Global exponential stability of positive periodic solutions for a cholera model with saturated treatment ∗

Cholera is an acute intestinal infectious disease caused by infection of the bacterium Vibrio cholerae, such as Vibrio cholerae serogroups O1 and O139, which is the major public health problem and affect primarily developing world populations with no proper access to adequate water and sanitation resources. Once they colonise the intestinal gut, then produce enterotoxin (which stimulates water and electrolyte secretion by the endothelial cells of the small intestine) that leads to copious, painless, and watery diarrhoea that can quickly lead to severe dehydration and death if treatment isn’t promptly given [4]. Up to now, the control of deadly outbreaks remains a challenge. In recent years, the number of cholera cases reported to World Health Organization (WHO) is on the increase. In 2015, 172454 cases and 1304 deaths of cholera were reported to WHO worldwide [25]. Outbreaks continued to affect several countries [25]. Overall, 41% of cases were reported from Africa, 37% from Asia, and 21% from the Americas [25]. So, cholera is also a global threat to public health, and it is one of the important indicators of social development.


Introduction
Cholera is an acute intestinal infectious disease caused by infection of the bacterium Vibrio cholerae, such as Vibrio cholerae serogroups O1 and O139, which is the major public health problem and affect primarily developing world populations with no proper access to adequate water and sanitation resources.Once they colonise the intestinal gut, then produce enterotoxin (which stimulates water and electrolyte secretion by the endothelial cells of the small intestine) that leads to copious, painless, and watery diarrhoea that can quickly lead to severe dehydration and death if treatment isn't promptly given [4].Up to now, the control of deadly outbreaks remains a challenge.In recent years, the number of cholera cases reported to World Health Organization (WHO) is on the increase.In 2015, 172454 cases and 1304 deaths of cholera were reported to WHO worldwide [25].Outbreaks continued to affect several countries [25].Overall, 41% of cases were reported from Africa, 37% from Asia, and 21% from the Americas [25].So, cholera is also a global threat to public health, and it is one of the important indicators of social development.
Mathematical models have been proven to be central importance for understanding dynamical behavior of the epidemic spreading in the infectious diseases [12,21,27].The mathematical model of cholera epidemics pandemic was first proposed by Capasso et al in 1979 [2].Some researchers considered a cholera model with imperfect vaccination, which studied the stability of a disease-free equilibrium and an endemic equilibrium [4,14,22,23,26,29,30].Also, the literature [22] analysed control strategies of cholera.Mwasa et al. formulated a mathematical model that captures some essential dynamics of cholera transmission to study the impact of some control strategies, such as public health educational campaigns, vaccination and treatment in reducing the incidence of disease [15,20].In [16], Safi presented a new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population.In [10], Khan et al. studied the dynamical behavior of cholera epidemic model with nonlinear incidence rate.To the best of our knowledge, these is no paper to consider a cholera model with both periodic incidence rate and saturated treatment function.
As it is well known, many infectious diseases exhibit seasonal fluctuations, and there is a saturation phenomenon during the treatment process.Therefore, the coefficients in the differential equations of ecology, epidemics, and population problems are usually timevarying.Usually, we use the periodic coefficients.So, to describe the dynamics of the cholera, we consider the following model: Here S represents the number of individuals susceptible to the disease, I represents the number of infected individuals infectious and able to spread the disease by contacting with the susceptibles, R is the number of the infectives removed or recovered, and B is the number of the pathogen population.In this paper, A, K, µ The notation R and R + refers to the space of real number and nonnegative real number, respectively.The notation R n and R n + refers to the space of n-dimensional real column vector and n-dimensional nonnegative real column vector, respectively.The notation R n×n refers to the n×n nonnegative real matrix space.For any x = (x 1 , x 2 , . . ., x n ) ∈ R n , let |x| denotes the absolute-value vector given by |x| = (|x 1 |, |x 2 |, . . ., |x n |), "T" denotes the transpose (x T = (x 1 , x 2 , . . ., x n ) T ), and we define x = max i∈{1,2,...,n} |x i |.If A ∈ R m×n , A T refers to the transpose of A.
https://www.mii.vu.lt/NAThe initial conditions associated with (1) are as follows: For simplicity, we first assume that a bounded continuous function g defined on R given by g + = sup t∈R g(t) and g − = inf t∈R g(t) .
In the following, we will always assume that

Preliminaries and lemmas
Firstly, we show that the existence of the disease free periodic solution of (1).To find the disease free periodic solution of (1), we consider the following equation: with initial condition S(0) = S 0 ∈ R + .(4) admits a unique positive ω-periodic solution S * (t) > 0, which is globally attractive in R + and hence, (1) has a unique disease free periodic solution (S * (t), 0, 0, 0).Let us define the basic reproduction number of (1), by applying the theory in Wang and Zhang [28] with where x = (I, B, S, R) T .For our purpose, we check conditions (A1)-(A7) in Section 1 of [28].( 1) is equivalent to the following form: where It is easy to see that conditions (A1)-(A5) are satisfied.We know that (6) has the disease free periodic solution x * (t) = (S * (t), 0, 0, 0).Now, we define f (t, x(t)) = F(t, x(t)) − V(t, x(t)) and M (t) = (∂f i (t, x * (t))/∂x j ) 3 i,j 4 , where f i (t, x(t)) and x i is the ith component of f (t, x(t)) and x, respectively.From (5) we can get and hence, r(Φ M (ω)) < 1, which implies that x * (t) is linearly asymptotically stable in the disease free subspace X s = (0, 0, S, R) ∈ R 4 + .Thus, the condition (A6) also holds.Next, we set F (t) and V (t) are 2 × 2 matrices defined by F (t) = (∂F i (t, x * (t))/ ∂x j ) 1 i,j 2 and V (t) = (∂V i (t, x * (t))/∂x j ) 1 i,j 2 , where F i (t, x) and V i (t, x) is the ith component of F(t, x) and V(t, x), respectively.Then from (2.4) it follows that for any t s, Y (s, s) = I, where I is 2 × 2 identity matrix.Therefore, condition (A7) holds.
Let C ω be the ordered Banach space of all ω-periodic function from R → R 2 , which is equipped with maximum norm • and the positive cone C ω + = {φ ∈ C ω : φ(t) 0 for any t ∈ R}.Consider the following linear operator L : Finally, we can define the basic reproduction number 0 of (1) as follows: From the above discussion, we obtain the following result for the local asymptotic stability of the disease free periodic solution (S * (t), 0, 0, 0) for (1).
Assume, by way of contradiction, that (7) doesn't hold.Then there must exist But from the first equation of ( 1), we have Next, we claim that From the second equation of (1) we obtain and hence, This contradicts I(T 2 ) = 0 and the claim is proved.Now, we prove that R(t) > 0 for all t ∈ [t 0 , T * ).If R(t) > 0, then by continuity we can choose a small positive constant ρ * such that If R(t 0 ) = 0, then which implies that (8) also holds.Now, we claim that Otherwise, there must exist From ( 1) and ( 10) we have which is a contradiction, and hence, (9) holds.Finally, we prove that We will prove it by the way of contradiction.Assume that (11) doesn't hold.Then there must exist T 4 ∈ [t 0 , T * ) such that But in view of the first equation of (1), we have a contradiction.Hence, (11) holds.
Lemma 3. Assume that and the assumptions of Lemma 2 hold.Let S(t), I(t), R(t), B(t) , Š(t), Ǐ(t), Ř(t), B(t) be the solutions of system (1) with initial conditions (2).Then there exist ť0 t 0 and positive constants ζ and k such that, for all t ť0 , Moreover, there exist constants t R ť0 and k R such that Proof.Let, for all t ∈ [t 0 , ∞).
Then (1) gives x 4 (t), x 4 (t), which implies dv, (23) and for all t ť0 .Let < min{L I , L S } be a positive constant such that This can be achieved because of ( 18), (19), and (20).Consequently, we can choose positive constants ζ and τ such that https://www.mii.vu.lt/NA From Lemma 2 we can choose ť0 t 0 such that, for all t ť0 , |}, and k 0 > 1 be a constant.It is obvious that In the following, we will show Otherwise, one of the following three cases must occur: Case 1.There exists θ 1 > 0 such that x(t) < k 0 x 0 + e ζ ť0 e −ζt for all t ∈ [ ť0 , θ 1 ).

Main results
Theorem 2. Under the assumptions of Lemma It follows from Lemma 3 that there exist t 0 > t 0 and k such that, for an nonnegative integer h and t + hT t 0 , Now, we show that ( S(t + hT ), I(t + hT ), R(t + hT ), B(t + hT )) q is convergent on any compact interval as q → ∞.Let [a, b] ⊂ R be an arbitrary interval.Choose a nonnegative integer q 0 such that t + q 0 T t 0 for t ∈ [a, b].Then for t ∈ [a, b] and q > q 0 we have which, together with (41), implies that {( S(t+hT ), I(t+hT ), R(t+hT ), B(t+hT ))} q converges uniformly to a continuous function, say (S * (t), I * (t), R * (t), B * (t)), on [a, b] ⊂ R.  Remark 3. To the best of our knowledge, there is no result on the global exponential stability of positive periodic solutions for the cholera model with periodic incidence rate and saturated treatment function.We also mention that the results in (see [1,5,9,13]) can not be applied to the global exponential stability of positive periodic solutions for system (1).Here we employ a novel proof to establish some criteria, which guarantee the existence and global exponential stability of positive periodic solutions for the cholera model.

Discussion
In this paper, we considered a non-autonomous cholera epidemic model, which involves almost periodic incidence rate and saturated treatment function.By using the differential inequality technique and Lyapunov functional method, we obtained the existence and global exponential stability of almost periodic solutions for the addressed SIR model, which improve and supplement existing ones.Also, an example and its numerical simulations are given to demonstrate our theoretical results.
As we all known, spatial diffusion plays an important role in epidemic spread [11,18,19,24].We will study the cholera models with spatial diffusion in the future.