Mathematical and Numerical Investigations for a Cholera Dynamics With a Seasonal Environment

.


Introduction
Cholera is a highly contagious diarrheal disease caused by a Gram-negative bacillus: vibrio cholerae.
This pathogenic bacterium has an exclusively digestive tropism and lives in a saprophytic state in water and estuaries.The bacterial strains responsible for cholera are transmitted either orally from water or food that it is contaminated with vibrio cholerae or by close contact with people infected with cholera.Cholera, known since Greek antiquity, was first time identified in the Ganges Delta, India.It remained there, for centuries, limited to Bangladesh, occasionally spilling over into the bordering territories of the Far East.The marked seasonality of cholera, or intra-annual variability, and the simultaneous appearance of cases in different geographically distant places are at the origin of the search for "environmental or climatic forcing" to explain these parallel emergence or resurgence phenomena.and independent.Physical factors (i.e.extrinsic factors), such as water temperature, can explain the seasonality of disease either by exerting a direct influence on the abundance and/or toxicity of vibrio cholerae in the environment, or by exerting an indirect influence, for example, on reservoirs or even on parameters having an impact on the latter.Other physical parameters influencing water levels, such as precipitation, have been invoked to explain the distribution of cholera cases around the world.Indeed, floods and droughts not only affect the concentration of the bacteria in the environment but also its survival through effects on salinity, pH or the concentration of nutrients in the environment.
The cholera/environment links are not only observed in the seasonal variability of cholera cases but also in the inter-annual variability, that is to say on an average time step of around 3 to 8 years.
The "Modeling" skill, if we take it in its broadest sense, refers for the mathematician to the fact of using a set of concepts, methods, mathematical theories that will make it possible to describe, understand and predict the evolution of phenomena external to mathematics.Modeling is a way to make the 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]).In particular, the modeling of the behavior of cholera bacteria in an aquatic environment [7] by proving the stability of both, the disease-free and the endemic steady sates by ignoring the human-to-human infection by vibrio cholerae.Later, in [8], the authors present the influence of phages in cholera control by extending the proposed model in [7] by adding a new compartment for phages in the model, and deduced that phages decrease the bacteria concentration of bacteria which reduces the infection.
Several other mathematical models of cholera including the phages as compartments are developed and analysed [9,10].
Note that seasonality in infectious is very repetitive [11].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.Cholera 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 cholera.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 [12][13][14].In such mathematical models, the basic reproduction number can be calculated either using the time-averaged system (autonomous) as in [15,16] or other definition as in [1,12,17,18] where all these definitions are different from the one defined for time-averaged system.In [19], the authors analysed the seasonal behaviour of an SVEIR epidemic model with vaccination.Similarly, in [20,21], the authors studied the seasonal behaviour of some epidemic models related HIV and chikungunya virus spread.We aim in this paper to study the dynamics of vibrio cholerae 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 generalised cholera epidemic model taking into of the seasonality.In Section 3, we concentrate on the case of fixed environment, and we calculated R 0 and we studied the global analysis of both, the disease-free and the endemic equilibrium points.However, in section 4, we focus on the stability of phage-free and phage-present periodic solutions for the case of seasonal environment.Several numerical examples are given to confirm the theoretical findings in Section 5. Finally, in section 6, we provide some conclusions.

Generalised Cholera Epidemic Model
The mathematical model for vibrio cholerae spread that we proposed here is a compartmental one.Let t be the time variable, and we denote by S(t) and I(t) the quantities of susceptible and infected hosts, respectively.We denote also by V (t) and P (t) to be the quantities of vibrio cholerae and phages, respectively.Therefore, we are interested by the dynamical behaviors of susceptible, infected, vibrio cholerae, and phages.Therefore, the model is given by the fourth dimensional system of differential equations hereafter.
The susceptible hosts have a periodic recruited rate d(t)Λ 1 (t), and a periodic death rate d(t) and a periodic incidence rate ρ 1 (t)S(t)f 1 (I(t)) + ρ 2 (t)S(t)f 2 (V (t)), where ρ 1 (t) and ρ 2 (t) are the periodic contact rates.The periodic parameters µ(t) and δ(t) describe the periodic death rates of the vibrio cholerae and the phages, respectively.η(t) is the periodic production rate from infected hosts to vibrio cholerae.The phages have a periodic proliferation rate given by δ(t)Λ 2 (t) + θ(t)ρ 3 (t)f 3 (V (t))P (t).More details concerning the significance of the model parameters are given in Table 1.The incidence rates f 1 , f 2 and f 3 and the model parameters satisfy the following assumptions:
Assumption 2.1 means that the vibrio cholerae-to-host and host-to-host incidence rates increase when susceptible hosts number increase and that no vibrio cholerae-to-host nor host-to-host infection can be in the absence of infected hosts and vibrio cholerae, respectively.

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).
3.1.Basic properties.In this subsection, we give some classical properties for epidemiological models.

3.2.
Basic reproduction number and steady states.As our model has several compartments, the next-generation matrix method [22][23][24] will be used to calculate the basic reproduction number as follows.
Then, the inverse matrix of V is given by and the next-generation matrix is given by Thus, the spectral radius of F V −1 which is the basic reproduction number is expressed by: as a steady state.
Proof.Consider E = (S, I, V, P ) to be a steady state then it satisfies: From Eq (3. 3) we obtain the vibrio cholerae-free steady state E 0 = (Λ 1 , 0, 0, Λ 2 ).Furthermore, we have We define the function Then, we obtain Let us define V to be the solution of , then V exists and is unique.Now, one has The derivative of the function g is given by (3.7) By Assumption 2.1 and Lemma 2.1, we have Therefore, we deduce that g (V ) ≤ 0 for all V ∈ (0, V ).Then, the function g(V ) admits a unique root V * ∈ (0, V ).Therefore, one obtains Therefore, the infected equilibrium E * = (S * , I * , V * , P * ) exists and is unique if R 0 > 1.

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.
The trace of the matrix M 0 is: Then, E 0 is locally asymptotically stable once R 0 < 1, however, it is unstable once R 0 > 1.
Proof.The linearisation of the dynamics (3.1) at the steady state E * = (S * , I * , V * , P * ) is: 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 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 L 1 (S, I, V, P ) given by: Clearly, L 1 (S, I, V, P ) > 0 for all variables S, I, V, P > 0 and L 1 (S * , I * , V * , P * ) = 0.The derivative of L 1 with respect to time is given by: we get .

Case of Seasonal 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).In order to define the disease-free periodic trajectory of model (2.1), let us consider the subsystem with the initial condition (S 0 , P 0 ) ∈ R 2 + .The dynamics (4.2) has a unique T -periodic trajectory (S * (t), P * (t)) such that S * (t) > 0 and P * (t) > 0. This solution is globally attractive in R 2 + ; therefore, the dynamics (2.1) admits a unique disease-free periodic trajectory (S * (t), 0, 0, P * (t)).
Let us define σ(t) = min t≥0 (µ(t), δ(t)) and then we have Proof.Using the dynamics (2.1), we obtain Let 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 I(t) and V (t) 4.2.Disease-free trajectory.By using the definition of R 0 given by the theory in [18].
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).
Theorem 4.3.Assume that R 0 > 1.The dynamics (2.1) admits at least one periodic solution such that there exists γ > 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 [27,Theorem 3.1.1].From Theorem 4.1, we have r (ϕ F −V (T )) > 1.Therefore, there exists Define the system of equations: P associated with the dynamics (4.10) admits a unique fixed point ( S0 α , P 0 α ) which is globally attractive in R 2 + .By using the implicit function theorem, α → ( S0 α , P 0 α ) is continuous.Thus, α > 0 can be chosen small enough such that Sα (t) > S(t) − η, and Pα (t) > P (t) − η, ∀ t > 0. Using the continuity property of the solution with respect to the initial condition, ∃α * such that Y 0 ∈ Ω 0 with We prove by contradiction that Suppose that lim sup for all k > 0. Therefore u(t, P k (Y 0 )) − u(t, Y 1 ) < α ∀ k > 0 and 0 ≤ t ≤ T.

Numerical Examples
Let us consider Holling type-II functions as examples that can describe the incidence rates in the dynamics (2.1).These function satisfy Assumption 2.1.
Three scenarios were consider here.The first one was allocated to the case of fixed environment.However, the second was allocated to the case where only the contact rates are seasonal.Finally, the last case were allocated to the case where all parameters are periodic.The numerical resolution was done using explicit Runge-Kutta formulas of orders 4 and 5 under Matlab.
However, in Figure 3, the calculated trajectories of the dynamics (5.2) converge to the disease-free steady state E 0 , then confirming the global asymptotic stability of E 0 if R 0 ≤ 1.

5.2.
Case of seasonal contact.The second was allocated to the case where only the contact rates, ρ 1 , ρ 2 and ρ 3 are seasonal functions.All the rest of parameters are fixed.We obtain the following system.
We give the results of some numerical simulations confirming the stability of the steady states of system (5.3).The approximation of the basic reproduction number R 0 was performed using the time-averaged system.
In Figure 4, the calculated trajectories of the dynamics (5.3) converge asymptotically to the periodic solution corresponding to the disease persistence.In Figure 5, we display a magnified view of the limit cycle for the case where R 0 > 1.In Figure 6, the calculated trajectories of the dynamics (5.3) converge to the disease-free trajectory if R 0 < 1.

5.3.
Case of periodic parameters.In the third step, we performed numerical simulations for the system (2.1)where all parameters were set as T -periodic functions.Thus the model is given by (5.4) 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 (5.4).The basic reproduction number R 0 was approximated by using the time-averaged system.
In Figure 12, the calculated trajectories of the dynamics (5.4) converge asymptotically to the periodic solution corresponding to the disease persistence if R 0 > 1.In Figures 11 and 13, we displayed a magnified view of the limit cycle for the case where R 0 > 1.In Figure 14, different initial conditions were considered and for each one of them, the solution converge to the same periodic solution.In Figure 15, the calculated trajectories of the dynamics (5.4) converge to the disease-free periodic solution E 0 (t) = (S * (t), 0, 0, P * (t)) for the case where R 0 ≤ 1.

Conclusions
In order to more understand the vibrio cholerae dynamics when describing the contamination of uninfected hosts, an important way is to take into account of both, contact with vibrio cholerae (vibrio cholerae-to-host transmission) and contact with infected hosts (host-to-host transmission).
The marked seasonality of cholera, impose the consideration of this property when modelling its dynamics.In this article, we proposed and analysed a mathematical model for vibrio cholerae 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 (5.2) 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.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.

Figure 1 .
Figure 1.Diagram explaining the transition between the model compartments.

4. 1 .Lemma 4 . 1 .
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 ofẇ (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[26], we have:[26].(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 ))).

Table 2 .
Used values for d 0 Case of fixed environment.Let us start by the simple case where there is no influence of the seasonality on the dynamics.Thus, we restrict our attention on the autonomous dynamics (3.1), i.e., all parameters are positive constants.