We now normalize the Poisson process to get where for 1, 2, , 9 i   .Then, as 0 t   , chastic the system erge llowing Itô sto differential equation [24] (26) conv s to the fo and  are the same as in system (24), W is the nine-dimensional Wiener process and G is defined by where

Computational Results
In this section, computational results are given for the stochastic system (28).We use the Euler-Maruy method with 1000 sample to solve the SDE model Le yama numerical method for system (28) is given by: ama (28).t h be a specified time step.The Euler-Maru , , ,

Computational Simulations
In this section, computational simulations are given for the models described in the previous sections.Here, two different cases of computational simulations were studied: in the first case while in the second case .In the co , the functions .The initial values , was taken as 300 years.These figures are produced by Matlab.  2 and Table 3 below that supports the comparison

Conclusion
An ordinary differential equation model and a corresponding equivalent stochastic differential equation model for schistosomiasis were studied.Four subpopulations were modelled: susceptible human, infected human, susceptible snails and infected snails.A threshold number was exhibited.Computational simulations were presented.The behavior of the deterministic and equivalent stochastic models are approximately the same.For the deterministic and equivalent stochastic models ) cannot have roots with positive real part.Hence, DFE E 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 .

Figure 2 Figure 3
Figure 2 illustrates the deterministic model (1) and t equivalent stochastic model (28) when We t, in the Figure 2 the trajectory of inistic and stochastic graphs are approxim y the same behaviour.Indeed, the parasite extinction i effective if

Figure 2 Figure 3 .
Figure 3. Deterministic and Equivalent stochastic models for

Table 1 . Possible changes in the population. Change Probability
the maximum time is reached.

Table 2 . Mean and standard deviation for the ODE epi- demic model (1) and the SDE epidemic model (28) at t = 300 years where 0 R = 0.97 .
i S stic models for .