Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

Abstract

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.

1. 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.

2. 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).

Figure 1 

The transfer diagram for the model (2.1).

The schematic diagram leads to the following system of ordinary differential equations:𝑆^′𝑉=𝑞𝜇𝐴−𝜇+𝑝𝑆−𝛽𝑆𝐼+𝜀𝑉,_′=𝐸1−𝑞𝜇𝐴+𝑝𝑆−𝛽𝜎𝑉𝐼−𝜇+𝜀𝑉,_′()−()𝐼=𝛽𝐼𝑆+𝜎𝑉𝜇+𝛾𝐸+1−𝑘𝛿𝑇,_′=𝛾𝐸−𝜇+𝛼₁𝑇+𝜉𝐼+𝑘𝛿𝑇,_′=𝜉𝐼−𝜇+𝛼₂+𝛿𝑇. (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; 𝛼₁ and 𝛼₂ 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 form𝑋^′=𝐺(𝑋), (2.2) where 𝑋=(𝑆,𝑉,𝐸,𝐼,𝑇)^𝑇∈𝑅⁵ and 𝐺(𝑋) is given by𝐺(𝑋)=𝐺₁(𝑋)𝐺₂(𝑋)𝐺₃(𝑋)𝐺₄(𝑋)𝐺₅(𝑋)=𝑉()−()𝑞𝜇𝐴−𝜇+𝑝𝑆−𝛽𝑆𝐼+𝜀𝑉1−𝑞𝜇𝐴+𝑝𝑆−𝛽𝜎𝑉𝐼−𝜇+𝜀𝛽𝐼𝑆+𝜎𝑉𝜇+𝛾𝐸+1−𝑘𝛿𝑇𝛾𝐸−𝜇+𝛼₁+𝜉𝐼+𝑘𝛿𝑇𝜉𝐼−𝜇+𝛼₂+𝛿𝑇. (2.3) It is easy to check that𝐺_𝑖(𝑋)_{𝑋𝑖(𝑡)=0,𝑋𝑡∈𝐶+}≥0,𝑖=1,2,3,4,5. (2.4) Due to Lemma??2 in [19], any solution of (2.1) is such that 𝑋(𝑡)∈𝑅⁵₊ for all 𝑡≥0.

Summing equations in (2.1) yields(𝑆+𝑉+𝐸+𝐼+𝑇)^′=𝜇[𝐴−(𝑆+𝑉+𝐸+𝐼+𝑇)]−𝛼₁𝐼−𝛼₂𝑇≤𝜇[𝐴−(𝑆+𝑉+𝐸+𝐼+𝑇)], (2.5) then it follows that lim_𝑡→+∞sup[𝑆(𝑡)+𝑉(𝑡)+𝐸(𝑡)+𝐼(𝑡)+𝑇(𝑡)]≤𝐴, so the setΩ=(𝑆,𝑉,𝐸,𝐼,𝑇)∈𝑅⁵₊∶𝑆+𝑉+𝐸+𝐼+𝑇≤𝐴 (2.6) is positively invariant for (2.1). Therefore, we will consider the global stability of (2.1) on the set Ω.

3. The Basic Reproduction Number and Existence of Equilibria

The model has a disease-free equilibrium 𝑃₀(𝑆₀,𝑉₀,0,0,0), where𝑆₀=𝐴𝜀+𝑞𝜇𝑝+𝜇+𝜀,𝑉₀=𝐴𝑝+1−𝑞𝜇.𝑝+𝜇+𝜀 (3.1) 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 as𝑑𝑥(𝑥)(𝑥)𝑑𝑡=ℱ−𝒱, (3.2) whereℱ(𝑥)=()0000(𝑥)=𝛽𝐼𝑆+𝜎𝑉(),𝒱𝜇+𝛾𝐸−1−𝑘𝛿𝑇𝜇+𝛼₁+𝜉𝐼−𝛾𝐸−𝑘𝛿𝑇𝜇+𝛼₂+𝛿𝑇−𝜉𝐼𝛽𝑆𝐼+𝜇+𝑝𝑆−𝑞𝜇𝐴−𝜀𝑉𝛽𝜎𝑉𝐼+𝜇+𝜀𝑉−1−𝑞𝜇𝐴−𝑝𝑆. (3.3) The Jacobian matrices of ℱ(𝑥) and 𝒱(𝑥) at the disease-free equilibrium 𝑃₀ are, respectively,𝐷ℱ𝑃₀=𝐹_3×30𝑃00,𝐷𝒱₀=𝑉_3×3000𝛽𝑆₀0𝜇+𝑝−𝜀0𝛽𝜎𝑉₀0−𝑝𝜇+𝜀, (3.4) where𝑆𝐹=0𝛽₀+𝜎𝑉₀0()𝛿000000,𝑉=𝜇+𝛾0−1−𝑘−𝛾𝜇+𝛼₁+𝜉−𝑘𝛿0−𝜉𝜇+𝛼₂+𝛿. (3.5) The model reproduction number, denoted by 𝑅₀ is thus given by𝑅₀=𝜌𝐹𝑉⁻¹=𝛽𝛾𝜇+𝛼₂+𝛿𝑆₀+𝜎𝑉₀𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝛿𝜉(1−𝑘)𝛾+𝜇+𝛾𝑘. (3.6)

The endemic equilibrium 𝑃^∗(𝑆^∗,𝑉^∗,𝐸^∗,𝐼^∗,𝑇^∗) of system (2.1) is determined by equations()−()𝑞𝜇𝐴−𝜇+𝑝𝑆−𝛽𝑆𝐼+𝜀𝑉=0,1−𝑞𝜇𝐴+𝑝𝑆−𝛽𝜎𝑉𝐼−𝜇+𝜀𝑉=0,𝛽𝐼𝑆+𝜎𝑉𝜇+𝛾𝐸+1−𝑘𝛿𝑇=0,𝛾𝐸−𝜇+𝛼₁+𝜉𝐼+𝑘𝛿𝑇=0,𝜉𝐼−𝜇+𝛼₂+𝛿𝑇=0. (3.7) The first two equations in (3.7) lead to𝑞𝑆=𝜇𝐴𝜇+𝜀+𝛽𝜎𝐼+𝜀1−𝑞𝑝+𝜇+𝛽𝐼𝜇+𝜀+𝛽𝜎𝐼−𝑝𝜀,𝑉=𝜇𝐴1−𝑞𝑝+𝜇+𝛽𝐼+𝑝𝑞𝑝+𝜇+𝛽𝐼𝜇+𝜀+𝛽𝜎𝐼−𝑝𝜀. (3.8) From the last equation in (3.7), we have𝜉𝑇=𝜇+𝛼₂+𝛿𝐼. (3.9) Substituting (3.9) into the fourth equation in (3.7) gives𝐸=𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉𝜇+𝛼₂+𝛿𝛾𝐼. (3.10) For 𝐼≠0, substituting (3.9), (3.10) into the third equation in (3.7) gives1𝑆+𝜎𝑉=𝛽𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝜇+𝛼₂+𝛿𝛾. (3.11) Substituting (3.8) into (3.11) yields𝐻(𝐼)𝑞∶=𝜇+𝜀+𝛽𝜎𝐼+𝜀1−𝑞+𝜎1−𝑞𝑝+𝜇+𝛽𝐼+𝑝𝑞−𝑝+𝜇+𝛽𝐼𝜇+𝜀+𝛽𝜎𝐼−𝑝𝜀𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂=𝜇+𝛿𝜇𝐴𝛽𝜎𝐼+𝜀+𝑞𝜇+𝜎𝑝+1−𝑞−𝑝+𝜇+𝛽𝐼𝜇+𝜀+𝛽𝜎𝐼−𝑝𝜀𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂+𝛿𝜇𝐴=0. (3.12) Direct calculation shows𝐻^′(𝐼)𝛽=−𝛽𝜎𝐼²+2𝛽𝜎𝐼𝜀+𝑞𝜇+𝜎𝑝+1−𝑞𝜇+𝜎₀𝑝+𝜇+𝛽𝐼𝜇+𝜀+𝛽𝜎𝐼−𝑝𝜀²<0, (3.13) where𝜎₀=𝑝+𝜇𝑝+1−𝑞𝜇𝜎²+𝜎𝑝𝜀+𝑞𝜇+𝜀𝑝+1−𝑞𝜇+𝜇+𝜀𝜀+𝑞𝜇, (3.14) 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, then𝐻(𝐼)<1−𝛽𝐼𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂+𝛿𝜇𝐴. (3.15) Thus,𝐻(0)=𝜀+𝑞𝜇+𝜎𝑝+1−𝑞𝜇𝜇−𝑝+𝜇+𝜀𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂=+𝛿𝜇𝐴𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂𝑅+𝛿𝜇𝐴⁰,𝐻(𝐴)<1−1−𝛽𝐴𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉−(1−𝑘)𝛿𝜉𝛾𝛽𝛾𝜇+𝛼₂=+𝛿𝜇𝐴𝜇+𝛼₂+𝛿𝛾𝜇−𝜇+𝛾𝜇+𝛼₁+𝜉+𝛿𝜉𝛾+𝜇𝑘𝛽𝐴𝛾𝜇𝜇+𝛼₂=−+𝛿𝜇+𝛼₂+𝛿𝜇𝜇+𝛼₁+𝜉+𝛾𝛼₁+𝜉+𝛿𝜉𝛾+𝜇𝑘𝛽𝐴𝛾𝜇𝜇+𝛼₂+𝛿=−𝜇+𝛼₂+𝛿𝜇𝜇+𝛼₁+𝛾𝛼₁+𝜉𝜇+𝛾𝜇+𝛼₂+𝜇𝛿𝜉(1−𝑘)𝛽𝐴𝛾𝜇𝜇+𝛼₂.+𝛿<0. (3.16) Therefore, by the monotonicity of function 𝐻(𝐼), for (3.12) there exists a unique positive root in the interval (0,𝐴) when 𝑅₀>1; there is no positive root in the interval (0,𝐴) when 𝑅₀≤1. We summarize this result in Theorem 3.1.

Theorem 3.1. For system (2.1), there is always the disease-free equilibrium 𝑃₀(𝑆₀,𝑉₀,0,0,0). When 𝑅₀>1, besides the disease-free equilibrium 𝑃₀, system (2.1) also has a unique endemic equilibrium 𝑃^∗(𝑆^∗,𝑉^∗,𝐸^∗,𝐼^∗,𝑇^∗), where 𝑆^∗=𝜇𝐴𝑞𝜇+𝜀+𝛽𝜎𝐼^∗+𝜀1−𝑞𝑝+𝜇+𝛽𝐼^∗𝜇+𝜀+𝛽𝜎𝐼^∗,𝑉−𝑝𝜀_∗=𝜇𝐴1−𝑞𝑝+𝜇+𝛽𝐼^∗+𝑝𝑞𝑝+𝜇+𝛽𝐼^∗𝜇+𝜀+𝛽𝜎𝐼^∗,𝐸−𝑝𝜀_∗=𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝑘𝛿𝜉𝜇+𝛼₂𝛾𝐼+𝛿^∗,𝑇^∗=𝜉𝜇+𝛼₂𝐼+𝛿^∗, (3.17) and 𝐼^∗ is the unique positive root of equation 𝐻(𝐼)=0.

4. Global Stability of Equilibria

Theorem 4.1. For system (2.1), the disease-free equilibrium 𝑃₀ is globally stable if 𝑅₀≤1; the endemic equilibrium 𝑃^∗ is globally stable if 𝑅₀>1.

4.1. The Proof Global Stability of the Disease-Free Equilibrium

For the disease-free equilibrium 𝑃₀(𝑆₀,𝑉₀,0,0,0), 𝑆₀ and 𝑉₀ satisfy equations𝑞𝜇𝐴−𝜇+𝑝𝑆−𝛽𝑆𝐼+𝜀𝑉=0,1−𝑞𝜇𝐴+𝑝𝑆−𝛽𝜎𝑉𝐼−𝜇+𝜀𝑉=0, (4.1) then (2.1) can be rewritten as follows:𝑆^′1=𝑆𝑞𝜇𝐴𝑆−1𝑆₀𝑉+𝜀𝑆−𝑉₀𝑆₀,𝑉−𝛽𝐼_′1=𝑉1−𝑞𝜇𝐴𝑉−1𝑉₀𝑉+𝑝𝑆−𝑉₀𝑆₀,𝑆−𝛽𝜎𝐼𝐸′=𝛽𝐼₀+𝜎𝑉₀+𝑆−𝑆₀+𝜎𝑉−𝑉₀−()𝐼𝜇+𝛾𝐸+1−𝑘𝛿𝑇,_′=𝛾𝐸−𝜇+𝛼₁𝑇+𝜉𝐼+𝑘𝛿𝑇,_′=𝜉𝐼−𝜇+𝛼₂+𝛿𝑇. (4.2) Define the Lyapunov function𝑉₁=𝑆−𝑆₀−𝑆₀𝑆ln𝑆₀+𝑉−𝑉₀−𝑉₀𝑉ln𝑉₀+𝐸+𝜇+𝛾𝛾𝛿(𝐼+𝜇+𝛾𝑘+𝛾1−𝑘)𝛾𝜇+𝛼₂+𝛿𝑇. (4.3) The derivative of 𝑉₁ is given by𝑉^′₁=𝑆−𝑆₀1𝑞𝜇𝐴𝑆−1𝑆₀𝑉+𝜀𝑆−𝑉₀𝑆₀+−𝛽𝐼𝑉−𝑉₀11−𝑞𝜇𝐴𝑉−1𝑉₀𝑉+𝑝𝑆−𝑉₀𝑆₀𝑆−𝛽𝜎𝐼+𝛽𝐼₀+𝜎𝑉₀+𝑆−𝑆₀+𝜎𝑉−𝑉₀−()+𝜇+𝛾𝐸+1−𝑘𝛿𝑇𝜇+𝛾𝛾𝛾𝐸−𝜇+𝛼₁+𝛿(+𝜉𝐼+𝑘𝛿𝑇𝜇+𝛾𝑘+𝛾1−𝑘)𝛾𝜇+𝛼₂+𝛿𝜉𝐼−𝜇+𝛼₂𝑇=+𝛿𝜇+𝛾𝜇+𝛼₁+𝜉𝜇+𝛼₂+𝛿−𝛿𝜉(1−𝑘)𝛾+𝜇+𝛾𝑘𝛾𝜇+𝛼₂𝑅+𝛿⁰−1𝐼+𝐹(𝑆,𝐼), (4.4) where𝐹(𝑆,𝐼)=𝜇𝑞𝐴𝑆−𝑆₀1𝑆−1𝑆₀+1−𝑞𝜇𝐴𝑉−𝑉₀1𝑉−1𝑉₀+𝜀𝑆−𝑆₀𝑉𝑆−𝑉₀𝑆₀+𝑝𝑉−𝑉₀𝑆𝑉−𝑆₀𝑉₀. (4.5) Denote 𝑥=𝑆/𝑆₀,𝑦=𝑉/𝑉₀; then𝐹()()1𝑆,𝐼=𝜇𝑞𝐴𝑥−1𝑥+1−11−𝑞𝜇𝐴𝑦−1𝑦−1+𝜀𝑉₀()𝑦𝑥−1𝑥−1+𝑝𝑆₀𝑥𝑦−1𝑦−1=∶𝐹𝑥,𝑦. (4.6) Applying (4.1) to function 𝐹(𝑥,𝑦) yields𝐹𝑥,𝑦=2𝜇𝑞𝐴+2𝜇1−𝑞𝐴+𝜀𝑉₀+𝑝𝑆₀−𝜇𝑆₀1𝑥−𝜇𝑞𝐴𝑥−𝜇𝑉₀𝑦𝐴1−𝜇1−𝑞𝑦−𝜀𝑉₀𝑦𝑥−𝑝𝑆₀𝑥𝑦. (4.7) By [11], we have 𝐹(𝑥,𝑦)≤0 for 𝑥,𝑦>0 and 𝐹(𝑥,𝑦)=0 if and only if 𝑥=𝑦=1. Since (𝜇+𝛾)(𝜇+𝛼₁+𝜉)(𝜇+𝛼₂+𝛿)>𝛿𝜉[(1−𝑘)𝛾+(𝜇+𝛾)𝑘] and 𝑅₀≤1, then 𝑉^′₁≤0. It follows from LaSalle invariance principle [21] that the disease-free equilibrium 𝑃₀ is globally asymptotically stable when 𝑅₀≤1.

4.2. 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+𝑝𝑆−𝛽𝑆𝐼+𝜀𝑉=0,1−𝑞𝜇𝐴+𝑝𝑆−𝛽𝜎𝑉𝐼−𝜇+𝜀𝑉=0,𝛽𝐼𝑆+𝜎𝑉𝜇+𝛾𝐸+1−𝑘𝛿𝑇=0,𝛾𝐸−𝜇+𝛼₁+𝜉𝐼+𝑘𝛿𝑇=0,𝜉𝐼−𝜇+𝛼₂+𝛿𝑇=0, (4.8) which will be used many times in the following inference.

By applying (4.8) and denoting𝑆𝑆^∗𝑉=𝑥,𝑉^∗𝐸=𝑦,𝐸^∗𝐼=𝑧,𝐼^∗𝑇=𝑢,𝑇^∗=𝑤, (4.9) we have𝑥^′=𝑥𝑞𝜇𝐴𝑆^∗1𝑥−1−𝛽𝐼^∗(𝑢−1)+𝜀𝑉^∗𝑆^∗𝑦𝑥,𝑦−1_′=𝑦1−𝑞𝜇𝐴𝑉^∗1𝑦−1+𝑝𝑆^∗𝑉^∗𝑥𝑦−1−𝛽𝜎𝐼^∗(,𝑧𝑢−1)_′=𝑧𝛽𝐼^∗𝐸^∗𝑆^∗𝑥𝑢𝑧−1+𝜎𝑉^∗𝑦𝑢𝑧()𝛿𝑇−1+𝑧1−𝑘^∗𝐸^∗𝑤𝑧,𝑢−1_′𝛾𝐸=𝑢^∗𝐼^∗𝑧𝑢𝑇−1+𝑘𝛿^∗𝐼^∗𝑤𝑢,𝑤−1_′=𝑤𝜉𝐼^∗𝑇^∗𝑢𝑤−1. (4.10) Define the Lyapunov function𝑉₂=𝑆^∗(𝑥−1−ln𝑥)+𝑉^∗𝑦−1−ln𝑦+𝐸^∗(𝑧−1−ln𝑧)+𝜇+𝛾𝛾𝐼^∗()+𝛿(𝑢−1−ln𝑢𝜇+𝛾𝑘+𝛾1−𝑘)𝛾𝜇+𝛼₂𝑇+𝛿^∗(𝑤−1−ln𝑤). (4.11)

The derivative of 𝑉₂ is given by 𝑉^′₂=𝑆^∗𝑥−1𝑥𝑥^′+𝑉^∗𝑦−1𝑦𝑦^′+𝐸^∗𝑧−1𝑧𝑧^′+𝜇+𝛾𝛾𝐼^∗𝑢−1𝑢𝑢^′+𝛿𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝜇+𝛼₂𝑇+𝛿^∗𝑤−1𝑤𝑤^′=(1𝑥−1)𝑞𝜇𝐴𝑥−1−𝛽𝑆^∗𝐼^∗(𝑢−1)+𝜀𝑉^∗𝑦𝑥+1−1𝑦−11−𝑞𝜇𝐴𝑦−1+𝑝𝑆^∗𝑥𝑦−1−𝛽𝜎𝑉^∗𝐼^∗(𝑢−1)+𝛽𝐼^∗(𝑧−1)𝑆^∗𝑥𝑢𝑧−1+𝜎𝑉^∗𝑦𝑢𝑧−1+(𝑧−1)(1−𝑘)𝛿𝑇^∗𝑤𝑧+−1𝜇+𝛾𝛾(𝑢−1)𝛾𝐸^∗𝑧𝑢−1+𝑘𝛿𝑇^∗𝑤𝑢+𝛿(−1𝜇+𝛾𝑘+𝛾1−𝑘)𝛾𝜇+𝛼₂+𝛿𝜉𝐼^∗(𝑢𝑤−1)𝑤()1−1=𝑞𝜇𝐴𝑥−1𝑥−1−𝛽𝑆^∗𝐼^∗(𝑥−1)(𝑢−1)+𝜀𝑉^∗()𝑦𝑥−1𝑥+1−11−𝑞𝜇𝐴𝑦−1𝑦−1+𝑝𝑆^∗𝑥𝑦−1𝑦−1−𝛽𝜎𝑉^∗𝐼^∗𝑦−1(𝑢−1)+𝛽𝑆^∗𝐼^∗(𝑧−1)𝑥𝑢𝑧−1+𝛽𝜎𝑉^∗𝐼^∗(𝑧−1)𝑦𝑢𝑧−1+(1−𝑘)𝛿𝑇^∗()𝑤𝑧−1𝑧+𝐸−1𝜇+𝛾^∗()𝑧𝑢−1𝑢−1+𝜇+𝛾𝛾𝑘𝛿𝑇^∗()𝑤𝑢−1𝑢+(−1𝜇+𝛾𝑘+𝛾1−𝑘)𝛿𝑇^∗𝛾×()𝑢𝑤−1𝑤1−1=𝑞𝜇𝐴2−𝑥−𝑥+11−𝑞𝜇𝐴2−𝑦−𝑦−𝛽𝑆^∗𝐼^∗(𝑥𝑢−𝑥−𝑢+1)+𝜀𝑉^∗𝑦𝑦−𝑥−𝑥+1+𝑝𝑆^∗𝑥𝑥−𝑦−𝑦+1−𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢−𝑦−𝑢+1+𝛽𝑆^∗𝐼^∗𝑥𝑢−𝑧−𝑥𝑢𝑧+1+𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢−𝑧−𝑦𝑢𝑧+1+(1−𝑘)𝛿𝑇^∗𝑤𝑤−𝑧−𝑧+1+𝜇+𝛾𝐸^∗𝑧𝑧−𝑢−𝑢++1𝜇+𝛾𝛾𝑘𝛿𝑇^∗𝑤𝑤−𝑢−𝑢+(+1𝜇+𝛾𝑘+𝛾1−𝑘)𝛿𝑇^∗𝛾𝑢𝑢−𝑤−𝑤+1=2𝑞𝜇𝐴+21−𝑞𝜇𝐴+𝜀𝑉^∗+𝑝𝑆^∗+(1−𝑘)𝛿𝑇^∗+𝜇+𝛾𝐸^∗+𝜇+𝛾𝛾𝑘𝛿𝑇^∗+𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝛿𝑇^∗−𝜇𝑆^∗1𝑥−𝑞𝜇𝐴𝑥−𝜇𝑉^∗1𝑦−1−𝑞𝜇𝐴𝑦+𝑧−𝛽𝑆^∗𝐼^∗−𝛽𝜎𝑉^∗𝐼^∗−(1−𝑘)𝛿𝑇^∗+𝜇+𝛾𝐸^∗+𝑢𝛽𝑆^∗𝐼^∗+𝛽𝜎𝑉^∗𝐼^∗−𝜇+𝛾𝐸^∗−𝜇+𝛾𝛾𝑘𝛿𝑇^∗+𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝛿𝑇^∗+𝑤(1−𝑘)𝛿𝑇^∗+𝜇+𝛾𝛾𝑘𝛿𝑇^∗−𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝛿𝑇^∗−𝜀𝑉^∗𝑦𝑥−𝑝𝑆^∗𝑥𝑦−𝛽𝑆^∗𝐼^∗𝑥𝑢𝑧−𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢𝑧−𝜇+𝛾𝐸^∗𝑧𝑢−(1−𝑘)𝛿𝑇^∗𝑤𝑧−𝜇+𝛾𝛾𝑘𝛿𝑇^∗𝑤𝑢−𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝛿𝑇^∗𝑢𝑤=2𝑞𝜇𝐴+21−𝑞𝜇𝐴+𝜀𝑉^∗+𝑝𝑆^∗+𝛽𝐼^∗𝑆^∗+𝜎𝑉^∗+3(1−𝑘)𝛿𝑇^∗+2𝜇+𝛾𝛾𝑘𝛿𝑇^∗−𝜇𝑆^∗𝑥1−𝑞𝜇𝐴𝑥−𝜇𝑉^∗1𝑦−1−𝑞𝜇𝐴𝑦−𝜀𝑉^∗𝑦𝑥−𝑝𝑆^∗𝑥𝑦−𝛽𝑆^∗𝐼^∗𝑥𝑢𝑧−𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢𝑧−𝛽𝐼^∗𝑆^∗+𝜎𝑉^∗𝑧𝑢−(1−𝑘)𝛿𝑇^∗𝑧𝑢−(1−𝑘)𝛿𝑇^∗𝑤𝑧−𝜇+𝛾𝛾𝑘𝛿𝑇^∗𝑤𝑢−𝜇+𝛾𝑘+𝛾(1−𝑘)𝛾𝛿𝑇^∗𝑢𝑤=2𝑞𝜇𝐴+21−𝑞𝜇𝐴+𝜀𝑉^∗+𝑝𝑆^∗+𝛽𝐼^∗𝑆^∗+𝜎𝑉^∗−𝜇𝑆^∗1𝑥−𝑞𝜇𝐴𝑥−𝜇𝑉^∗1𝑦−1−𝑞𝜇𝐴𝑦−𝜀𝑉^∗𝑦𝑥−𝑝𝑆^∗𝑥𝑦−𝛽𝑆^∗𝐼^∗𝑥𝑢𝑧−𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢𝑧−𝛽𝐼^∗𝑆^∗+𝜎𝑉^∗𝑧𝑢+(1−𝑘)𝛿𝑇^∗𝑧3−𝑢−𝑤𝑧−𝑢𝑤+𝜇+𝛾𝛾𝑘𝛿𝑇^∗𝑤2−𝑢−𝑢𝑤=∶𝐹𝑥,𝑦,𝑧,𝑢+(1−𝑘)𝛿𝑇^∗𝑧3−𝑢−𝑤𝑧−𝑢𝑤+𝜇+𝛾𝛾𝑘𝛿𝑇^∗𝑤2−𝑢−𝑢𝑤. (4.12) Following [11], 𝛽𝐼^∗(𝑆^∗+𝜎𝑉^∗)=(𝜇+𝛾)𝐸^∗ and𝐹𝑥,𝑦,𝑧,𝑢=2𝑞𝜇𝐴+21−𝑞𝜇𝐴+𝜀𝑉^∗+𝑝𝑆^∗+𝜇+𝛾𝐸^∗−𝜇𝑆^∗1𝑥−𝑞𝜇𝐴𝑥−𝜇𝑉^∗𝑦−11−𝑞𝜇𝐴𝑦−𝜀𝑉^∗𝑦𝑥−𝑝𝑆^∗𝑥𝑦−𝛽𝑆^∗𝐼^∗𝑥𝑢𝑧−𝛽𝜎𝑉^∗𝐼^∗𝑦𝑢𝑧−𝜇+𝛾𝐸^∗𝑧𝑢≤0, (4.13) 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 𝑅₀>1 by LaSalle invariance principle [21].

5. Numerical Simulation

In this section, some numerical results of system (2.1) are presented for supporting the analytic results obtained above. All of the parameter values are estimated. The model parameters are taken as: 𝐴=55.496, 𝜇=0.0143, 𝑞=0.6, 𝛽=0.05, 𝜎=0.2, 𝑝=0.05, 𝜀=0.03, 𝛾=0.00368, 𝛿=1.5, 𝛼₁=0.3, 𝛼₂=0.05. First, we choose 𝜉=0.9, 𝑘=0.3, numerical simulation gives 𝑅₀=0.8809<1, then the disease-free equilibrium 𝑃₀ is globally asymptotically stable (Figure 2). Second, we choose 𝜉=0.8, 𝑘=0.36, numerical simulation gives 𝑅₀=1, then the disease-free equilibrium 𝑃₀ is globally asymptotically stable (Figure 3). At last, we choose 𝜉=0.6, 𝑘=0.5, numerical simulation gives 𝑅₀=1.2938>1, then the endemic equilibrium 𝑃^∗ is globally asymptotically stable (Figure 4).

Figure 2 

When 𝑅0<1, the disease-free equilibrium 𝑃0 is globally asymptotically stable.

Figure 3 

When 𝑅0=1, the disease-free equilibrium 𝑃0 is globally asymptotically stable.

Figure 4 

When 𝑅0>1, the endemic equilibrium 𝑃∗ is globally asymptotically stable.

6. 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, 𝑅₀, 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 𝜕𝑅₀/𝜕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, 𝜕𝑅₀/𝜕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 𝑅₀, the treatment failure coefficient 𝑘, and the per-capita treatment rate 𝜉.

Figure 5 

The relationship among 𝑅0, 𝑘 and 𝜉.

Acknowledgments

This work was partially supported by the NNSF of China (10961018), the NSF of Gansu Province of China (1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China, and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.