Mathematical Modeling for a CHIKV Transmission Under the Inﬂuence of Periodic Environment

. We studied a simple mathematical model for the chikungunya virus ( CHIKV ) spread under the inﬂuence of a seasonal environment with two routes of infection. We investigated the existence and the uniqueness of a bounded positive solution, and we showed that the system admits a global attractor set. We calculated the basic reproduction number R 0 for the both cases, the ﬁxed and seasonal environment which permits us to characterise both, the extinction and the persistence of the disease with regard to the values of R 0 . We proved that the virus-free equilibrium point is globally asymptotically stable if R 0 ≤ 1, while the disease will persist if R 0 > 1. Finally, we gave some numerical examples conﬁrming the theoretical ﬁndings.


Introduction
Arboviruses constitute a group of viruses which are transmitted to humans or animals by bites from blood-sucking vectors (mosquitoes, ticks and sandflies).Certain viruses have had a renewed medical importance in these two recent decades, notably the Dengue virus, the Yellow Fever virus, the virus Zika disease and Chikungunya virus.The adaptation of the Chikungunya virus to new vectors (Aedes albopictus), the adaptation of these vectors to new environments, and the severe clinical forms associated with these arboviruses mean that they have become emerging and urgent issues around the world, particularly in South America and Europe.Chikungunya and Zika virus infection have had renewed medical interest following massive epidemics which started respectively in Kenya in 2004 and in the Yap Islands (Micronesia) in 2007.Chikungunya and Zika viruses are mainly transmitted to man during a blood meal of Aedes mosquitoes whose entomological surveillance (population of the environment, resistance of mosquitoes to insecticides) remains almost completely absent in Mali compared to Anopheles mosquitoes (malaria vectors).The medical interest of Chikungunya is in particular linked to serious forms in newborns (encephalitis, dermatological bullous lesions in particular), severe forms (hepatitis, neurological forms -uncommon), complications linked to comorbidities and finally to long-term rheumatological forms.
The mathematical modeling permits for mathematician to use a set of concepts, methods, mathematical theories that facilitate the description, the understand and the prediction of the evolution of phenomena external to mathematics which make a link between reality and mathematics.For several centuries, mathematics has not only been a tool extremely important for acting on and modifying nature, one of the main pillars of technique and technology, but also (and perhaps above all) a major instrument to understand it.In this sense, they are not only a source of utility but also of "truth".In particular, mathematical modeling is a way for studying the disease, predicting its behavior in the future, and then proposing suitable strategies.Several researchers worked on some mathematical models for several infectious diseases [1][2][3][4][5][6][7].In particular, the modeling of the behavior of CHIKV dynamics was studied in several recent works [8][9][10][11][12][13][14][15].
Note that seasonality in infectious is very repetitive [16].In particular, each year with the return of cold weather, infectious diseases spread among the population.Although they are often temporary and harmless, they can nevertheless be much more serious, particularly in the weakest people.CHIKV epidemics occur in a context marked by seasonal rains and tropical storms which have caused heavy flooding.Seasonal factors such as the monsoon or rainy season affect the development of an epidemic.We then talk about seasonality of CHIKV.Climate changes linked to global warming can interact with seasonal climatic factors, particularly through climatic anomalies (drought, floods) and be the cause of significant epidemic outbreaks Several sand simple mathematical models of infectious diseases that take into account of the seasonality were proposed [17][18][19].In such mathematical models, the basic reproduction number can be calculated either using the time-averaged system (autonomous) as in [20,21] or other definition as in [22,23] where all these definitions are different from the one defined for time-averaged system.In [24], the authors analysed the seasonal behaviour of an SVEIR epidemic model with vaccination.Similarly, in [25][26][27][28][29], the authors studied the seasonal behaviour of some epidemic models related to HIV, chikungunya virus and Typhoid Fever spread.We aim in this paper to study the dynamics of CHIKV in relation with phages and hosts when it is considered in both, fixed and seasonal environment and with a nonlinear general incidence rate.We calculated the basic reproduction number as the spectral radius of an integral operator.We analysed the global stability of the disease-free solution where we proved that it is globally asymptotically stable if R 0 < 1.However, R 0 > 1, we proved that the dynamics is persistent and so the disease-present solution converges to a limit cycle.We confirmed the theoretical findings by using an intense numerical examples.
The rest of this article is organized as follows.In Section 2, we present a simple epidemic model of CHIKV taking into of the seasonality.In Section 3, we considered firstly the case of fixed environment, and we calculated R 0 and we investigated the global analysis of both, the diseasefree and the endemic steady states.However, in section 4, we focus on the stability of virus-free and virus-present periodic trajectories for the case of seasonal environment.Some numerical tests are given in Section 5 confirming the theoretical findings.Finally, in section 6, we give some concluding remarks.

CHIKV Epidemic Model
We consider a compartmental mathematical model for the dynamics of a CHIKV.Let us denote by X s (t), X i (t), X v (t) and X p (t) the quantities of susceptible hosts, infected hosts, CHIKV and phages, respectively.Therefore, the model is given by the fourth dimensional system of differential equations hereafter.
with initial conditions given by (X s (0), X i (0), X v (0), X p (0)) ∈ R 4  + .The susceptible hosts have a periodic recruited rate d(t)Θ 1 (t), and a periodic death rate d(t) and a periodic incidence rate τ , where τ 1 (t) and τ 2 (t) are the periodic contact rates.The periodic parameters µ(t) and m(t) describe the periodic death rates of the CHIKV and the phages, respectively.ξ(t) is the periodic production rate from infected hosts to CHIKV.The phages have a periodic proliferation rate given by m(t)Θ 2 (t) + (t)τ 3 (t)X v (t)X p (t).More details concerning the significance of the model parameters are given in Table 1.

Notation Definition Notation Definition
X s (t)

Case of Fixed Environment
In this section, we assume that all parameters are positive constant reflecting the case of fixed environment.Therefore, we obtain the the autonomous form of the dynamics (2.1).
with initial conditions (X s (0), 3.1.Basic properties.In this subsection, we give some classical properties for epidemiological models.Let σ = min(µ, m), then we obtain the following results.
a positively invariant and attractor of the dynamics (3.1).

Basic reproduction number and steady states.
As our model has several compartments, the next-generation matrix method [30][31][32] will be used to calculate the basic reproduction number as follows.
. Then, the next-generation matrix is given by . Thus, the spectral radius of FV −1 which is the basic reproduction number is expressed by: • If R 0 ≤ 1, then (3.1) admits only E 0 = (Θ 1 , 0, 0, Θ 2 ) as a steady state.
• If R 0 > 1, then the autonomous dynamics (3.1) admits two steady states; E 0 and an endemic steady state Proof.Consider E = (X s , X i , X v , X p ) to be a steady state then it satisfies: From Eq (3. 3) we obtain the CHIKV-free steady state E 0 = (Θ 1 , 0, 0, Θ 2 ).Furthermore, we have We define the function Then, we obtain lim Therefore, we deduce that g (X v ) ≤ 0 for all X v ∈ (0, m τ 3 ).Then, the function g(X v ) admits a unique root ).Therefore, one obtains Therefore, the infected equilibrium 3.3.Local analysis.We aim, in this section, to analyse the local stability of the equilibria of the dynamics (3.1).
Theorem 3.1.In the case where R 0 < 1, the phage-free E 0 is locally asymptotically stable, and in the case where R 0 > 1, E 0 is unstable.
Proof.The linearisation of the dynamics (3.1) at the steady state E 0 is: .
Proof.The linearisation of the dynamics (3.1) at the steady state The characteristic polynomial is then given by: The characteristic polynomial Q(λ) = 0 if, and only if or if Suppose that the eigenvalue λ is with positive real part.Therefore, since , then, by considering the left-hand side, we obtain |λ + d| however, by considering the right-hand side, we obtain This is a contradiction and then λ has non-positive real-part and then the endemic equilibrium point E * should be locally asymptotically stable.
3.4.Global analysis.Our aim, in this section, is to prove the global stability of the equilibria of the dynamics (3.1).Consider the function G(x) = x − 1 − ln x that we will use is this section.
Proof.Let us define the Lyapunov function F 0 (X s , X i , X v , X p ) given by: Note that F 0 (X s , X i , X v , X p ) > 0 for all X s , X i , X v , X p > 0 and F 0 (Θ 1 , 0, 0, Θ 2 ) = 0. Furthermore, we have [33], one can deduces that E 0 is globally asymptotically stable if Theorem 3.4.E * is globally asymptotically stable for the dynamics (3.1) once R 0 > 1.
Proof.Let us define the Lyapunov function F * (X s , X i , X v , X p ) given by: The derivative of F * with respect to time is given by: Since the steady state Using the rule that 1 n n i=1 a i ≥ n n i=1 a i , we get 1 5 One can deduce easily that E * is globally stable by using the LaSalle's invariance principle [33].

Influence of Periodic Environment
Let return to the main dynamics (2.1) for a seasonal environment.For any continuous, positive T-periodic function g(t), we define g u = max t∈[0,T) g(t) and g l = min t∈[0,T) g(t).

Preliminary.
Let A(t) to be a T-periodic m × m matrix continuous function that it is irreducible and cooperative.Let β A (t) to be the fundamental matrix with positive entries, solution of ẇ(t) = A(t)w(t). (4.1) Let us denote the spectral radius of the matrix β A (T) by r(β A (T)).By using the Perron-Frobenius theorem, one can define r(β A (T)) to be the principal eigenvalue of β A (T).According to [34], we have: Lemma 4.1.[34].(4.1) admits a positive T-periodic function x(t) such that w(t) = x(t)e at with a = 1 T ln(r(β A (T))).
is a positively invariant and attractor of trajectories of dynamics (2.1) with Proof.Using the dynamics (2.1), we obtain and In section 4.2, we aim to define the basic reproduction number; R 0 , the disease-free and then its global stability for R 0 ≤ 1.Later, in section 4.3, we aim to prove that compartments X i (t) and X v (t) 4.2.Disease-free trajectory.By using the definition of R 0 given by the theory in [23].
The dynamics (4.4) has a disease-free periodic solution Y * (t) = (0, 0, where and Y i are the i-th components of f (t, Y(t)) and Y, respectively.A simple calculation give us + .Therefore, the condition (A6) in [23, Section 1] is also fulfilled.
Let us define F(t) and V(t) to be two matrices defined by where F i (t, Y) and V i (t, Y) are the i-th components of F (t, Y) and V(t, Y), respectively.A simple calculation by using (4.4) give us the expressions of matrices F(t) and V(t) as the following: Consider Z(t 1 , t 2 ) to be the two by two matrix solution of the system for any t 1 ≥ t 2 , with Z(t 1 , t 1 ) = I 2 , i.e., the 2 × 2 identity matrix.Therefore, condition (A7) is also fulfilled.
Denote by C T the ordered Banach space of T-periodic functions that are defined on R → R 2 , with the maximum norm .∞ and the positive cone C + T = {ψ ∈ C T : ψ(s) ≥ 0, for any s ∈ R}.Consider the linear operator K : C T → C T given by Therefore, the basic reproduction number, R 0 , of dynamics (2.1) is given by R 0 = r(K).Thus, the local stability of the disease-free periodic trajectory, E 0 (t) = (X * s (t), 0, 0, X * p (t)), of the dynamics (2.1) with respect to R 0 is given hereafter.
Therefore, it remains to satisfy the global attractivity of E 0 (t) once R 0 < 1.Using (4.3) in Proposition 4.1, for any for t > T 1 .Let M 2 (t) be the two by two matrix function given hereafter using the equivalences in Theorem 4.1, one has r(ϕ F−V (T)) < 1.By choosing m 1 > 0 satisfying r(ϕ F−V+m 1 M 2 (T)) < 1 and we consider the dynamics hereafter, Using Lemma 4.1, there exists a positive T-periodic function x 1 (t) such that w(t) ≤ x 1 (t)e a 1 t with Furthermore, we have that lim Then, we deduce that the disease-free periodic trajectory E 0 (t) is globally attractive.4.3.Endemic trajectory .Note that the dynamics (2.1) admits Γ 2 as an invariant compact set.
Theorem 4.3.Assume that R 0 > 1.The dynamics (2.1) admits at least one periodic solution such that there exists ε > 0 that satisfies Proof.We aim to prove that P is uniformly persistent with respect to (Ω 0 , ∂Ω 0 ) which permits to prove that the solution of the dynamics (2.1) is uniformly persistent with respect to (Ω 0 , ∂Ω 0 ) by using [35,Theorem 3.1.1].From Theorem 4.1, we have r(ϕ F−V (T)) > 1.Therefore, there exists ξ > 0 such that r(ϕ F−V−ξM 2 (T)) > 1. Define the system of equations: where X0 ∃ n such that nT > T 3 for all n > n.Therefore for all n > n which contradicts the fact that Xs (nT) = 0.Then, S0 > 0 and Ỹ0 is a positive T-periodic solution of the dynamics (2.1).

Case of seasonal contact.
The second was allocated to the case where only the contact rates, τ 1 , τ 2 and τ 3 are seasonal functions.All the rest of parameters are fixed.We obtain the following system. (5.2) with the positive initial condition (S 0 , I 0 , V 0 , P 0 ) ∈ R 4 + .We give the results of some numerical simulations confirming the stability of the steady states of system (5.2).The approximation of the basic reproduction number R 0 was performed using the time-averaged system.Figure 5. Behavior of the dynamics (2.1) for τ 0 1 = 1.2, τ 0 2 = 0.8 and τ 0 3 = 0.9 then R 0 ≈ 2.57 > 1. Figure 6.Behavior of the dynamics (2.1) for τ 0 1 = 1.2, τ 0 2 = 0.8 and τ 0 3 = 0.9 then R 0 ≈ 2.57 > 1.
In Figures 5 and 6, the trajectories of the dynamics (5.2) converge asymptotically to the periodic solution corresponding to the disease-persistence.In Figures 7 and 8, the trajectories of the dynamics (5.2) converge to the disease-free trajectory if R 0 < 1.
(5.3) with the positive initial condition (S 0 , I 0 , V 0 , P 0 ) ∈ R 4  + .We give the results of some numerical simulations confirming the stability of the steady states of system (5.3).The basic reproduction number R 0 was approximated by using the time-averaged system.In Figures 9 and 10, the trajectories of the dynamics (5.3) converge asymptotically to the periodic solution corresponding to the disease persistence if R 0 > 1.In Figures 11 and 12, the trajectories of the dynamics (5.3) converge to the disease-free periodic solution E 0 (t) = (X * s (t), 0, 0, X * p (t)) for the case where R 0 ≤ 1.

Conclusions
In order to more understand the CHIKV dynamics when describing the contamination of uninfected hosts, an important way is to take into account of both, contact with CHIKV (CHIKV-to-host transmission) and contact with infected hosts (host-to-host transmission).The marked seasonality of CHIKV, impose the consideration of this property when modelling its dynamics.In this article, we proposed and analysed a mathematical model for CHIKV dynamics reflecting the seasonality observed in real life.The basic reproduction number was defined and the steady states of the dynamics were calculated for the first step when considering the autonomous dynamics.We characterised the existence and uniqueness of the steady states.We characterised also the stability conditions for these steady states.Later, we concentrated on the non-autonomous dynamics and we defined the basic reproduction number, R 0 by using an integral operator.It is proved that once R 0 ≤ 1, all solution of the dynamics converge to the disease-free periodic trajectory and that the disease persists if R 0 > 1.We performed the theoretical findings by some numerical examples using explicit Runge-Kutta formulas of orders 4 and 5 under Matlab for three cases, the autonomous dynamics, the seasonal contact dynamics and the fully seasonal dynamics.As it is seen in the numerical simulations and proved theoretically that for the first case, the solution converge to one of the equilibria of the dynamics (5.1) regarding Theorems 3.3 and 3.4.However, for the second and third cases, the solutions converge to a limit cycle regarding Theorems 4.2 and 4.3.