DYNAMIC BEHAVIOR OF A DELAY CHOLERA MODEL WITH CONSTANT INFECTIOUS PERIOD∗

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number R0 > 1, the system is permanent. If R0 < 1, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If R0 > 1, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.


Introduction
Cholera is an acute bacterial illness caused by infection of the intestinal tract with the bacterium Vibrio cholerae. Cholera may produce severe gastrointestinal symptoms, including profuse, watery diarrhea, as well as vomiting and dehydration [13,17,22,29]. It has long been, and continues to be, a world health issue. Cholera usually occurs in areas where there's poor sanitation, over-crowding, war or famine [30].
Mathematical models can describe the dynamic character of infectious diseases to show the likely outcome of an epidemic. And they have played an important role in the disease control in epidemiological aspect and help inform public health interventions. Waterborne diseases such as cholera, diarrheal disease, dysentery, giardia, are caused by pathogenic microorganisms that most commonly are transmitted in contaminated fresh water. Few researchers have contributed towards the mathematical study of the eradication of waterborne diseases, for example, distributed delay model [27], spatially explicit model [9,10], time-varying model [2,25], case studies model [24]. Recently, there have been several efforts in the mathematical modelling of cholera dynamics. The first mathematical model of cholera was developed by Capasso and Paveri-Fontana [5]. In [5], they proposed a mathematical model to describe the 1973 cholera epidemic in Bari (a city in Italy). In their version, two equations describe the dynamics of infected people in the community and the dynamics of the aquatic population of pathogenic bacteria. In 2001, Codeço [6] extended the model in reference [5]. She added an equation for the dynamics of the susceptible population. She studied the role of the aquatic reservoir in the endemicepidemic dynamics of cholera. In [23], Pascual et al. generalized Codeço's model by including a fourth equation for the volume of water in which the formative live following Codeço's [6]. In [32], Zhou X.Y. et al. considered a cholera model with vaccination on the base of the model of Codeço [6]. They added an equation for the dynamics of the vaccinated populations. They analyzed the locally and globally asymptotical stability of the disease-free and endemic equilibria of their system. In [28], Jianjun Paul Tian et al. presented several nonlinear ordinary differential systems of cholera, which incorporated both human population and pathogen Vibrio cholerae concentration. They employed three different techniques, including the monotone dynamical systems, the geometric approach, and Lyapunov functions, to investigate the endemic global stability for several biologically important cases. We may find other mathematical studies on modeling cholera dynamics in references [18][19][20]33]. To the best of our knowledge, these studies do not explicitly confider a delay cholera model with constant infectious period.
In the natural world, there are many diseases which the infected population recover and become susceptible or removed population by itself after they are infected by some certain time. The phenomenon was studied by Hethcote et. al. [14]. For cholera, the incubation period ranges from a few hours to 5 days, usually 2-3 days [15]. Hence, in this paper, we will present a delay cholera model with constant infectious period. We consider the total human population sizes denoted by N (t), which including susceptible individuals S(t), infected individuals I(t) and recovered individuals R(t). The pathogen population at time t, is given by B(t). The susceptible human population is increased by births and/or immigration at a constant rate A (> 0). Natural death occurs in the human classes at a rate µ 1 (> 0). Infected individuals may die due to cholera at a rate δ (> 0). Infected people contribute to the concentration of vibrios at a rate η (> 0). The pathogen population is generated at a rateμ (> 0) and the cholera pathogen has a natural death rateμ (> 0) in the aquatic environment, which in this case, is the set of untreated water consumed by the population. According to Islam [16], we know that Vibrio cholerae population decay does not necessarily imply death but also the transition towards a non-culturable state. Hence, we assumeμ >μ, and vibrios have a net death rate We assume that susceptible people becomes infected at a rate βλ(B), where β is the rate of contact with untreated water and λ(B) is the probability of such person to catch cholera. And λ(B) depends on the concentration of Vibrio cholerae, B, which is given by the dose-response function B K+B , where K is the concentration of V. cholera in water that yields 50% chance of catching cholera [6]. We also assume that when a susceptible individual is infected, there is a time τ (> 0) during which the infectious individual develops, and only after that time the infected individual becomes the removed one [8,31]. The time τ is called infection time. The probability that an individual remains in the infectious period at least t time units before developing Cholera is given by a step function with value 1 for 0 ≤ t ≤ τ and value zero for t > τ . The probability that an individual in the infectious period time t units has survived to develop cholera is e −(µ1+δ)τ .
The model is given in the following: (1.1) The second equation of system (1.1) can be rewritten by formally integrating the delay differential equations for I(t) as follows: Furthermore, from the last two equations of system (1.1), we can obtain and The initial conditions for system (1.1) take the form In this paper, we will discuss the dynamical behavior of system (1.1). The remainder of this paper is originated as follows. In the next section, we present some basic results, for example, the positive invariance of system (1.1), the existence of equilibria, the boundedness of solutions. In Section 3, we derive the local and global stability of the disease-free equilibrium. A set of conditions which assure the permanence of the system (1.1) are obtained in Section 4. In Section 5, we derive the local and global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results in Section 6. A brief discussion is given in Section 7 to conclude this work.

Some basic results
In this section, we present some basic results, such as the positive invariance of system (1.1), the existence of equilibria, the boundedness of solutions. It is important to show positivity and boundedness for the system (1.1) as they represent populations. Positivity implies that populations survives and boundedness may be interpreted as a natural restriction to growth as a consequence of limited resources.

Positivity
Since all the state variables (i.e. S, I, R, B) in system (1.1); susceptibles, infectious, recovered and the v. cholerae population are number densities, then they are required to be non-negative. Hence, we need to show the positivity of solutions of the system (1.1).
Clearly, T > 0, and if T < ∞ then one of S(t), I(t), R(t), B(t) must be zero. We have from system Thus, From the second equation of system (1.1), we have which is strictly positive in [0, ϵ] for small ϵ > 0. From (1.3) and (1.4), we can conclude that R(t) > 0 and B(t) > 0. Thus we can conclude that solutions of system (1.1) remain positive for all t > 0. Proof. From the first equations of system (1.1), we can obtain

Boundedness
for t ≥ 0 with initial condition S(0) + I(0) + R(0) > 0. Thus, we can get From the last equation of system (1.1), we can get We can obtain that We complete the proof of the theorem.

Equilibria
It is easy to see that the system (1.1) always exists a disease-free equilibrium E 0 (S 0 , 0, 0, 0) (where S 0 = A µ1 ), which exists for all values of the parameters. In order to consider the existence of the endemic equilibrium E * , we need define the basic reproduction number R 0 according to the definition in [7], extended to a delay epidemic model.

Characteristic equation
LetĒ(S,Ī,R,B) be arbitrarily equilibrium of system (1.1). To study the locally asymptotic stability of the steady statesĒ, let us define Then the linearized system of (1.1) atĒ is given by (2.1) We then express system (2.1) in matrix form as follows: where A 1 and A 2 are 4 × 4 matrices given by The characteristic equation of system (1.1) is given by where I is the 4 × 4 identity matrix.

Stability of disease-free equilibrium E 0
In this section, we will discuss the local and global stability of the disease-free equilibrium E 0 of system (1.1), respectively. For the disease-free equilibrium E 0 , (2.2) becomes Clearly, (3.1) always has two negative roots λ 1 = λ 2 = −µ 1 . Other roots of (3.1) are determined by the equation it is apparent that (3.3) has two negative roots −µ 2 and −µ 1 − δ if R 0 < 1. Now, when R 0 < 1, we need to show that all the eigenvalues in f 1 (λ, 0) = 0 have negative real parts. First, note that any eigenvalue in f 1 (λ, 0) = 0 satisfies which is equivalent to Assume that there exists a zero in Hence, all the eigenvalues in (3.1) have negative real parts, implying E 0 is locally asymptotically stable.
Then we have the following theorem.
In the following, we will consider the attraction of the disease-free equilibrium for system (1.1). In order to consider the attraction of the equilibria and the permanence of the solutions of system (1.1), we need the following important lemmas.  [11] Let {f n } n∈N0 be a measurable sequence of non-negative function defied on a measurable set Ω. If there exists a non-negative integrable function g defined on Ω and such that f n ≤ g on Ω for all n, then Proof. From the last equation of system (1.1), we can obtain Similarly, we can get lim inf be any positive solution of system (1.1) with initial conditions (1.5). It follows from the first equations of system (1.1) that It follows from the second equation of system (1.1) that By Lemmas 3.2 and 3.3, we can obtain (3.5) Noting that R 0 < 1, it follows from (3.6) that For ε > 0 sufficiently small, we obtain from the third equation of system (1.1) that, for t > T 2 + τ , Hence, .
From the first equation of system (1.1), for t > T 2 , we can obtain By comparison it follows that Therefore, This completes the proof. From Theorems 3.1 and 3.2, we can obtain the following result.
Proof. Let If R 0 > 1, then it is easy to show that, for λ real, Hence, (3.2) has a positive root at least. Accordingly, the disease-free equilibrium E 0 is unstable if R 0 > 1.

Permanence
In this section, we will investigate the permanence of system (1.1).
A standard comparison argument shows that Hence, for ε > 0 sufficiently small, there is a T 1 > 0 such that if t > T 1 , It follows from the last equation of system (1.1) that, for t > T 1 , which yields Since the above inequality holds for arbitrary ε > 0 sufficiently small, it follows that Hence, for ε > 0 sufficiently small there is a It follows from the first equation of system (1.1) that, for t > T 2 , Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that Choose positive constants B 0 large enough and d small enough such that We now claim that if R 0 > 1, there does not exist anyt 1 > 0 such that B(t) ≤ B 0 for all t >t 1 . Otherwise, there exists a t 0 such that (4.4) It follows from the first equation of system (1.1) and (4.4) that, for t > t 0 , Thus, for t ≥ t 0 + d, we have For t > 0, define a differentiable function Calculating the derivative of V (t) along solutions of system (1.1) we derive that It follows from (4.4), (4.5) and (4.7) that The second equation of (1.1) can be rewritten as It follows from (4.9) that From the last equation of system (1.1) we can get Hence, Since B 0 l ≤ B 0 , we derive from (4.2) that (4.11) We deduce from (4.10) and (4.11) that B(t * ) > B 0 l , which is a contradiction. This proves the claim. Therefore, we obtain from (4.3) and (4.8) that On the other hand, it follow from (4.1) and (4.6) that A contradiction occurs. Hence, the claim is proved.
By the claim, we are left to consider two possibilities. First, B(t) ≥ B 0 for all t sufficiently large. Second, B(t) oscillates about B 0 for all t sufficiently large.
For the second case, we assume that where t 1 sufficiently large such that S(t) ≥ v 1 − ε for ε > 0 being sufficiently small. Since B(t) is uniformly continuous, there is a 0 < T < τ (independent of the choice of t 1 ) such that B(t) > B 0 2 for t 1 < t < 1 +T. If γ ≤ T , there is nothing to prove. Let us consider the case that T < γ <≤ τ . For t 1 + T < t ≤ t 1 + γ, we have (4.12) From the last equation of system (1.1), we have Hence, We get I(t) ≥ I 1 for t ∈ [t 1 , t 1 + γ]. For t ∈ (t 1 + τ, t 1 + 3τ 2 ], from (4.12), we have (4.14) From the last equation of system (1.1) and (4.14), we have Hence, For t ∈ (t 1 + τ, t 1 + 3τ 2 ], from (4.15), we have Hence, Continuing the process above, we derive that is the minimum integer being greater than or equal to x), and B i (1 ≤ i ≤ 2k − 1) are defined in (4.13), (4.15), (4.17) and (4.18). Obviously From the last equation of system (1.1) and (4.19), we have Hence, Noting v 2 ≤ B 0 , we derive from (4.3) that Hence, we can deduce from (4.20) and (4.21) that B(t 1 + d + T * ) > v 2 , which is a contradiction. Therefore, we have that Since this kind of interval [t 1 + d, t 1 + γ] is chosen in an arbitrary way (we only need t 1 to be large), we conclude that B(t) ≥ v 2 for all t sufficiently large in the second case.
For ε > 0 sufficiently small, it follows from the third equation of system (1.1) that, for t > T 4 , Since the inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that From the proof we can see that lim inf t→+∞ This completes the proof.

(5.6)
Let z = ω 2 . Then (5.5) becomes In order to consider the existence of positive zeros of the above third degree polynomials, we need the following lemma.

Proof. Let (S(t), I(t), R(t), B(t))
Hence, for ε > 0 sufficiently small there is a It follows from the second equation of system (1.1) that We derive from (5.9) that lim sup t→+∞ . (5.10) From (5.10), we can obtain Hence, for ε > 0 sufficiently small there is a T 2 > T 1 > 0 such that if t > T 2 , I(t) ≤ M I 1 + ε. From the last equation of (1.1), we can get Hence, for ε > 0 sufficiently small there is a T 3 > T 2 > 0 such that if t > T 3 , B(t) ≤ M B 1 + ε. It follows from the third equation of system (1.1) that, for t > T 2 + τ, Hence, Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that Therefore, for ε > 0 sufficiently small there is a T 4 > T 3 + τ > 0 such that if t > T 4 , It follows from the first equation of system (1.1) that, for t > T 4 By comparison we derive that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that lim inf ].
Hence, for ε > 0 sufficiently small there is a . (5.11) By Theorem 4.1, we see that if R 0 > 1, lim inf t→+∞ I(t) > 0. Therefore, we derive from Hence, for ε > 0 sufficiently small there is a T 6 > T 5 such that if t > T 6 , From the last equation of (1.1), we can get Hence, for ε > 0 sufficiently small there is a T 7 > T 6 > 0 such that if t > T 7 , Clearly, we have that Noting that M S n > S * and µ 1 > β, it follows from (5.13) that Therefore, the sequence M S n is monotonically non-increasing. Hence, lim Hence, the endemic equilibrium E * is globally attractive. From Theorems 5.1 and 5.2, we can get the following result. Theorem 5.3. If R 0 > 1, µ 1 > β, ϑ 3 ≥ 0 and ∆ < 0 hold true, then the endemic equilibrium E * is globally asymptotically stable.

Numerical simulations
In the previous sections, we introduced the analytical tools proposed and used them for a qualitative analysis of the system obtaining some results about the dynamics of the system. In this section, we perform a numerical analysis of the model based on the previous results. In order to illustrate feasibility of the main results of Theorems 5.2 and 5.3, we perform some numerical simulations by using the software Matlab 7.0.
Our model involves 8 parameters, including the delay τ . We choose a set of parameters which are listed in Table 1. In order to support our results about instability switches, we computed the numerical solution of system (1.1) for different Natural death rate of human 5.48 × 10 −5 /day [21] µ Rate of loss of V. cholera 1.06 /day [6] µ Growth rate of V. cholera 0.73/day [6] δ Disease-induced death rate 0.015/day [12] K Concentration of V. cholera in water 9.5 × 10 6 cells/L [12] τ Infectious period Varied Assumed values of τ . Since the zeros of τ 0 occur at τ 01 = 9.1974 and τ 02 = 11.0892, we considered the values τ = 9 in the stability region, τ = 9.5 in the instability region and τ = 12 again in the stability region. In the first and third cases ( Fig. 1 and Fig. 3), the solution shows dumped oscillations revealing the asymptotic stability of equilibrium E * , whereas in the second case (Fig. 2) the oscillations are sustained, thus confirming that E * is unstable.
Although the conditions of Theorem 5.3 (especially, µ 1 > β) are not satisfied, the endemic equilibrium E * will be asymptotically stable by numerical simulations (Fig. 1 and Fig. 3). Therefore, we can affirm that the conditions of Theorem 5.3 have room for improvement.

Discussion
In this paper, we formulate a delay cholera epidemic model with a constant infectious period. The model equations are delay differential equations with delay dependent parameters. We discuss the global attractivity of the disease-free equilibrium and the endemic equilibrium of system (1.1) by using iterative schemes and comparison principles, respectively. We also present the permanence of system (1.1). By using the geometric stability switch criteria in delay differential systems with delay dependent parameters, we obtain that there could exist stability switch about the endemic equilibrium. And we have confirmed it via the numerical simulations. We also find that the endemic equilibrium E * will be asymptotically stable by numerical simulations although the conditions of Theorem 5.2 are not satisfied. Perhaps, we may prove the globally asymptotical stability of the endemic equilibrium E * by using the method of constructing the appropriate Lyapunov function. We leave it in the future.

Kµ1µ2(µ1+δ)
> 0 and ∂I * ∂τ = Aβη(µ1+δ)e −(µ 1 +δ)τ µ1r+βr+µ1δ+βµ1+µ 2 1 +βδ > 0. Therefore, the number of secondary infections will increase when the infectious period increases. And the number of the infectives will increase when the infectious period increases. We can conclude that prolonging infectious period by medical interventions will have negative effect. The infectious period, plays a significant role in cholera surveillance, prevention, and control [1].   The long infectious period for diseases can give individuals a false sense of security. Cholera with long infectious periods are more likely to spread extensively. Hence, we should shorten infectious periods to intervene cholera. In [4], the authors considered an age-of-infection cholera model. Under some assumptions, the global dynamics of a PDE cholera model was shown to be determined completely by the basic reproduction number R 0 . The disease died out if R 0 was below or at the threshold value 1 and otherwise the disease persists. The global stability of the disease-free and endemic equilibria was proved by the construction of Lyapunov functionals. Our model is different from the one proposed in [4], which incorporates simultaneously the age-of-infection structure of individuals and the age structure of pathogen with infectivities given by kernel functions.
Lastly, we can improve the cholera model by several ways. For example, we can consider a cholera model with both constant latency time and constant infectious period. In this case, we may add an equation for the dynamics of the latented populations. We may also consider the vaccination effort of the cholera. And we can add an equation for the dynamics of the vaccinated populations. All of them will be left in the future.