Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is constructed. We establish that the global dynamics are completely determined by the basic reproduction number R0. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable. If R0 > 1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also given to explain our conclusions.


Introduction
Epidemiological models describing a directly transmitted viral or bacterial agent in a closed population and consisting of susceptibles S , infectives I , and recovers R were considered by Kermack and Mckendrick 1927 .For some diseases, such as influenza and tuberculosis, on adequate contact with an infectious individual, a susceptible becomes exposed for a while, that is, infected but not yet infectious.Thus it is realistic to introduce a latent compartment usually denoted by E , leading to an SEIR model 1 .Such type of models has been widely discussed in recent decades 2-8 .
Vaccination is important for the elimination of infectious disease.Usually, the vaccination process are different schedules for different disease and vaccines.For some disease, such as hepatitis B virus infection 9 , doses should be taken by vaccinees several times and there must be some fixed time intervals between two doses.Considering the time for vaccines to obtain immunity and possibility to be infected before vaccination, Liu et al. 10 studied the vaccination effects via two SVIR models according to continuous vaccination strategy and pulse vaccination strategy PVS , respectively.Li et al. 11 considered that the vaccine is available for both the susceptibles and the newborns, and the immunity of the vaccinated individuals is temporary and that the efficiency of vaccine is not complete.
In 12-15 , it is assumed that the treated individuals have partial immunity and can be infected through contacts with infectious individuals.Yang et al. 15 incorporated the incomplete treatment into TB epidemic model with treatment, it is assumed that the being treated individuals are kept at an isolated environment, therefore, individuals in treatment compartment are not able to infect others or be infected.Since individual's symptoms of TB may disappear after being treated, but a few of tubercle bacillus may still be left in the body of the treated individual 16, 17 , then the treated individual may still be a TB carrier and become latent or may enter the infectious compartment for treatment failure 18 .
Motivated by these works, in this paper, we consider an SVEIT epidemiological model with varying infectivity.The model assumes that the vaccine is available for both the susceptibles and the newborns and the immunity of the vaccinated individuals is temporary, and that the efficiency of vaccine is not complete.And we also incorporate the incomplete treatment into the epidemic model.
The organization of this paper is as follows.In the next section, the epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is formulated.In Section 3, the basic reproduction number and the existence of equilibria are investigated.The global stability of the disease-free and endemic equilibria are proved in Section 4, and some numerical simulations are given in Section 5.In the last section, we give some brief discussions.

The Model Formulation
In this section, we introduce an epidemic model with incomplete treatment and vaccination.The total population is partitioned into five compartments: the susceptible compartment S , the vaccinated compartment V , the latent compartment E , the infectious compartment I , and the treatment compartment T .The population flow among those compartments is shown in the following diagram Figure 1 .
The schematic diagram leads to the following system of ordinary differential equations:

2.1
Here, μA is the birth rate of the population; μ is the natural death rate of the population; q 0 < q < 1 is the fraction of the unvaccinated newborns, 1 − q is the fraction of the vaccinated newborns; p is the vaccinating rate coefficient for the susceptible individuals; ε is the rate coefficient of losing the immunity from the vaccination.β and σβ 0 < σ < 1 are the transmission coefficient of the infection for the susceptible and vaccinated individuals from the infectious individuals, where 0 < σ < 1 shows that the efficiency of the vaccine is not complete 100% ; γ is the rate coefficient of transfer from the latent compartment to the infectious one; ξ is the percapita treatment rate for the infectious individuals; δ is the rate coefficient at which a treated individual leaves compartment T ; α 1 and α 2 are the disease-induced death rate coefficients for individuals in compartments I and T , respectively; k 0 ≤ k ≤ 1 is the fraction of the drug-resistant individuals in the treated individuals.k reflects the failure of treatment, where k 0 means that all the treated individuals will become latent, while k 1 means that the treatment fails and all the treated individuals will still be infectious.
It is important to show positivity and boundedness for the system 2.1 as they represent populations.Firstly, we present the positivity of the solutions.System 2.1 can be put into the matrix form where X S, V, E, I, T T ∈ R 5 and G X is given by It is easy to check that Due to Lemma 2 in 19 , any solution of 2.1 is such that X t ∈ R 5 for all t ≥ 0. Summing equations in 2.1 yields is positively invariant for 2.1 .Therefore, we will consider the global stability of 2.1 on the set Ω.

The Basic Reproduction Number and Existence of Equilibria
The model has a disease-free equilibrium P 0 S 0 , V 0 , 0, 0, 0 , where In the following, the basic reproduction number of system 2.1 will be obtained by the next generation matrix method formulated in 20 .
Let x E, I, T, S, V T , then system 2.1 can be written as where The Jacobian matrices of F x and V x at the disease-free equilibrium P 0 are, respectively, where The model reproduction number, denoted by R 0 is thus given by The endemic equilibrium P * S * , V * , E * , I * , T * of system 2.1 is determined by equations

3.7
The first two equations in 3.7 lead to From the last equation in 3.7 , we have 3.9 Substituting 3.9 into the fourth equation in 3.7 gives

3.12
Direct calculation shows where then function H I is decreasing for I > 0. Since p μ βI μ ε βσI −pε > βI βσI ε σp μ , and it follows from σ ≤ 1 that βσI ε qμ σ p 1 − q μ < βσI ε σp μ, then Thus, Therefore, by the monotonicity of function H I , for 3.12 there exists a unique positive root in the interval 0, A when R 0 > 1; there is no positive root in the interval 0, A when R 0 ≤ 1.
We summarize this result in Theorem 3.1.

Global Stability of Equilibria
Theorem 4.1.For system 2.1 , the disease-free equilibrium P 0 is globally stable if R 0 ≤ 1; the endemic equilibrium P * is globally stable if R 0 > 1.

4.2
Define the Lyapunov function

4.3
The derivative of V 1 is given by where

4.6
Applying 4.1 to function F x, y yields

The Proof Global Stability of the Endemic Equilibrium
For the endemic equilibrium P * S * , V * , E * , I * , T * , S * , V * , E * , I * , and T * satisfy equations qμA − μ p S − βSI εV 0, which will be used many times in the following inference.
By applying 4.8 and denoting we have

4.10
Define the Lyapunov function

4.11
The derivative of V 2 is given by

4.12
Following 11 , βI * S * σV * μ γ E * and we have F x, y, z, u ≤ 0 for x, y, z, u > 0 and F x, y, z, u 0 if and only if x y 1 and z u.Therefore, V 2 ≤ 0 for x, y, z, u, w > 0 and V 2 0 if and only if x y 1, z u w, and the maximum invariant set of system 2.1 on the set { x, y, z, u, w : V 2 0} is the singleton 1, 1, 1, 1, 1 .Thus, for system 2.1 , the endemic equilibrium P * is globally asymptotically stable if R 0 > 1 by LaSalle invariance principle 21 .

Discussion
We have formulated an epidemic model with incomplete treatment and vaccination and investigated their dynamical behaviors.By means of the next-generation matrix, we obtained their basic reproduction number, R 0 , which play a crucial role.By constructing Lyapunov function, we proved the global stability of their equilibria: when the basic reproduction number is less than or equal to one, all solutions converge to the disease-free equilibrium, that is, the disease dies out eventually; when the basic reproduction number exceeds one,   the unique endemic equilibrium is globally stable, that is, the disease will persist in the population and the number of infected individuals tends to a positive constant.
For system 2.1 , k reflects the failure of treatment.Direct calculation shows that ∂R 0 /∂k > 0, then decreasing the treatment failure coefficient is helpful to reduce epidemic infection.The realization of decreasing k mainly depends on decreasing the appearance of drug resistance.On the other hand, ∂R 0 /∂ξ < 0 implies that increasing ξ has positive effect on epidemic control, since increasing ξ is to shorten the period of the infectious compartment in the nonisolated environment, that is, to start treating as soon as possible.This measure is effective to control the spread of the epidemic.Figure 5 shows the relation among the basic reproduction number R 0 , the treatment failure coefficient k, and the per-capita treatment ξ.

Figure 1 :
Figure 1: The transfer diagram for the model 2.1 .
and I * is the unique positive root of equation H I 0.