Periodic Solutions for Stochastic Differential Equations Driven by General Counting Processes: Application to Malaria

In this paper, a class of periodic stochastic differential equations driven by general counting processes (SDEsGp) is studied. First, an existence-uniqueness result for the solution of general SDEsGp based on Poisson processes with τperiodic stochastic intensity of time t has been given, for some τ > 0. Then, using the properties of periodic Markov processes, sufficient conditions for the existence and uniqueness of a periodic solution of the considered equations are obtained. We will then apply the obtained results to the propagation of malaria in a periodic environment.


Introduction
In stochastic modeling, a dynamical system is the set constituted by a system and stochastic evolution equations describing the random evolution of its trajectories. There are two types of systems in the literature: discrete systems and continuous systems. Discrete systems are generally described by jump processes which give a good representation of the studied phenomenon. Jump processes are stochastic counting processes J t t≥0 whose paths are constant except for jumps of size +1 or −1. The simplest exemple is, of course the Poisson process P t t≥0 which attracted much attention due to its wide applications in the fields where counting problems arise: Queues, population dynamic, chimical reaction, pharmacology, biology, medical sciences, celestial mechanics, statistical physics, and so on . Note also that the distribution of this process is determined by specifying the intensity parameter λ which gives where F t is the history for the process up to time t. Usually, more general counting processes are determined by specifying a stochastic intensity function λ t, P which as in Eq.(1), gives P (P(t + t) − P(t) > 0|F t ) = λ (t, P) t + o( t). (2) A number of authors have used this more generalized counting process to study stochastic differential equations of type; where X(t) = X i (t) , i = 1, 2, ..., d is a d-vector composed of jump processes and gives the state of the process at any time t and the integer k the total number of Poisson processes in the model.
For all and such that , the stochastic processes P (t) t≥0 and P (t) t≥0 are independent standard Poisson processes. The theoretical development of stochastic process based on Poisson processes has be seen in recent years and several results have been proven: Existence and uniqueness of the solution of Eq.(3) are proved under more general conditions (Anderson, D.F and Kurtz, T.G, 2015), (Guy,R and all, 2014), (Guy,R and all, 2014). This result is discussed in more detail in Kurtz (Kurtz, T.G, 1980). In the above equations, the general counting process counts the jumps of type and h gives the direction.
Taking expectation of the previous stochastic Eq.
(3) exhibe a great difficulty, which are caused by the jumps of Poisson processes random mesures when intensity functions are stochastic processes. As far as we know, there are no results on the existence and uniqueness of a periodic solutions for Eq.(3).
Let β K (X) denote the explosion time of the jump process X(t) t≥0 . In the present work, we are interested in the existence result of the solution of Eq.(3). Now we introduce the assumptions (H 1 ) -(H 2 ) which ensure the problem is well posed in our discussion.
Assumption 1 (H 1 )For all X ∈ J ([0, +∞[) d and each , we make those assumptions, we define a positive continue functionλ (t, x), which is τ−periodic in time t for some τ > 0, that is for all t ≥ 0 This assumption describes the impact of the variability of an environment on the intensity of the Poisson process, which will allow for periodic fluctuations in the environment on the dynamics of the system, In the remainder of our work, we now pay attention to this stochastic differential equation (SDEsGp), Eq.(6) is equivalent to the system of d equations where h i ∈ {0, −1, 1}.
In the first part of this article, we give the results of existence and uniqueness of the solution for more general (SDEsGp) type. Next, we propose sufficient conditions under which we prove the existence and uniqueness of periodic solution to Eq.(6). Finally, an example is provided to demonstrate the obtained results and apply the previous analytical results to study stochastic dynamic for malaria epidemic model under periodic environment.

Existence and Uniqueness
Existence and uniqueness of the solution to Eq.(6) follows by using (H 1 ) and (H 2 ) to solve this equation "from one jump to the next". In Kurtz (Kurtz, T.G.,1982) , the uniqueness of Eq.(4) implies t 0 λ (s, X(s))ds is a stopping time for X (t) t≥0 .
Let F t = σ X(s), 0 ≤ s ≤ t be the history for the stochastic process X(t) t≥0 up to time t or the natural filtration for the process. Define by M (t) = P t 0λ (s, X(s))ds , then by random time change properties the process is a standard Poisson process where σ (t) = inf s > 0, s 0λ (r, X(r))dr > t . We associate to the considered stochastic process M (t) t≥0 the process with random time change follow Therefore, Eq.(6) can be rewritten like the following equation where F (t, X(t)) = k =1 h λ (t, X(t)).
The Eq.(9) is equivalent to the following equation where N is a scale parameter and X N (t) = 1 N X(t) is the vector of proportions. In a case of models with fixe total population size, N = d ı=1 X i (t) is the total population size at time t. Then, X N (t) is the fractions vector of rescaled numbers of individuals in the various states at time t.
Proposition 2.1 Since P (t) − t is a martingale, then the normalized process is a F t −martingale and verifies lim N→∞ sup 0≤r T ∧β K (X) Q N (r) → 0 a.s ; for all T > 0.
Moment estimate is the most basic and useful technic of analyzing dynamic behavior of solution of stochastic functional differential equation. To study the moment estimate for studying stability (Mao, X.R., 2007), boundedness (Luo, Q. and all, 2011), existence and uniqueness of the solution to Eq.(6), we need the following Lemmas.
Lemme 2.3 Assume that there is a function L(t) ∈ L 1 ([0, T ]; R + ) such that L(t) < ∞ for any given t ∈ [0, T ], where T > 0 and F(t, x) verifies for all t ∈ [0, T ], x, y ∈ R n , and there exists p ≥ 2 such that lim Then there is a unique solution (X(t)) t∈[0,T ] to Eq.(6) and E sup 0≤r≤t Proof Set X 0 = X(t = 0) the initial value of the process (X(t)) t∈[0,T ] , and consider a sequence of process which can be inductively defined by where τ 1 is the first time of jump of the process and in general for n = 2, ...., define the following sequence "from one jump to the next" for any t ∈ [0, T ], where the sequence (τ n ) n≥0 define the time of the n th jump of the process and given by By an inductive argument and the properties of Poisson processes, we show that each X n is adapted and cadlàg.
By the Jensen's inequality (a 1 + a 2 + .... + a m ) p ≤ m p−1 a p 1 + a p 2 + ... + a p m , a i ≥ 0, ∀i = 1, 2, ..., m and p ≥ 2, we have the following inequality By Eq.(16), the stopped stochastic sequence X N n (t) t∈[0,T ] remain constant; i.e X N n (t) = X N n−1 (t) on [0, τ n+1 [ by construction and it is easy to show that τ n n≥0 is an increasing sequence which grow to infinity as n → ∞. Then the following inequality holds Journal of Mathematics Research Vol. 13,No. 4; 2021 where .
∞ is the infinit norm. By using Holderś inequality and the martingale inequality, together with condition (13), (14), we obtain for 1 where C(p) = p p − 1 p is a positive constant. Substituting the above inegalities in (19), we have the result Uniqueness holds by setting X N (t) = lim n→∞ X N n (t) and using Gronwall inequality. If X N n (t) t∈[0,T ] and Y N n (t) t∈[0,T ] are two solutions for some T > 0, and we obtain from inequality (19) for the case p = 2, . Taking X N 0 = Y N 0 the same initial condition of the processes, we get h n (t) ≡ 0, this lead to with probability one X(t) = Y(t) is the unique solution to Eq.(6) for all t > 0.
Lemme 2.4 Suppose that condition (14) hold and Proof This result can be proved by the standard truncation procedure. For each k ≥ 1, define the troncature function; where we set x ∧ k x x = 0 when x = 0. By Lemma 2.3, there exists a unique solution to the following equation with the initial value X N 0 . Define the stopping time We can observe that for t ∈ 0, β  To study the moment estimate of the process X N (t) t≥0 by Itô's formula, we need first the following result.
Proposition 2.5 The stochastic process X N (t) t≥0 is continuous in probability, i.e for all s, t and a positive real number b, has a cadlàg modification Z N (t) t≥0 and the stochastic process is a continuous process which can be define as follows. If Q is a standard Poisson random measures on R 2 + (i.e. with mean measure Lebesgue), then Then the stochastic process X N (t) t≥0 has the same distribution with the stochastic process Z N (t) t≥0 , solution of the following equation:Z Remark 2.6 We would assume more generally for the periodic stochastic intensityλ s, NX N (s) that: as N → ∞.

Remark 2.7 Define by Z (t)
t≥0 the solution to Eq.(21), and has a continuous limits and the Then the following result gives more details on the mode of convergence.
Proposition 2.8: Law of Lagre Numbers, (Britton, T. and Pardoux, E., 2019) Assume that the assumption (H 1 ) and denote the solution to Eq.(6). Then lim Corollary 2.9 Assume that the assumption of proposition (2.8) are satisfied and x(t)) and corollary 2.9 can be stated in terms of mτ-periodic solutions.
. To prove (ii), we consider an other solution q 1 (t) = x 1 (t + τ) with initial condition q 1 (0) = x 0 . By uniqueness of x 0 , we obtain uniqueness of q(t) and the proof is completed.
Remark 2.10 Skorokhod's Representation theorem (Billingsley, P., 1999, See p. 70) proves that there exists a probability space (Ω, F , P) such that lim Let's define the compensated random measureQ(ds, du) = Q(ds, du) − λ(du)dt. Then, the process Z (t) t≥0 solution of the Eq.(21) is differentiable and as the form written as follows Let C 1,2 (R × R n ; R), denote the family of all functions V (t, z) on R × R n , which are twice continuously differentiable in z and once in t. For each V (t, z) ∈ C 1,2 (R × R n ; R), we define an Itô operator LV (t, z) of the function V (t, z) associated with the Eq.(23) of (SDEsGp) by The Itô formula for (SDEsGp) type stochastic integrals is Then Eq.(23) has a unique solution Z (t) t≥0 , and the solution satisfies the following moment estimate Proof Assumption (24) gives nonexplosion condition for solution to a more general stochastic differential equation by Lyapunov functions method. The proof is motivated by (Zhang, X. and all, 2015). From Lemma 2.4, we obtain there is a unique maximal local solution Z (t) . We need to prove thatβ K (Z) = T and will proceed by absurd. If it is not, there exist two constants ε and S < T such that Thus we can find a sufficiently large integer k 0 such that By Itô's formula and inequality 25 given by Lemma 2.11, we have the following results for any t ∈ [0, S ] Thus we get Also, Define by Then by condition (24) of Lemma 2.11, we get lim k→∞ η (k) = ∞. From inequality (27) and inequality (29), it follows the Therefore, the stochastic process Z N (t) Now we prove the moment estimate. By the integration formula by part or more generally by Itô's formula applaying to the process exp where ., . t is the quadratic variation. Taking expectation, we get for every time t ∈ [0, T ].

Periodic Solution of Stochastic Differential Equations Driven by General Counting Processes (SDEsGp)
Periodic mouvement is very universal in nature, for exemple climate changes every three or four seasons. So for periodic stochastic intensity, an importante aspect is to ensure uniqueness of a periodic solution to Eq.(6).
In this paragraph, we discus the periodicity of the solution of Eq.(6). First, we recall some definitions of periodic Markov processes which are found in (Li, D and Xu, D, 2013).
Definition 2.12 A stochastic process z(t, ) with values in Banach space R n , defined for t ≥ 0 on a probability space ( where F v is the σ-algebra of events generated by all events of the form {z(u, ) ∈ A, u ≤ v} and B denotes the σ-algebra of Borel set in R n .

The transition function of a Markov process, p
Later on we shall often denote a family of Markov processes by z (t 0 ,φ) t ( ) for all t 0 ∈ R + and z t 0 = z(t 0 + s) = φ(s) a bounded F 0 -mesurable function.
Definition 2.14 The Markov families z (t 0 ,φ) t ( ) are said to be uniformly bounded, if for each α > 0, t 0 ∈ R + , there exists a positive constant θ = θ(α) which is independent of t 0 such that E z(t 0 ) ≤ α implies that E z(t, t 0 , z(t 0 ) ≤ θ, t ≥ t 0 . In a general way, the Markov families z (t 0 ,φ) t ( ) are said to be p-uniformly bounded if E . is replaced by E .
and there are two positive constants α and γ such that for all t ∈ [0, ∞[,  (31) and (32), we get the p-uniformly bounded of the solution. To fit together with assumption (H 1 ) the transition probabilities function to Eq.(23) is τ-periodic (Li, D and Xu, D, 2013), this yield there exits an τ-periodic solution to Eq.(23). This complete the proof.
Our last point is to give sufficient conditions for uniqueness of the periodic stochastic solution. We need first the following definitions.
Definition 2.16 The τ-periodic stochastic solutionZ * (t) to Eq.(23) is said to be asymptotically stable in p-th moment, if it is stable in p-th moment and satifies: We are going now to give sufficients condition on the asymptotic stability of solution to Eq.(23) which ensure the uniqueness of stochastic periodic solution. LetZ z 1 (t) andZ z 2 (t) be two periodic solutions to Eq.(23) with initial values z 1 and z 2 respectively. Then we have:Z (34) is defined as following: Theorem 2.17 Assume that all inequalities in theorem hold and there exists a function W ∈ C 1,2 ([0, ∞[ × R n ; R + ) such that: and where θ, ϑ are positive constants and α(t) has the property: Then the τ-periodic stochastic solutionZ * (t) of Eq. (23) is asymptotically stable in p-th moment. Therefore Eq.(23) has a unique stochastic periodic solution.
Proof Applying Itô's formula to the process exp Using (36) and (37), we get Then, which prove that the τ-periodic stochastic solutionZ * (t) to Eq.(23) is stable in p-th moment for α(t) ≥ 0. Next by condition (37) and taking the limit, we get lim Then, the τ-periodic stochastic solutionZ * (t) is asymptotically stable in p-th moment. Therefore, the τ-periodic stochastic solution is unique. This complete the proof.

Description of the Model
We considere a mathematical compartmental model to describe a Malaria epidemic under periodic environment and assume that the human total population size K varies little over the time, which comes down to consider it constant in a given region. We denote by S h (t) and I h (t) the total size of susceptible and infectious human population respectively at time t. Then to study the dynamics of malaria in human population, we need to know the dynamics of I h (t). For mosquitoes population, we considere two stages of mosquitoes' growth: the juvenile stage which size is denoted by L v (t) and the adult mosquitoes are divided in two compartments. Let S v (t) and I v (t) denote the total size of susceptible and infectious mosquitoes respectively at time t. We denote the biting rate of mosquitoes by β v (t) which is the number of bites per mosquito per unit time at time t and depend of environment parameters. Let p and be the probabilities that a mosquito arrives randomly at a human and picks the human if he is infectious and susceptible, respectively. Since infectious humans are more attractive to mosquitoes, we assume that p ≥ l. Several event happen during the transmission of malaria i) Obviously, we assume that the probability that an infectious mosquito picks a susceptible human is uniform and given by S h (t)+pI h (t) , the ratio between the total bitten susceptible humans over the total bitten humans. Thus, the counting process which count the cumulative occurred infectious humans up to time t is where c is the probability of transmission of infection from an infectious mosquito to a susceptible human given that the contact between the two occurs, β v (t) is a positive, continuous function τ-periodic in time t for some τ > 0 and P H S I (t) t≥0 is a standard Poisson process.
ii) Similary, the probability that a susceptible mosquito picks an infectious human is pI h (t) S h (t)+pI h (t) , the ratio between the total bitten infectious humans over the total bitten humans. The counting process which count the cumulative occurred infectious mosquitoes up to time t is where b is the probability of transmission of infection from an infectious human to a susceptible mosquito given that the contact between the two occurs and P V S I (t) t≥0 is a standard Poisson process. where P H I Mn (t) t≥0 and P H I Mm (t) t≥0 are standard Poisson processes. iv) If the treatment is effective, an infectious human can recover from malaria, this happen at a constant rate ρ and the counting process which count the cumulative occurred recovered infectious humans up to time t is where P H IG (t) t≥0 is a standard Poisson process. v) After laying eggs on waterfront by adult female mosquitoes, juvenile mosquitoes are born at a varying rate λ L (t) which varies according to the water temperature (Paaijmans, K.P. and all, 2009). Thus, the counting process which count the cumulative occurred new born juvenile mosquitoes up to time t is is the total size of adult female mosquitoes, λ L (t) is a positive, continuous function τ-periodic in time t for some τ > 0 and P V LN (t) t≥0 is a standard Poisson process. vi) Juvenile mosquitoes leave juvenile stage for adulte stage at the varying rate λ v (t) which varies according to the air temperature and define the birth rate of adult susceptible mosquitoes (Rubel, F. and all, 2008). The counting process which count the cumulative occurred new born adult susceptible mosquitoes up to time t is where λ v (t) is a positive, continuous function τ-periodic in time t for some τ > 0 and P V S N (t) t≥0 is a standard Poisson process. vii) During their development juvenile mosquitoes can die naturally at the varying rate µ L (t) or for predation at the density-dependent death rate α. The counting processes which count the cumulative occurred juvenile mosquitoes who died up to time t are where µ L (t) is a positive, continuous function τ-periodic in time t for some τ > 0, P V Lv Mn (t) t≥0 and P V Lv Mp (t) t≥0 are standard Poisson processes.
viii) Adult female mosquitoes, susceptible and infectious die naturally at the varying rate µ v (t). The counting processes which count the cumulative occurred adult female mosquitoes who died up to time t are where µ v (t) is a positive, continuous function τ-periodic in time t for some τ > 0, P V S M (t) t≥0 and P V I M (t) t≥0 are standard Poisson processes.
All the considered standard Poisson processes P H S I (t) t≥0 , P V S I (t) t≥0 , P H I Mn (t) t≥0 , P H I Mm (t) t≥0 , P H IG (t) t≥0 , P V LN (t) t≥0 , P V S M (t) t≥0 , P V I M (t) t≥0 , P V S N (t) t≥. , P V Lv Mn (t) and P V Lv Mp (t) are mutually independent.
Define by X(t) = I h (t), L v (t), S v (t), I v (t) the R 4 -vector which gives the state of the stochastic process of the dynamic of malaria at time t. Despite the continuous positive functions are τ-periodic, they introduse in the model the external randomness of the environment. Then, the stochastic dynamics of malaria is described by the following stochastic differential equations driven by general counting processes (SDEsGp) http://jmr.ccsenet.org Journal of Mathematics Research Vol. 13,No. 4; 2021 where the R 4 -vector (X(t)) t≥0 determine the born, the infection, the recovery and the death processes respectivelly of humans and mosquitoes populations in random environment.

Example
To illustrate our main results we consider an exemple based on actual data. We study Malaria transmission case in Abidjan based on average monthly temperature data from 1999 to 2016. In 2018, 6 countries accounted for more than half of all malaria cases worlwide and Côte d'Ivoire is in fourth place behind Nigeria (25%), the Democratic Republic of Congo (12%), Uganda (5%) with (4%) according to the latest World Malaria report released in December 2019. Related to this report children under 5 years of age are the most vulnerable group affected by malaria; in 2018, they accounted for (67%) (272 000) of all malaria deaths worldwide. This represents a major problem of public health for affected countries. Based on data from Côte d'Ivoire, we got actual data to have good estimations of environmental parameters. Then we examine the existence and uniqueness of stochastic periodic solutions for the following associate stochastic differential equation with Poisson random measurẽ Taking this number, we estimate the human natural death rate as : d h = 1 57, 422 × 12 = 0, 00145 month −1 . The values p and are probabilities and may vary from 0 to 1 with p ≥ (Chamchod F, Britton NF , 2011), (Lacroix R and all, 2005).
Environment parameters such as temperature, play a major role in the life cycle of mosquitoes, we evaluate the periodic fonctions relative to mosquitoes population by using the monthly mean temperature of Abidjan from 1991 to 2016 (see https://climateknowledgeportal. worldbank.org/country/cote-divoire/climate-data-historical) as show in the following table. From data for the Wordbank obtained from Climate change knowledge, the monthly mean temperature of Abidjan is very close to those of Port Harcourt from 1990 to 2012 for which we have good estimation of those periodic fonctions. We choose the same approximate periodic functions whithout loss of generality. According to Paaijmans and all (Paaijmans, K.P. and all, 2009), Wang, X and Zaho, X.Q (Wang, X and Zaho, X.Q, 2017); the natural biting rate of mosquitoes in Abidjan is approximate by  It follow from Wang, X and Zaho, X.Q (Wang, X and Zaho, X.Q, 2017) and Rubel, F and all (Rubel, F. and all, 2008) the birth rates of juvenile and adult mosquitoes and the death rate of juvenile mosquitoes in Abidjan is approximate by and µ L (t) = 0.2240 cos(πt/6) + 1.5699 sin(πt/6) − 0.4849 cos(2πt/6) − 0.4268 sin(2πt/6) − 0.0835 cos(3πt/6) − 0.3016 sin(3πt/6) − 0.0210 cos(4πt/6) − 0.2684 sin(4πt/6) − 0.0051 cos(5πt/6) − 0.0845 sin(5πt/6) + 9.0288 month −1 .

Journal of Mathematics
. Applaying Itô's formula, we proove next that all the existence conditions are satisfy and the limit process x(t) in Eq. (5) exist and has the following dynamics