Modeling and analyzing cholera transmission dynamics with vaccination age

Abstract

A new mathematical model is formulated to investigate the transmission dynamics of cholera under vaccination, with a focus on the impact of vaccination age. The basic reproduction number is derived and proved to be a sharp control threshold determining whether or not the infection is persistent. We conduct a rigorous analysis on the local and global stability properties of the equilibria in system. Meanwhile, we compare the results to those of the simplified model based on ordinary differential equations where the effects of vaccination age are not incorporated. Numerical simulation results verify our analytical findings.

1. Introduction

A large class of waterborne infectious diseases are caused by bacterial agents that spread the infections originated from the aquatic environment. Common routes of disease transmission start with human individuals contacting, or consuming food and water contaminated by urine and feces from, infected individuals. Then the pathogenic bacteria enter and grow inside the human body, produce toxins and lead to symptoms such as fever, rash, and diarrhea [51].

Typical examples of waterborne bacterial infections include dysentery, typhoid fever, Brainerd diarrhea, and cholera, all of which are easily transmitted through the ingestion of contaminated water and food. In particular, cholera is an acute intestinal disease caused by the bacterium Vibrio cholerae and characterized by severe watery diarrhea and frequent vomiting after a latency of one to two days. If not treated promptly, cholera can cause a serious imbalance of water and electrolytes inside the human body, leading to shock and even death within a few days. The infection can spread rapidly in areas with inadequate sanitation and hygiene resources, with poorly functioning municipal water distribution systems, and with limited health infrastructure and medical supplies [35]. Although an ancient disease, cholera remains a significant threat to public health for people living in poverty in developing countries and regions, with an estimate of 2–4 million cases every year [1,51]. For example, a large cholera outbreak started in the Republic of Haiti in mid-October 2010, and disease morbidity and mortality continued to increase throughout the year of 2011. As of December 2011, the Haitian government reported more than 520,000 cases, averaged at 200 new cases per day, and nearly 7000 deaths [11,12]. Additionally, a recent cholera epidemic in Yemen during 2016–2017 recorded the largest cholera outbreak in modern history, with more than 1 million cases reported by WHO at the end of 2017 [52].

In order to better understand the dynamics of cholera transmission and spread, a number of mathematical epidemic models have been published. The first indirect transmission model for cholera was proposed by Capasso and Paveri-Fontana in 1973 [15]. Codeco in 2001 extended this model by adding a pathogen compartment that represents the bacterial concentration in the environment [19]. The model in [19] has been further extended by a number of studies in recent years. For example, in paper [33], Mukandavire et al. investigated a model incorporating both the environmental-to-human and human-to-human transmission pathways to simulate the 2008–2009 cholera outbreak in Zimbabwe. In [47], Tuite et al. proposed a multi-patch model to investigate the cholera outbreak in Haiti during 2010–2012. More recently, He et al. [24] published a model to study the impact of limited medical resources on the Yemen cholera outbreak during 2016–2017. King et al. [26] proposed a two-patch cholera ODE model including classes for “inapparent” infections and the feature of varying periods of waning immunity. The results in [26] showed that asymptomatic carriers of cholera may be responsible for long range dissemination of the bacteria to distant communities, and their number may be much higher than those reported. In addition, Capssso and co-workers [2,3,14,16–18] employed reaction-diffusion systems to investigate the spatial spread of choler and other infectious diseases mediated by environmental pollution. Other related modeling studies for cholera can be found in [4,6,36,37,41,44–46,48–50,53] and references therein.

The frequent outbreaks and the high morbidity associated with cholera underscore the importance of effective cholera control, and tremendous efforts have been devoted to the management of cholera. Common intervention methods for cholera include rehydration therapy, antibiotics, water chlorination, and vaccination. Oral rehydration is widely used to treat individuals with minor or moderate infections, whereas antibiotics are recommended for severe cases (though the treatment could lead to antimicrobial resistance). Water sanitation is an effective way to reduce the concentration of pathogens in the environment so as to improve the quality of drinking water, which is critical for the prevention of cholera and other waterborne infections in the long run. In addition, with the introduction of low-cost oral vaccines based on live (but weakened) or killed whole-cells, vaccination has been an affordable yet effective means to fight cholera, and has been successfully implemented in cholera endemic places. Its use in epidemic and emergency settings was also conditionally recommended by WHO [13,31]. A successful demonstration is the deployment of cholera vaccines in Haiti which made significant contribution in containing the 2010–2012 Haiti cholera outbreak [7,39].

As a promising strategy for cholera control, vaccination has received much attention and has been the research topic of many clinical and theoretical studies. A number of mathematical models have incorporated cholera vaccination which exhibit rich dynamics (see, e.g., [5,9,21,27,34,38]). For example, it is shown in [5,27] that models with all susceptible individuals vaccinated can have multiple endemic equilibria when the basic reproduction number is lower than unity. In spite of these findings, the specific effects of vaccination on cholera transmission and the detailed impact of vaccination policy on a cholera epidemic are not well understood yet at present.

A critical issue of vaccination is concerned with the evolution of the induced host immunity. In general, an individual gains disease immunity upon receiving a vaccine that stimulates a primary response against the pathogen inside the human body without causing symptoms of the disease. However, such vaccine-induced immunity is not permanent and is typically decreasing over time, and, in the case of cholera, worn out in a few years. Meanwhile, the immunity waning process may not be a steady flow; the immunity waning rate typically depends on the time (i.e., the age since receiving the vaccination) instead of being a constant.

Another important factor of vaccination is its effectiveness. In practice, vaccines may not be 100% effective, and vaccinated people may still have a chance to contract the disease. For example, in a field trial in Bangladesh, it was found that the cholera vaccine based on the whole B cell recombinant subunit (WC/rBS) provided 85 to 90 percent of immune protection in all age groups within upon vaccination (and after 6 months this protection level dropped to 60 %) [31]. Nevertheless, since such individuals are already vaccinated and have at least some degree of immunity, their infection would be less severe, often not showing typical symptoms of cholera, compared to the majority of those who are not vaccinated. We will refer to their infection as asymptomatic infection in this work. On the other hand, asymptomatic carriers play a crucial role in shaping the spatiotemporal patterns of cholera epidemics [20]. Bacteria in their feces can enter and contaminate the environment, and spread the disease [12]. However, it has received little attention in the study of cholera epidemics [26,40].

In the present study, we propose a new mathematical model to investigate the detailed effects of vaccination on cholera transmission dynamics. We assume that cholera vaccines are administered for a portion of individuals that enter the host population. Our model takes into account the vaccination age structure, the imperfect vaccination, and the asymptomatic infection. We introduce a partial differential equation (PDE) to describe the vaccinated population where the age structure is incorporated, and employ ordinary differential equations (ODEs) for the susceptible, the asymptomatic infected, and the typical (i.e., symptomatic) infected, populations. Our model is thus a mixed ODE-PDE system that represents the complex interaction among the different host populations and the environmental pathogens.

We organize the remainder of this paper as follows. In Section 2, we describe the cholera model with vaccination age structure. In Section 3, we determine the existence and stability of the equilibria in the model and the basic reproduction number of the model. In Sections 4 and 5, we prove the disease persistence and global asymptotical stabilities, respectively, of the equilibria. In Section 6, we perform some numerical simulation to our proposed model. Finally, we draw the conclusions in Section 7.

2. Model formulation

We divide the host population into the susceptible individuals, the asymptomatic infective individuals, the (symptomatic) infective individuals, and the vaccinated individuals. We use S(t), A(t) and I(t) to represent the numbers of the susceptible, asymptomatic infective, and infective individuals, respectively, at time t. Meanwhile, we use v(t, a) to denote the density of the vaccinated individuals with vaccination age a at time t. Let $V(t)={\int }_0^{+\infty }v(a,t)da$ represent the total of the vaccinated people at time t. In addition, we let B(t) represent the concentration of the pathogenic bacteria in the environment at time t.

We denote the influx rate of the host population by Λ, the natural mortality rate by μ, and the effective contact rate between the susceptible individuals and the pathogen by β. We assume that a proportion p (0 < p < 1) of the individuals that enter the host population are vaccinated. We also assume that when the susceptible people contract the disease, a portion θ (0 < θ < 1) become asymptomatic whereas 1 − θ show symptoms. We let the functions α(a) and σ represent the waning rate of the vaccine-induced immunity and the probability that the vaccine has an immuno-protective effect, respectively. We denote the shedding rates of the asymptomatic infected and infected people by ω₁ and ω₂, and the removal rates of the asymptomatic infected and infected people by η₁ and η₂, respectively. We assume that individuals recovered from cholera infection will not contract the disease again [23]. In addition, m₀ denotes the rate of bacterial clearance, and γ represents the probability that an asymptomatic infected person becomes a (symptomatic) infected person.

Our model thus takes the form

$$\begin{gathered} \frac{dS(t)}{dt}=(1-p)\mathrm{\Lambda }-\beta B(t)S(t)-\mu S(t)+{\int }_0^{+\infty }\alpha (a)v(a,t)da, \\ \frac{dA(t)}{dt}=\theta \beta B(t)S(t)+{\int }_0^{\infty }\beta (1-\sigma (a))v(a,t)daB(t)-\left(\gamma +\mu +{\eta }_1\right)A(t), \\ \frac{dI(t)}{dt}=(1-\theta )\beta B(t)S(t)+\gamma A(t)-\left(\mu +{\eta }_2\right)I(t), \\ \frac{\partial v(a,t)}{\partial a}+\frac{\partial v(a,t)}{\partial t}=-(\mu +\alpha (a))v(a,t)-\beta (1-\sigma (a))v(a,t)B(t), \\ v(0,t)=p\mathrm{\Lambda }, \\ \frac{dB(t)}{dt}={\omega }_{1}A(t)+{\omega }_{2}I(t)-m_{0}B(t). \end{gathered}$$ (2.1)

Before we proceed, we introduce an assumption to reduce the number of parameters and facilitate the mathematical analysis. We assume that ω₁ = kη₁ and ω₂ = kη₂ for some positive constant k. Then the last equation in system (2.1) can be rewritten as

$$\frac{dB(t)}{dt}=k{\eta }_{1}A(t)+k{\eta }_{2}I(t)-m_{0}B(t).$$

By re-scaling the variable $\frac{1}{k}B(t)\to B(t)$ and the parameter kβ → β, we may re-write system (2.1) as

$$\begin{gathered} \frac{dS(t)}{dt}=(1-p)\mathrm{\Lambda }-\beta B(t)S(t)-\mu S(t)+{\int }_0^{+\infty }\alpha (a)v(a,t)da, \\ \frac{dA(t)}{dt}=\theta \beta B(t)S(t)+{\int }_0^{\infty }\beta (1-\sigma (a))v(a,t)daB(t)-\left(\gamma +\mu +{\eta }_1\right)A(t), \\ \frac{dI(t)}{dt}=(1-\theta )\beta B(t)S(t)+\gamma A(t)-\left({\eta }_2+\mu \right)I(t), \\ \frac{\partial v(a,t)}{\partial a}+\frac{\partial v(a,t)}{\partial t}=-(\mu +\alpha (a))v(a,t)-\beta (1-\sigma (a))v(a,t)B(t), \\ v(0,t)=p\mathrm{\Lambda }, \\ \frac{dB(t)}{dt}={\eta }_{1}A(t)+{\eta }_{2}I(t)-m_{0}B(t). \end{gathered}$$ (2.2)

In what follows we will study system (2.2) equipped with the initial conditions:

$$S(0)=S_0,A(0)=A_0,I(0)=I_0,v(0,a)=v_0(a),B(0)=B_0.$$

Assumption 2.1.

The parameters and functions in system (2.2) satisfy the following conditions

1. All the parameters are positive;

2. The functions α(a), σ(a) ∈ L^∞ (0, ∞) are bounded and uniformly continuous.

3. The function v₀ (a) is non-negative and integrable.

Using methods from Iannelli [25] and [30], we can show the existence and uniqueness of solutions to system (2.2) with the initial conditions and Assumption 2.1. Thus, we can define a solution semiflow

$$\mathrm{\Phi }:\left(t,S_0,A_0,I_0,v_0(\cdot ),B_0\right)=(S(t),A(t),I(t),v(t,\cdot ),B(t)),$$

$$R_{+}^3\times L_{+}^1\times R_{+}\to R_{+}^3\times L_{+}^1\times R_{+}$$

for t ∈ R₊ and $\left(S_0,A_0,I_0,v_0(\cdot ),B_0\right)\in R_{+}^3\times L_{+}^1\times R_{+}$.

Let $N(t)=S(t)+A(t)+I(t)+{\int }_0^{\infty }v(t,a)da$. It follows from (2.2) that

$$\frac{dN(t)}{dt}\le \mathrm{\Lambda }-\mu N(t),$$

which implies that $\text{lim}{\text{sup}}_{t\to \infty }N(t)\le \frac{\mathrm{\Lambda }}{\mu }$. Let

$$\mathrm{\Omega }=\left\{(S(t),A(t),I(t),v(t,\cdot ),B(t))\in R_{+}^3\times L_{+}^1\times R_{+}:N(t)\le \frac{\mathrm{\Lambda }}{\mu },B(t)\le \frac{\eta \mathrm{\Lambda }}{\mu m_0},\eta =\text{max}\left\{{\eta }_1,{\eta }_2\right\}\right\}.$$

Thus, it is straightforward to show Ω is positively invariant and attracts all positive solutions of Ω in system (2.2). In the following, we restrict our attention to solutions of (2.2) with the initial conditions and Assumption 2.1.

3. Existence and stability of Equilibria

We remark that system (2.2) can be reduced to a cholera model which consists of only ordinary differential equations, if the dependence on the vaccination age is removed. Analysis of such a simplified model, nevertheless, can offer some useful insight into cholera transmission dynamics. Details of this ODE model and its equilibrium analysis are provided in Appendix: ODE model. Below we will focus our attention on model (2.2).

System (2.2) always has a disease-free equilibrium E₀ = (S⁰, A⁰, I⁰, v⁰ (a), B⁰) with

$$S^0=\frac{\mathrm{\Lambda }}{\mu }\left(1-p+p{\int }_0^{+\infty }\alpha (a)\pi (a)da\right),A^0=0,I^0=0,v^0(a)=p\mathrm{\Lambda }\pi (a),B^0=0,$$ (3.1)

where

$$\pi (a)=e^{-{\int }_0^a(\mu +\alpha (\tau ))d\tau }.$$

The two disease-free components, S⁰ and v⁰ (a), are non-zero and their values are both shaped by the vaccination age. In the special case when α(a) ≡α and σ(a) ≡σ, it is straightforward to verify that the E₀ leads to the disease-free equilibrium x₀ associated with the ODE model (7.1); i.e., S⁰ = S₀, and ${\int }_0^{\infty }{v}^0(a)da=V_0$, where S₀ and V₀ are the corresponding components of ${\mathcal{E}}_0$ defined in Eq. (7.2).

Suppose that there exists an endemic equilibrium E* = (S*, A*, I*, v* (a), B*) where S*, A*, I*, v* (a) and B* are positive, which satisfy the following equations

$$0=(1-p)\mathrm{\Lambda }-\beta B^{*}{S}^{*}-\mu S^{*}+{\int }_0^{+\infty }\alpha (a)v^{*}(a)da,$$ (3.2)

$$0=\theta \beta B^{*}{S}^{*}+{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}da-\left(\gamma +\mu +{\eta }_1\right)A^{*},$$ (3.3)

$$0=(1-\theta )\beta B^{*}{S}^{*}+\gamma A^{*}-\left({\eta }_2+\mu \right)I^{*},$$ (3.4)

$$\frac{d{v}^{*}(a)}{da}=-(\mu +\alpha (a))v^{*}(a)-(1-\sigma (a)\beta v^{*}(a)B^{*},$$ (3.5)

$$v^{*}(0)=p\mathrm{\Lambda },$$ (3.6)

$$0={\eta }_1{A}^{*}+{\eta }_2{I}^{*}-m_0{B}^{*}.$$ (3.7)

By (3.5) and (3.6) we can obtain

$$v^{*}(a)=p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }.$$ (3.8)

Substituting (3.8) into (3.2) we have

$$S^{*}=\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }.$$ (3.9)

Substitution (3.8) and (3.9) into (3.3) yields

$$\begin{gathered} A^{*}=\frac{\beta \theta B^{*}}{\gamma +\mu +{\eta }_1}\left[\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }\right] \\ +\frac{B^{*}}{\gamma +\mu +{\eta }_1}{\int }_0^{+\infty }(1-\sigma (a))\beta p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da. \end{gathered}$$ (3.10)

Meanwhile, substituting (3.8) and (3.9) into (3.4), we obtain

$$\begin{gathered} I^{*}=\frac{\beta (1-\theta )}{\mu +{\eta }_2}\left[\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }\right]B^{*} \\ +\frac{\gamma }{{\eta }_2+\mu }\frac{\beta \theta }{\gamma +\mu +{\eta }_1}\left[\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }\right]B^{*} \\ +\frac{\gamma }{{\eta }_2+\mu }\frac{1}{\gamma +\mu +{\eta }_1}{\int }_0^{+\infty }(1-\sigma (a))\beta p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da{B}^{*}. \end{gathered}$$ (3.11)

In addition, substituting (3.10) and (3.11) into (3.7), we obtain an equation in terms of B*:

$$\begin{gathered} R\left(B^{*}\right)\triangleq \left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{\beta \theta }{\gamma +\mu +{\eta }_1}\left[\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }\right] \\ +\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{1}{\gamma +\mu +{\eta }_1}{\int }_0^{+\infty }(1-\sigma (a))\beta p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da \\ +{\eta }_2\frac{\beta (1-\theta )}{\mu +{\eta }_2}\frac{(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }{e}^{{\int }_0^a-(\mu +\alpha (\tau ))d\tau }{e}^{{\int }_0^a-(1-\sigma (\tau ))\beta B^{*}d\tau }da}{\beta B^{*}+\mu }-m_0 \end{gathered}$$

By analyzing the function R(B*), we obtain R′(B*) < 0, R(+∞) = −m₀ <0. Hence, when $R(0)=\underset{B^{*}\to 0}{\text{lim}}R\left(B^{*}\right)>\text{0}$, the equation R(B*) = 0 has a unique solution B* > 0.

Direct calculation yields $R(0)=\underset{B^{*}\to 0}{\text{lim}}R\left(B^{*}\right)=m_0\left(R_v-1\right)$, where

$$\begin{gathered} R_v=\frac{1}{m_0}\left[\left({\eta }_1+\frac{{\eta }_2\gamma }{{\eta }_2+\mu }\right)\frac{\theta }{\gamma +\mu +{\eta }_1}+\frac{{\eta }_2(1-\theta )}{{\eta }_2+\mu }\right]\beta S^0 \\ +\frac{1}{m_0}\left(\frac{{\eta }_1}{\gamma +\mu +{\eta }_1}+\frac{\gamma {\eta }_2}{{\eta }_2+\mu }\frac{1}{\gamma +\mu +{\eta }_1}\right){\int }_0^{\infty }(1-\sigma (a))\beta v^0(a)da. \end{gathered}$$ (3.12)

Clearly, only when R_v > 1, Eq. (3.11) has a unique solution B* > 0. Consequently, by (3.7)–(3.10), v* (a) > 0, S* > 0, A* > 0 and I* > 0 can be uniquely determined. Thus, when R_v > 1, system (2.2) has a unique endemic equilibrium E*. We have obtained the following result:

Theorem 3.1.

1. If R_v < 1, system (2.2) has a unique disease-free equilibrium E_0.

2. If R_v > 1, system (2.2) has a unique disease-free equilibrium E₀ and a unique endemic equilibrium E*.

We comment that Eq. (3.12) defines the basic reproduction number for system (2.2). The equation expresses R_v in terms of quantities at the disease-free state: the first part is related to the disease-free component of the susceptible population, and the second part is related to that of the vaccinated population; both parts are dependent on the vaccination age. Alternatively, we may re-write Eq. (3.12) as

$$\begin{gathered} R_v=\frac{{\eta }_1}{m_0\left(\gamma +\mu +{\eta }_1\right)}\left[\theta \beta S^0+{\int }_0^{\infty }(1-\sigma (a))\beta v^0(a)da\right] \\ +\frac{\gamma {\eta }_2}{m_0\left(\gamma +\mu +{\eta }_1\right)\left(\mu +{\eta }_2\right)}\left[\theta \beta S^0+{\int }_0^{\infty }(1-\sigma (a))\beta v^0(a)da\right] \\ +\frac{{\eta }_2(1-\theta )}{m_0\left(\mu +{\eta }_2\right)}\beta S^0. \end{gathered}$$ (3.13)

Eq. (3.13) represents the basic reproduction number in three parts: the first and last parts describe the contributions from the asymptomatic infection and symptomatic infection, respectively, whereas the middle part comes from the interaction between the two types of infections. It is clear to observe that vaccination age plays a direct role in shaping all the three parts which, combined together, depict the overall risk of cholera in a population under the impact of vaccination. In the simplified case with α(a) ≡α and σ(a) ≡σ, we have S⁰ = S₀ and ${\int }_0^{\infty }{v}^0(a)da=v_0$, and we can easily observe that there is a one-to-one correspondence between the three terms in Eq. (3.13) and those in Eq. (7.5). That is, the reproduction number R_v of the age-dependent model (2.2) is reduced to the reproduction number ${\mathcal{R}}_0$ of the ODE model (7.1) when the age structure is removed, as can be naturally expected.

With the incorporation of vaccination age, the value of R_v is in general different from that of ${\mathcal{R}}_0$. Fig. 2 illustrates a comparison between these two reproduction numbers. We set $\sigma (a)=e^{-c_{1}a}$ and $\alpha (a)=1-e^{-c_{2}a}$ with positive constants c₁ and c₂ for the age-dependent model (2.2). Meanwhile, we evaluate σ(a) and α(a) at a specific number a > 0 to obtain the corresponding parameters σ and α in the ODE model (7.1); here we show the results for a = 1, and the pattern is similar for other values of a. Fig. 2(a) plots the values of R_v and ${\mathcal{R}}_0$ with respect to the vaccination rate p, where p ranges between 0 and 1. When p = 0, the vaccinated compartment is effectively eliminated from each of the two models, and so R_v and ${\mathcal{R}}_0$ coincide with each other. When p > 0, however, the two curves deviate from each other, and the value of R_v is always higher than that of ${\mathcal{R}}_0$. Meanwhile, Fig. 2 (b) plots R_v and ${\mathcal{R}}_0$ with respect to the asymptomatic infection rate θ, where θ ranges between 0 and 1. We again observe that the curve of R_v is above that of ${\mathcal{R}}_0$ throughout 0 ≤ θ ≤1. In each of the two figures, there is a range of parameter values (for p or θ) where we clearly observe that ${\mathcal{R}}_0<1$ but R_v > 1. These results demonstrate that the two models (2.2) and (7.1) lead to two different reproduction numbers, and indicate that using the ODE model (7.1) might under-estimate the disease risk due to the neglect of the impact of vaccination age.

Fig. 2.

Comparison between the basic reproduction numbers R_v of the age-dependent model (2.2) and ${\mathcal{R}}_0$ of the ODE model (7.1): (a) when the vaccination rate p varies from 0 to 1; (b) when the asymptomatic infection rate θ varies from 0 to 1.

Below we conduct stability analysis to the two equilibria of system (2.2) and establish R_v = 1 as a sharp threshold for disease eradication and disease persistence.

Let x(t) = S(t) − S*, y(t) = A(t) − A*, z = I(t) − I*, w(t, a) = v(t, a) − v*(a), m(t) = B(t) − B*. Linearizing system (2.2) around an equilibrium (S*, A*, I*, v*(·), B*), we obtain a perturbation linear system and consider the eigenvalue problem for the linear system. For simplicity, we denote the time-independent perturbations corresponding to an eigenvalue λ using the same letters. Thus, we have the following eigenvalue problem

$$\begin{gathered} \lambda x=-\beta m{S}^{*}-\beta B^{*}x-\mu x+{\int }_0^{\infty }\alpha (a)w(a)da, \\ \lambda y=\theta \beta m{S}^{*}+\theta \beta B^{*}x+\beta B^{*}{\int }_0^{\infty }(1-\sigma (a))w(a)da+\beta m{\int }_0^{\infty }(1-\sigma (a))v^{*}(a)da-\left(\gamma +\mu +{\eta }_1\right)y, \\ \frac{dw(a)}{da}=-(\lambda +\mu +\alpha (a))w(a)-\beta B^{*}(1-\sigma (a))w(a)-\beta m(1-\sigma (a))v^{*}(a), \\ w(a)\text{=}\text{0,} \\ \lambda z=(1-\theta )\beta m{S}^{*}+(1-\theta )\beta B^{*}x+\gamma y-\left({\eta }_2+\mu \right)z, \\ m\lambda ={\eta }_{1}y+{\eta }_{2}z-m_{0}m. \end{gathered}$$ (3.14)

Now we first consider the local stability of the disease-free equilibrium. In this case, we have A* = I* = B* = 0. Thus, system (3.14) can simplify significantly to the following form

$$\begin{gathered} \lambda x=-\beta m{S}^0-\mu x+{\int }_0^{\infty }\alpha (a)w(a)da, \\ \lambda y=\theta \beta m{S}^0+\beta m{\int }_0^{\infty }(1-\sigma (a))v^0(a)da-\left(\gamma +\mu +{\eta }_1\right)y, \\ \frac{dw(a)}{da}=-(\lambda +\mu +\alpha (a))w(a)-\beta m(1-\sigma (a))v^0(a), \\ w\text{(}a\text{)}\text{=}\text{0,} \\ \lambda z=(1-\theta )\beta m{S}^0+\gamma y-\left({\eta }_2+\mu \right)z, \\ m\lambda ={\eta }_{1}y+{\eta }_{2}z-m_{0}m. \end{gathered}$$ (3.15)

Solving the second equation for y in terms of m, we have

$$y=\frac{\theta \beta m{S}^0+\beta m{\int }_0^{\infty }(1-\sigma (a)v^0(a)da}{\lambda +\mu +\gamma +{\eta }_1}.$$

Solving the differential equation for z in terms of m, we obtain

$$z=\frac{(1-\theta )\beta m{S}^0}{\lambda +\mu +{\eta }_2}+\frac{\gamma m(\theta \beta S^0+\beta {\int }_0^{\infty }\left(1-\sigma (a)v^0(a)da\right)}{\left(\lambda +\mu +{\eta }_2\right)\left(\lambda +\mu +\gamma +{\eta }_1\right)}.$$

Substituting into the last equation and canceling m, we obtain the following equation for the eigenvalue λ of the linear operator:

$$\frac{{\eta }_1(\theta \beta S^0+\beta {\int }_0^{\infty }\left(1-\sigma (a)v^0(a)da\right)}{\left(\lambda +m_0\right))\left(\lambda +\mu +\gamma +{\eta }_1\right)}+\frac{{\eta }_2(1-\theta )\beta S^0}{\left(\lambda +m_0\right)\left(\lambda +\mu +{\eta }_2\right)}+\frac{{\eta }_2\gamma (\theta \beta S^0+\beta {\int }_0^{\infty }\left(1-\sigma (a)v^0(a)da\right)}{\left(\lambda +m_0\right)\left(\lambda +\mu +{\eta }_2\right)\left(\lambda +\mu +\gamma +{\eta }_1\right)}=1.$$ (3.16)

Define a function $\mathcal{G}(\lambda )$ to be the right-hand side in Eq. (3.16). Obviously, $\mathcal{G}(\lambda )$ is a continuously differentiable function with ${\text{lim}}_{\lambda \to \infty }\mathcal{G}(\lambda )=0$. By direct computation, it is easy to show that ${\mathcal{G}}^{\prime }(\lambda )<0$, therefore, $\mathcal{G}(\lambda )$ is a decreasing function. Hence, any real solution of Eq. (3.16) is negative if $\mathcal{G}(0)<1$, and positive if $\mathcal{G}(0)>1$. From Eq. (3.16), it follows that $\mathcal{G}(0)=R_v$. In addition, through direct calculation, it is easy to obtain that $\left|\mathcal{G}(\lambda )\right|\le \mathcal{G}(Re\lambda )$ for any λ with Reλ≥0. On the other hand, for the case R_v < 1, suppose that λ is a solution to the equation $\mathcal{G}(\lambda )=1$ with Reλ≥0. Then we have

$$1=\left|\mathcal{G}(\lambda )\right|\le \mathcal{G}(Re\lambda )\le \mathcal{G}(0)=R_v<1.$$

This is a contraction. Thus, all solutions to Eq. (3.16) have a negative real part. Applying the methods in [10,30] (where the authors provide connections between the roots of the characteristic equation with PDEs and the stability of the equilibrium), we can conclude that the disease-free equilibrium E₀ is locally stable. In the case R_v > 1, $\mathcal{G}(0)>1$. Since $\mathcal{G}(0)\to 0$ as λ→ ∞, Eq. (3.16) only has solutions with posit real parts. Similar to the arguments in [10,30], the disease-free equilibrium E₀ is unstable. Summarizing the above discussion, we have

Theorem 3.2.

If R_v < 1, then the disease-free equilibrium E₀ is locally asymptotically stable. If R_v > 1, then the disease-free equilibrium E₀ is unstable.

4. Persistence

We shall proceed to analyze the global stability of the equilibria in system (2.2). Inspired by recent studies of Magal et al. [28], McCluskey [32], Shuai et al [8,41], and Martcheva and Li [29], we will construct suitable Lyapunov functions to establish the global stability of the equilibria. Here, we shall apply a generic form of the following function

$$g(x)=x-1-\text{ln}x\ge 0\text{for}x>0,\text{and}g(x)=0\text{if}\text{and}\text{only}\text{if}x=1.$$

One difficulty with the construction of a Lyapunov functional is that it may not be defined when some variables in the system become zero [28]. To overcome this difficulty, we need to show that the disease persists for R_v > 1 in system (2.2). To achieve it, we first consider the uniform ρ− persistence by using the theory and methods developed in [30,42].

Integrating the third equation in system (2.2) along the characteristic lines, we obtain

$$v(a,t)=\{\begin{array}{ll}p\mathrm{\Lambda }\pi (a){\pi }_1(a),\hfill & t>a,\hfill \\ v_0(a-t)\frac{\pi (a)}{\pi (a-t)}\frac{{\pi }_1(a)}{{\pi }_1(a-t)},\hfill & t<a,\hfill \end{array}$$ (4.1)

where ${\pi }_1(a)=e^{-\beta {\int }_0^a(1-\sigma (\tau ))B(\tau )d\tau }$.

For a bounded real-valued function f on [0, ∞), we introduce the following notation $f_{\infty }=\underset{t\to \infty }{\text{lim}\text{inf}}f(t)$, $f^{\infty }=\underset{t\to \infty }{\text{lim}\text{sup}}f(t)$.

We also need the following Fluctuation Lemma, which comes from [43].

Lemma 4.1.

Let f : [0, ∞) → R be bounded and twice differentiable with bounded second derivative. Let t_n and f(t_n) converge to f_∞ or f^∞. Then f′(t_n) → 0 as n → ∞.

Lemma 4.2.

If R_v > 1, then system (2.2) is uniformly weakly ρ − persistent; i.e., there exists an ε₀ > 0 such that

$$\text{lim}\underset{t\to \infty }{\text{sup}}A(t)\ge {\epsilon }_0,\text{lim}\underset{t\to \infty }{\text{sup}}I(t)\ge {\epsilon }_0,\text{lim}\underset{t\to \infty }{\text{sup}}B(t)\ge {\epsilon }_0.$$

Proof.

First, notice that if R_v > 1, we can Choose ${\tilde{\epsilon }}_0>0$ small enough such that

$${\eta }_1\left(\theta \beta S^0+{\int }_0^{\infty }(1-\sigma (a))\beta v^0(a)da\right)+\gamma {\eta }_2\left(\theta \beta S^0+{\int }_0^{\infty }(1-\sigma (a))\beta v^0(a)da\right)+{\eta }_2(1-\theta )\beta S^0-{\tilde{\epsilon }}_0>m_0\left(\gamma +\mu +{\eta }_1\right)\left(\mu +{\eta }_2\right).$$ (4.2)

By way of contraction, there exists a solution (S(t), A(t), I(t), v(t, ·), B(t)) ∈ Ω in system (2.2) such that lim sup_t→∞ B(t) < ε₀. Thus, there exists a sequence {t_n}. It follows from the the first equation of (2.2) that

$$\frac{dS\left(t_n\right)}{dt}\ge (1-p)\mathrm{\Lambda }-\beta S\left(t_n\right){\epsilon }_0-\mu S\left(t_n\right)+p\mathrm{\Lambda }{\int }_0^{t_n}\alpha (a)\pi (a)da+{\epsilon }^{\prime },$$

where we have used the inequality e^x > 1 + x, for x≠0, and where ${\epsilon }^{\prime }=-p\mathrm{\Lambda }\beta {\epsilon }_0{\int }_0^{t_n}\alpha (a)\pi (a){\int }_0^a(1-\sigma (\tau ))d\tau da$.Using the Fluctuation Lemma 4.1, and choosing the above t_n → ∞ such that S(t_n) → S_∞ and $\frac{dS\left(t_n\right)}{dt}\to 0$, we obtain

$$S_{\infty }\ge \frac{\mathrm{\Lambda }\left(1-p+p{\int }_0^{\infty }\alpha (a)\pi (a)da\right)}{\mu +\beta {\epsilon }_0}.$$

Thus, there exits t₁ > t₀ such that $S(t)\ge \frac{\mathrm{\Lambda }\left(1-p+p{\int }_0^{\infty }\alpha (a)\pi (a)da\right)}{\mu +\beta {\epsilon }_0}-{\epsilon }_0\stackrel{def}{=}{S}_0^{{\epsilon }_0}$ for t≥t₁. Without loss of generality, we can assume that $S(t)\ge S_0^{{\epsilon }_0}$ Thus, from system (2.2), we have

$$\begin{gathered} \frac{dA(t)}{dt}\ge \theta \beta B(t)S_0^{{\epsilon }_0}+p\mathrm{\Lambda }{\int }_0^{\infty }\beta (1-\sigma (a))\pi (a)daB(t)-\left(\gamma +\mu +{\eta }_1\right)A(t), \\ \frac{dI(t)}{dt}\ge (1-\theta )\beta B(t)S_0^{{\epsilon }_0}+\gamma A(t)-\left({\eta }_2+\mu \right)I(t), \\ \frac{dB(t)}{dt}\ge {\eta }_{1}A(t)+{\eta }_{2}I(t)-m_{0}B(t), \end{gathered}$$ (4.3)

which can be rewritten in the following linearized system:

$$\frac{dX(t)}{dt}\ge C_{{\epsilon }_0}{X}^T(t),$$

where X(t) = (A(t), I(t), B(t)), and matrix $C_{{\epsilon }_0}$ satisfies

$$C_{{\epsilon }_0}=\left(\begin{array}{ccc}-\left(\gamma +\mu +{\eta }_1\right)& 0& \theta \beta S_0^{{\epsilon }_0}+p\mathrm{\Lambda }{\int }_0^{\infty }\beta (1-\sigma (a))\pi (a)da\\ \gamma & -\left({\eta }_2+\mu \right)& (1-\theta )\beta S_0^{{\epsilon }_0}\\ {\eta }_1& {\eta }_2& -m_0\end{array}\right).$$

By direct computation, using the relation (4.2), we can determine that there is at least one positive eigenvalue of matrix $C_{{\epsilon }_0}$. By the standard comparison principle of a linear system, from (4.3), we have A(t) → +∞, I(t) → +∞, B(t) → + ∞, t → +∞ as R_v > 1. This contradicts with the boundedness of the solutions in system (2.2). Thus, we have completed the proof of Lemma 4.2. □

We now show that system (2.2) has a global compact attractor ${\mathcal{M}}_0$. A set M₀ in Ω is called a global compact attractor for the solution semiflow Φ, if M₀ is a maximal compact invariant set, and if for all open sets U containing M₀ and all bounded sets K of Ω, there exists some t₀ > 0 such that Φ(t, K) ⊆U for all t ≥t₀ (see [22], Section 3.4). In order to prove our Theorem, we need the following two results, which come from Lemma 3.2.3 and Theorem 3.4.6 in [22]. These methods and techniques have been recently employed in [30].

Lemma 4.3.

For each t ≥0, suppose $T(t)=\mathcal{S}(t)+\mathcal{U}(t):\mathrm{\Omega }\to \mathrm{\Omega }$ has the property that $\mathcal{U}(t)$ is complete continuous and there is a continuous function $k:{ℝ}^{+}\times {ℝ}^{+}\to {ℝ}^{+}$ such that k(t, r) → 0 as t → 0 and $\left|\mathcal{S}(t)x\right|\le k(t,r)$, if |x| < r. Then T(t) (t ≥0) is asymptotically smooth.

Lemma 4.4.

Let T (t) be semigroup acting on $\mathrm{\Omega }=R_{+}^3\times L_{+}^1\times R_{+}$. If T(t) : Ω → Ω, t ∈ R₊, is asymptotically smooth, point dissipative and orbits of bounded sets are bounded, then there exists a global attractor.

Now we first establish the following result:

Lemma 4.5.

Assume that R_v > 1, then there exists ${\mathcal{M}}_0$ (a compact subset of Ω), which is a global attractor for the solution semiflow Φ of system (2.2) in Ω.

Proof.

Set

$$\mathrm{\Phi }\left(t;S_0,A_0,I_0,v_0(\cdot ),B_0\right)=(S(t),A(t),I(t),v(\cdot ,t),B(t)),$$

Ψ: [0, ∞), × Ω → Ω, with Φ(t, Φ(s,.)) = Φ(t + s, ·) for all t, s ≥0 and Φ(0, ·) being the identity map. Our goal is to show that Φ satisfies the assumptions of Lemmas 4.3 and 4.4. To this end, we split the solution semiflow Φ into two components $\mathrm{\Phi }=\widehat{\mathrm{\Phi }}\left(t,x^0\right)+\tilde{\mathrm{\Phi }}\left(t,x^0\right)$ such that $\widehat{\mathrm{\Phi }}\left(t,x^0\right)\to 0$ as t → ∞ for every x⁰ ∈ Ω, and for a fixed t and any bounded set K in Ω, the set $\left\{\tilde{\mathrm{\Phi }}\left(t,x^0\right):x^0\in K\right\}$ is precompact. The two summands are defined as follows:

$$\begin{gathered} \widehat{\mathrm{\Phi }}\left(t,S_0,A_0,I_0,v_0(\cdot ),B_0\right)=(0,0,0,\widehat{v}(\cdot ,t),0); \\ \tilde{\mathrm{\Phi }}\left(t,S_0,A_0,I_0,v_0(\cdot ),B_0\right)=(S(t),A(t),I(t),\tilde{v}(\cdot ,t),B(t)). \end{gathered}$$

Notice that S(t), A(t), I(t), B(t) satisfy system (2.2) with $v(a,t)=\widehat{v}(a,t)+\tilde{v}(a,t)$, where

$$\tilde{v}(a,t)=\{\begin{array}{ll}p\mathrm{\Lambda }\pi (a)e^{-\beta B{\int }_0^a(1-\sigma (\tau ))d\tau },\hfill & \text{for}0\le a\le t,\hfill \\ 0\hfill & \text{for}t<a,\hfill \end{array}$$ (4.4)

and

$$\widehat{v}(a,t)=\{\begin{array}{ll}0,\hfill & \text{for}0\le a\le t,\hfill \\ v_0(a-t)\frac{\pi (a)}{\pi (a-t)}\frac{{\pi }_1(a)}{{\pi }_1(a-t)},\hfill & \text{for}t<a.\hfill \end{array}$$ (4.5)

Obviously, $\tilde{v}(a,t)$ and $\widehat{v}(a,t)$ are nonnegative. Using (4.5), it is easy to obtain that

$$\begin{gathered} \Vert \widehat{\mathrm{\Phi }}\left(t,S_0,A_0,I_0,v_0(\cdot ),B_0\right)\Vert =\Vert \widehat{v}(\cdot ,t){\Vert }_1={\int }_t^{+\infty }{v}_0(a-t)\frac{\pi (a)}{\pi (a-t)}\frac{{\pi }_1(a)}{{\pi }_1(a-t)}da \\ ={\int }_0^{+\infty }{v}_0(a)\frac{\pi (a)}{\pi (a+t)}\frac{{\pi }_1(a)}{{\pi }_1(a+t)}da \\ \le e^{-\mu t}{\int }_0^{+\infty }{v}_0(a)da=e^{-\mu t}{\Vert v_0\Vert }_1 \\ \le e^{-\mu t}{\Vert \left(S_0,A_0,I_0,v_0(\cdot ),B_0\right)\Vert }_1. \end{gathered}$$

Thus, the function k(t, r) in Lemma 4.3 can be chosen as k(t, r) = e^−μtr, and $\widehat{\mathrm{\Phi }}$ satisfies Lemma 4.3.

Now we are in position to show that $\tilde{\mathrm{\Phi }}$ is completely continuous, that is, for any t ≥ 0 and a bounded subset K ⊂ Ω, the set $\tilde{\mathrm{\Phi }}(t,K)$ is compact. As $\tilde{\mathrm{\Phi }}\le \mathrm{\Phi }$ and Φ(t, K) is bounded, we only need to show that the family of functions $\{\tilde{v}(\cdot ,t)\}$ is compact in L¹ (0, ∞), which is done by using the Frèchet-Kolmogorov theorem to show the compactness in L¹ (0, ∞) (see [54]). From our previous analysis, we have the set {Φ(t, x₀): t≥0, ∥x₀∥≤K}, and K is some constant. This implies that for any fix t ∈ R₊, and any bounded set K ⊂ Ω, the set ${\mathrm{\Omega }}_t=\left\{\tilde{\mathrm{\Phi }}\left(t,S_0,A_0,I_0,v_0(\cdot ),B_0\right):\left(S_0,A_0,I_0,v_0(\cdot ),B_0\right)\in K\right\}$ is precompact. Thus, it is sufficient to show that ${\mathrm{\Omega }}_{t,v}=\{\tilde{v}(\cdot ,t)):(S(t),A(t),I(t),v(\cdot ,t),B(t))\in {\mathrm{\Omega }}_t\}$. Since $\tilde{v}(a,t)=0$, for a > t, the third condition is trivial in Frèchet-Kolmogorov theorem. Now we show the following relation holds

$$\underset{h\to 0+}{\text{lim}}{\int }_0^{\infty }|\tilde{v}(a+h,t)-\tilde{v}(a,t)|da=0,\mathit{\text{uniformly}}\mathit{\text{in}}{\mathrm{\Omega }}_{t,v}.$$ (4.6)

By (4.4), $\tilde{v}\text{(0,}t\text{)}$ is defined when t = 0. Given ε, let h be small and consider

$$\begin{gathered} {\int }_0^{\infty }|\tilde{v}(a+h,t)-\tilde{v}(a,t)|da=p\mathrm{\Lambda }{\int }_0^{t-h}\left|\pi (a+h){\pi }_1(a+h)-\pi (a){\pi }_1(a)\right|da \\ +p\mathrm{\Lambda }{\int }_{t-h}^t\pi (a){\pi }_1(a)da \\ \le p\mathrm{\Lambda }{\int }_0^{t-h}{\pi }_1(a+h)\mid \pi (a+h)-\pi (a))\mid da \\ +p\mathrm{\Lambda }{\int }_0^{t-h}\pi (a)\mid {\pi }_1(a+h)-{\pi }_1(a))\mid da+p\mathrm{\Lambda }{\int }_{t-h}^t\pi (a){\pi }_1(a)da. \end{gathered}$$ (4.7)

Since B(t) ∈ Ω is bounded, and the function π(a), π₁(a) are uniformly continuous and bounded, we see that the right integral in the above inequality (4.7) is sufficiently small and is independent of the family of functions. Thus, (4.6) holds. This completes the proof of the compactness of $\tilde{\mathrm{\Phi }}$ and hence the semi-flow Φ has a global attractor in Ω. This complete the proof of Lemma 4.5. □

By now, we have verified all the conditions of Theorem 5.2 in [42] and hence we obtain the following result.

Theorem 4.6.

Suppose that R_v > 1, then the solution semi-flow Φ in system (2.2) is uniformly ρ−persistent

By using Thorem 4.6 and further analysis, we can establish the disease persistence in system (2.2). The lemma below [25] will be helpful in the following discussion.

Lemma 4.7.

Let $k:{ℝ}_{+}\to {ℝ}_{+}$ be L¹-integrable and $f:{ℝ}_{+}\to ℝ$ be bounded. Then

$$\underset{t\to \infty }{\text{lim}\text{sup}}{\int }_0^{t}k(\theta )f(t-\theta )d\theta \le \Vert k{\Vert }_1\underset{t\to \infty }{\text{lim}\text{sup}}f(t).$$

It follows from Lemma 4.7 easily that

$$\underset{t\to \infty }{\text{lim}\text{inf}}{\int }_0^{t}k(\theta )f(t-\theta )d\theta \ge \Vert k{\Vert }_1\underset{t\to \infty }{\text{lim}\text{inf}}f(t).$$ (4.8)

Theorem 4.8.

Suppose that R_v > 1, then the infection of the disease in system (2.2) is persistent; that is, there exists an ε > 0 (independent of initial conditions), such that any solution (S(t), A(t), I(t), v(a, t), B(t)) of (2.2) with (S₀, A₀, I₀, v₀ (·), B₀) ∈ Ω satisfies

$$\underset{t\to +\infty }{\text{lim}\text{inf}}S(t),\underset{t\to +\infty }{\text{lim}\text{inf}}A(t),\text{lim}\underset{t\to \infty }{\text{inf}}I(t),\underset{t\to +\infty }{\text{lim}\text{inf}}\Vert v(a,t){\Vert }_1,\underset{t\to +\infty }{\text{lim}\text{inf}}B(t)\ge \epsilon .$$

Proof.

By Theorem 4.6, there exists an η> 0 such that

$$\underset{t\to +\infty }{\text{lim}\text{inf}}B(t)\ge \eta ,\text{for}\text{all}\left(S_0,A_0,I_0,v_0(\cdot ),B_0\right)\in \mathrm{\Omega }.$$

Now let (S₀, A₀, I₀, v₀ (·), B₀) ∈ Ω. First, recall that $\text{lim}{\text{sup}}_{t\to +\infty }B(t)\le \frac{\eta \mathrm{\Lambda }}{\mu m_0}$. This combined with the first equation of system (2.2) gives

$$\underset{t\to +\infty }{\text{lim}\text{inf}}S(t)\ge \frac{(1-p)\mathrm{\Lambda }}{\frac{\beta \eta \mathrm{\Lambda }}{\mu m_0}+\mu }\stackrel{def}{=}{\epsilon }_1.$$

In fact, by Lemma 4.1, there exists a sequence {t_n} such that t_n → ∞, $S\left(t_n\right)\to \underset{t\to \infty }{\text{lim}\text{inf}}S(t)$ and $\frac{dS\left(t_n\right)}{dt}\to 0$ as n → ∞. Without loss of generality, we assume that {B(t_n)} is convergent, otherwise, we take a subsequence of {t_n}. Then $\underset{n\to \infty }{\text{lim}}B\left(t_n\right)\le \frac{\eta \mathrm{\Lambda }}{\mu m_0}$. Taking n → ∞ in

$$\frac{dS\left(t_n\right)}{dt}\ge (1-p)\mathrm{\Lambda }-\beta B\left(t_n\right)S\left(t_n\right)-\mu S\left(t_n\right)$$

Produces

$$0\ge (1-p)\mathrm{\Lambda }-\left(\beta \frac{\eta \mathrm{\Lambda }}{\mu m_0}+\mu \right)\underset{t\to \infty }{\text{lim}\text{inf}}S(t),$$

which gives $\underset{t\to \infty }{\text{lim}\text{inf}}S(t)\ge {\epsilon }_1$.

Second, it follows from (4.1) that

$$\Vert v(\cdot ,t){\Vert }_1\ge {\int }_0^{t}v(a,t)da\ge {\int }_0^{t}p\mathrm{\Lambda }\pi (a){\pi }_1(a)da.$$

Note that $\pi (a)\ge e^{-\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)a}$, ${\pi }_1(a)\ge e^{-\beta \Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)a}$. Thus we have

$$\underset{t\to \infty }{\text{lim}\text{inf}}\Vert v(\cdot ,t){\Vert }_1\ge \frac{\beta }{\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)(\Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)}\stackrel{def}{=}{\epsilon }_2.$$

Similarly, from the second equation of system (2.2), we have

$$\begin{gathered} \frac{dA(t)}{dt}\ge {\int }_0^t\beta (1-\sigma (a))v(a,t)daB(t)-\left(\gamma +\mu +{\eta }_1\right)A(t) \\ \ge p\mathrm{\Lambda }{\int }_0^t\beta (1-\sigma (a))e^{-\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)a}{e}^{-\beta \Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)a}daB(t)-\left(\gamma +\mu +{\eta }_1\right)A(t). \end{gathered}$$

By the fluctuation lemma there exists a sequence {s_n} such that s_n → ∞, $y\left(s_n\right)\to \underset{t\to \infty }{\text{lim}\text{inf}}y(t)$, and $\frac{dy\left(s_n\right)}{dt}\to 0$ as n → ∞. Letting n → ∞ in

$$\frac{dA\left(s_n\right)}{dt}\ge {\int }_0^{s_n}p\mathrm{\Lambda }\beta (1-\sigma (a))e^{-\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)a}{e}^{-\beta \Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)a}daB\left(s_n\right)-\left(\gamma +\mu +{\eta }_1\right)A\left(s_n\right)$$

and using (4.8), we get

$$0\ge p\mathrm{\Lambda }\beta {\Vert (1-\sigma (\cdot ))e^{-\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)}{e}^{-\beta \Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)}\Vert }_1\underset{t\to +\infty }{\text{lim}\text{inf}}B(t)-\left(\gamma +\mu +{\eta }_1\right)\underset{t\to +\infty }{\text{lim}\text{inf}}A(t),$$

which implies that

$$\underset{t\to +\infty }{\text{lim}\text{inf}}A(t)\ge \frac{p\mathrm{\Lambda }\beta \eta {\Vert (1-\sigma (\cdot ))e^{-\left({\mu }_1+\Vert \alpha {\Vert }_{\infty }\right)}{e}^{-\beta \Vert B{\Vert }_1\left(1-\Vert \sigma {\Vert }_{\infty }\right)}\Vert }_1}{\gamma +\mu +{\eta }_1}\stackrel{def}{=}{\epsilon }_3.$$

Finally, from system (2.2), we have

$$\frac{dI(t)}{dt}\ge \gamma A(t)-\left({\eta }_2+\mu \right)I(t).$$

Similar as for obtaining lim inf_t→+∞ A(t) ≥ ε₃, we can get

$$\underset{t\to +\infty }{\text{lim}\text{inf}}I(t)\ge \frac{\gamma {\epsilon }_3}{{\eta }_2+\mu }\stackrel{def}{=}{\epsilon }_4.$$

Let ε = min {η, ε₁, ε₂, ε₃, ε₄}. This completes the proof. □

5. Global stability

In this section, we shall establish the global stability of the equilibria in system (2.2). We give the following Theorem:

Theorem 5.1.

(i) If R_v < 1, then the disease-free equilibrium E₀ of system (2.2) is globally asymptotically stable; (ii) If R_v > 1, then the endemic equilibrium E* of system (2.2) is globally asymptotically stable provided that η₁ = η₂ and θ = 1.

Proof.

Case (i). From the first equation in (2.2), we have

$$\frac{dS(t)}{dt}\le (1-p)\mathrm{\Lambda }-\mu S(t)+{\int }_0^{+\infty }\alpha (a)v(a,t)da.$$

By Lemma 4.1, we can choose a sequence {t_n}, such that

$$\frac{dS\left(t_n\right)}{dt}\le (1-p)\mathrm{\Lambda }-\mu S\left(t_n\right)+{\int }_0^{t_n}\alpha (a)p\mathrm{\Lambda }\pi (a)da+{\int }_{t_n}^{\infty }{v}_0(a-t)\frac{\pi (a)}{\pi \left(a-t_n\right)}\frac{{\pi }_1(a)}{{\pi }_1\left(a-t_n\right)}da.$$

When n → ∞, such that t_n → ∞, S(t_n) → S^∞, $\frac{dS\left(t_n\right)}{dt}\to 0$, we have

$$S^{\infty }\le (1-p)\frac{\mathrm{\Lambda }}{\mu }+\frac{1}{\mu }{\int }_0^{+\infty }\alpha (a)p\mathrm{\Lambda }\pi (a)da,$$

where we apply $\underset{n\to \infty }{\text{lim}}{\int }_{t_n}^{\infty }{v}_0(a-t)\frac{\pi (a)}{\pi \left(a-t_n\right)}\frac{{\pi }_1(a)}{{\pi }_1\left(a-t_n\right)}=0$. Thus, we have S^∞ ≤S⁰. Consider the following Lyapunov function

$$W\text{(}t\text{)}\text{=}{b}_{\text{1}}{L}_{\text{1}}\text{(}t\text{)}\text{+}{b}_{\text{2}}{L}_{\text{2}}\text{(}t\text{)}\text{+}{b}_{\text{3}}{L}_{\text{3}}\text{(}t\text{),}$$

where

$$\begin{gathered} L_1(t)=A(t),L_2(t)=I(t),L_3(t)=B(t), \\ {b}_1=\frac{1}{\gamma +\mu +{\eta }_1}\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right),b_2=\frac{{\eta }_2}{{\eta }_2+\mu },b_3=1. \end{gathered}$$

Differentiating W(t) along the trajectory of system (2.2), we obtain

$$\begin{gathered} \frac{dW(t)}{dt}=\left(\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{\theta }{\gamma +\mu +{\eta }_1}+\frac{{\eta }_2(1-\theta )}{\mu +{\eta }_2}\right)\beta B(t)S(t) \\ +\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{\beta B(t)}{\gamma +\mu +{\eta }_1}{\int }_0^{\infty }(1-\sigma (a))v(a,t)da-m_{0}B(t). \end{gathered}$$ (5.1)

Similarly, by Lemma 4.1, choose a sequence t_n such that W(t_n) → W^∞, $\frac{dW\left(t_n\right)}{dt}\to 0$. From Eq. (5.1), we have

$$\begin{gathered} 0\le \left(\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{\theta }{\gamma +\mu +{\eta }_1}+\frac{{\eta }_2(1-\theta )}{\mu +{\eta }_2}\right)\beta B^{\infty }{S}^{\infty } \\ +\left({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2}\right)\frac{\beta B^{\infty }}{\gamma +\mu +{\eta }_1}{\int }_0^{\infty }(1-\sigma (a))p\mathrm{\Lambda }\pi (a)da-m_0{B}^{\infty } \\ \le m_0\left(R_v-1\right)B^{\infty }. \end{gathered}$$

This implies that B^∞ ≤0 (since R_v < 1). But since B_∞ ≥0, we have B^∞ = B_∞ = 0, and B(t) → 0 as t → ∞.

Using the equation for $\frac{dB(t)}{dt}$ in system (2.2) and Lemma 4.1, we have

$${\eta }_1{A}^{\infty }+{\eta }_2{I}^{\infty }\le {\mu }_0{B}^{\infty }.$$

Since A_∞ ≥0, I_∞ ≥0 we also have that A(t) → 0 and I(t) → 0 as t → ∞.

Similarly, using the equation for $\frac{dS(t)}{dt}$ in system (2.2) and Lemma 4.1, we have

$$0\ge (1-p)\mathrm{\Lambda }-\beta B_{\infty }{S}_{\infty }-\mu S_{\infty }+{\int }_0^{\infty }\alpha (a)p\mathrm{\Lambda }\pi (a)e^{-\beta {\int }_0^a(1-\sigma (\tau ))B_{\infty }(\tau )d\tau }da.$$

Thus, we obtain that S_∞ ≥S⁰. This implies that S(t) → S⁰ as t → ∞.

We also have v(a, t) → pΛπ(a) as t → ∞. Hence, when R_v < 1, the disease-free equilibium E₀ of system (2.2) is globally asymptotically stable.

Case (ii).

We now investigate the global stability of the infected equilibrium E* of model (2.2) by constructing Lyapunov function methods. We shall use the following generic form of a Lyapunov function

$$g(x(t))=x(t)-1-\text{ln}x(t),$$

where g(x) ≥0 for all x > 0.

Let us consider the following Lypunov functional G(t):

$$G\text{(}t\text{)}\text{=}{b}_{\text{1}}{G}_1\left(t\right)+b_{\text{2}}{G}_2\left(t\right)+b_{\text{3}}{G}_3\left(t\right)+b_{\text{4}}{G}_4\left(t\right)+b_2{G}_5\left(t\right),$$

where

$$G_1(t)=S^{*}g\left(\frac{S(t)}{S^{*}}\right),G_2(t)=A^{*}g\left(\frac{A(t)}{A^{*}}\right),G_3(t)=I^{*}g\left(\frac{I(t)}{I^{*}}\right),$$

$$G_4(t)=B^{*}g\left(\frac{B(t)}{B^{*}}\right),G_5(t)={\int }_0^{+\infty }\left[v^{*}(a)g\left(\frac{v(a,t)}{v^{*}(a)}\right)\right]da,$$

$$b_1=({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2})\frac{\theta }{\gamma +\mu +{\eta }_1}+\frac{{\eta }_2}{{\eta }_2+\mu }(1-\theta ),$$

$$b_2=({\eta }_1+\frac{\gamma {\eta }_2}{\mu +{\eta }_2})\frac{1}{\gamma +\mu +{\eta }_1},b_3=\frac{{\eta }_2}{{\eta }_2+\mu },b_4=1.$$

Differentiating $\frac{d{G}_1(t)}{dt}$ and using the equilibrium $\mu S^{*}=(1-p)\mathrm{\Lambda }+{\int }_0^{+\infty }\alpha (a)v^{*}(a)da-\beta B^{*}{S}^{*}$, we have

$$\begin{gathered} \frac{d{G}_1(t)}{dt}=\left(1-\frac{S^{*}}{S(t)}\right)\left[(1-p)\mathrm{\Lambda }-\beta B(t)S(t)-\mu S(t)+{\int }_0^{+\infty }\alpha (a)v(a,t)da\right] \\ =\left(1-\frac{S^{*}}{S(t)}\right)\left[(1-p)\mathrm{\Lambda }\left(1-\frac{S(t)}{S^{*}}\right)+{\int }_0^{+\infty }\alpha (a)v^{*}(a)da\left(\frac{v(a,t)}{v^{*}(a)}-\frac{S(t)}{S^{*}}\right)+\beta B^{*}{S}^{*}\left(\frac{S(t)}{S^{*}}-\frac{B(t)S(t)}{B^{*}{S}^{*}}\right)\right] \\ =(1-p)\mathrm{\Lambda }\left[-g\left(\frac{S^{*}}{S(t)}\right)-g\left(\frac{S(t)}{S^{*}}\right)\right]+{\int }_0^{+\infty }\alpha (a)v^{*}(a)da\left[-g\left(\frac{S(t)}{S^{*}}\right)+g\left(\frac{v(a,t)}{v^{*}(a)}\right)-g\left(\frac{v(a,t)S^{*}}{v^{*}(a)S(t)}\right)\right]+\beta B^{*}{S}^{*}\left[g\left(\frac{S(t)}{S^{*}}\right)-g\left(\frac{B(t)S(t)}{B^{*}{S}^{*}}\right)+g\left(\frac{B(t)}{B^{*}}\right)\right] \\ =-(1-p)\mathrm{\Lambda }g\left(\frac{S^{*}}{S(t)}\right)+{\int }_0^{+\infty }\alpha (a)v^{*}(a)da\left[g\left(\frac{v(a,t)}{v^{*}(a)}\right)-g\left(\frac{v(a,t)S^{*}}{v^{*}(a)S(t)}\right)\right]+\beta B^{*}{S}^{*}\left[-g\left(\frac{B(t)S(t)}{B^{*}{S}^{*}}\right)+g\left(\frac{B(t)}{B^{*}}\right)\right]-\mu S^{*}g\left(\frac{S(t)}{S^{*}}\right). \end{gathered}$$ (5.2)

Similarly, differentiating $\frac{d{G}_2(t)}{dt}$, we have

$$\begin{gathered} \frac{d{G}_2(t)}{dt}=\left(1-\frac{A^{*}}{A(t)}\right)\left[\theta \beta B(t)S(t)+{\int }_0^{\infty }(1-\sigma (a))\beta v(a,t)B(t)da-\left(\gamma +\mu +{\eta }_1\right)A(t)\right] \\ =\left(1-\frac{A^{*}}{A(t)}\right)\left[\theta \beta B^{*}{S}^{*}\left(\frac{B(t)S(t)}{B^{*}{S}^{*}}-1\right)+{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}da\left(\frac{v(a,t)B(t)}{v^{*}(a)B^{*}}-1\right)+\left(\gamma +\mu +{\eta }_1\right)A^{*}\left(1-\frac{A(t)}{A^{*}}\right)\right] \\ =\theta \beta B^{*}{S}^{*}\left[g\left(\frac{B(t)S(t)}{B^{*}{S}^{*}}\right)-g\left(\frac{B(t)S(t)A^{*}}{B^{*}{S}^{*}A(t)}\right)+g\left(\frac{A^{*}}{A(t)}\right)\right]+\left(\gamma +\mu +{\eta }_1\right)A^{*}\left[-g\left(\frac{A(t}{A^{*}}\right)-g\left(\frac{A^{*}}{A(t)}\right)\right] \\ +{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}da\left[g\left(\frac{B(t)v(a,t)}{v^{*}(a)B^{*}}\right)+g\left(\frac{A^{*}}{A(t)}\right)-g\left(\frac{B(t)v(a,t)A^{*}}{v^{*}(a)B^{*}A(t)}\right)\right]. \end{gathered}$$ (5.3)

By direct computation, $\frac{d{G}_3(t)}{dt}$ gives

$$\begin{gathered} \frac{d{G}_3(t)}{dt}=\left(1-\frac{I^{*}}{I(t)}\right)\left[(1-\theta )\beta B(t)S(t)+\gamma A(t)-\left({\eta }_2+\mu \right)I(t)\right] \\ =\left(1-\frac{I^{*}}{I(t)}\right)\left[(1-\theta )\beta B(t)S(t)-(1-\theta )\beta B^{*}{S}^{*}+\gamma A(t)-\gamma A^{*}-\left({\eta }_2+\mu \right)I(t)+\left({\eta }_2+\mu \right)I^{*}\right] \\ =(1-\theta )\beta B^{*}{S}^{*}\left[g\left(\frac{B(t)S(t)}{B^{*}{S}^{*}}\right)-g\left(\frac{B(t)S(t)I^{*}}{B^{*}{S}^{*}I(t)}\right)+g\left(\frac{I^{*}}{I(t)}\right)\right] \\ +\gamma A^{*}\left[g(\frac{A(t}{A^{*}})-g(\frac{I^{*}A(t)}{I(t)A^{*}})+g(\frac{I^{*}}{I(t)})\right] \\ +\left({\eta }_2+\mu \right)I^{*}\left[-g\left(\frac{I^{*}}{I(t)}\right)-g\left(\frac{I(t)}{I^{*}}\right)\right]. \end{gathered}$$ (5.4)

By directing computing the derivative of $\frac{d{G}_4(t)}{dt}$ with respect to t, we have

$$\begin{gathered} \frac{d{G}_4(t)}{dt}=(1-\frac{B^{*}}{B(t)})[{\eta }_{1}A(t)+{\eta }_{2}I(t)-m_{0}B(t)] \\ ={\eta }_1{A}^{*}[g(\frac{A(t)}{A^{*}})-g(\frac{B^{*}A(t)}{B(t)A^{*}})+g(\frac{B^{*}}{B(t)})]+{\eta }_2{I}^{*}[g(\frac{I(t)}{I^{*}})-g(\frac{B^{*}I(t)}{B(t)I^{*}})+g(\frac{B^{*}}{B(t)})]+m_0{B}^{*}[-g(\frac{B^{*}}{B(t)})-g(\frac{B(t)}{B^{*}})]. \end{gathered}$$ (5.5)

From Eqs. (5.2)–(5.5), we obtain

$$\begin{gathered} b_1\frac{d{G}_1(t)}{dt}+b_2\frac{d{G}_2(t)}{dt}+b_3\frac{d{G}_3(t)}{dt}+b_4\frac{d{G}_4(t)}{dt} \\ =-b_1(1-p)\mathrm{\Lambda }g(\frac{S^{*}}{S(t)})-b_1\mu S^{*}g(\frac{S(t)}{S^{*}})-b_2\theta \beta B^{*}{S}^{*}g(\frac{B(t)S(t)A^{*}}{B^{*}{S}^{*}A(t)}) \\ -b_2{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}dag(\frac{B(t)v(a,t)A^{*}}{v^{*}(a)B^{*}A(t)})-b_3(1-\theta )\beta B^{*}{S}^{*}g(\frac{B(t)S(t)I^{*}}{B^{*}{S}^{*}I(t)}) \\ -b_3\gamma A^{*}g(\frac{1^{*}A(t)}{I(t)A^{*}})-{\eta }_1{A}^{*}g(\frac{B^{*}A(t)}{B(t)A^{*}})-{\eta }_2{I}^{*}g(\frac{B^{*}I(t)}{B(t)I^{*}}) \\ +b_1{\int }_0^{+\infty }\alpha (a)v^{*}(a)da(\frac{v(a,t)}{v^{*}(a)}-\frac{v(a,t)S^{*}}{v^{*}(a)S(t)}+ln\frac{S^{*}}{S(t)}) \\ +b_2{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}dag(\frac{B(t)v(a,t)}{v^{*}(a)B^{*}})+(b_1\beta B^{*}{S}^{*}-m_0{B}^{*})g(\frac{B(t)}{B^{*}}). \end{gathered}$$ (5.6)

By directly computing the derivative of G₅(t) with respect to t, we obtain

$$\begin{gathered} \frac{d{G}_5(t)}{dt}={\int }_0^{+\infty }[(1-\frac{v^{*}(a)}{v(a,t)})\frac{\partial v(a,t)}{\partial t}]da \\ =-{\int }_0^{+\infty }(1-\frac{v^{*}(a)}{v(a,t)})[\frac{\partial v(a,t)}{\partial a}+(\mu +\alpha (a))v(a,t)+(1-\sigma (a))\beta v(a,t)B(t)]da \\ =-{\int }_0^{+\infty }{v}^{*}(a)(\frac{v(a,t)}{v^{*}(a)}-1)[\frac{\partial v(a,t)}{\partial a}\frac{1}{v(a,t)}+(\mu +\alpha (a))+(1-\sigma (a))\beta B(t)]da \\ =-{\int }_0^{+\infty }{v}^{*}(a)(\frac{v(a,t)}{v^{*}(a)}-1)[\frac{\partial v(a,t)}{\partial a}\frac{1}{v(a,t)}+(\mu +\alpha (a))+(1-\sigma (a))\beta B^{*}]da \\ +\beta B^{*}{\int }_0^{+\infty }{v}^{*}(a)(1-\sigma (a))(\frac{v(a,t)}{v^{*}(a)}-1-\frac{v(a,t)B(t)}{v^{*}(a)B^{*}}+\frac{B(t)}{B^{*}})da. \end{gathered}$$ (5.7)

In addition, by direct computation, we have

$$\frac{\partial }{\partial a}g(\frac{v(a,t)}{v^{*}(a)})=(1-\frac{v^{*}(a)}{v(a,t)})(\frac{{\partial }_{a}v(a,t)}{v^{*}(a)}-\frac{{\partial }_a{v}^{*}(a)v(a,t)}{v^{*}(a)v^{*}(a)}).$$ (5.8)

$${\partial }_a{v}^{*}(a)=-(\mu +\alpha (a))v^{*}(a)-(1-\sigma (a))\beta v^{*}(a)B^{*}.$$ (5.9)

Substituting Eq. (5.9) into Eq. (5.8) yields

$$v^{*}(a)\frac{\partial }{\partial a}g(\frac{v(a,t)}{v^{*}(a)})=(1-\frac{v^{*}(a)}{v(a,t)})({\partial }_{a}v(a,t)+(\mu +\alpha (a)+(1-\sigma (a))\beta B^{*})v(a,t)).$$ (5.10)

Substituting Eq. (5.10) into (5.7) gives

$$\begin{gathered} \frac{d{G}_5(t)}{dt}=-{\int }_0^{+\infty }{v}^{*}(a)\frac{\partial }{\partial a}g(\frac{v(a,t)}{v^{*}(a)})da \\ +\beta B^{*}{\int }_0^{+\infty }{v}^{*}(a)(1-\sigma (a))(\frac{v(a,t)}{v^{*}(a)}-1-\frac{v(a,t)B(t)}{v^{*}(a)B^{*}}+\frac{B(t)}{B^{*}})da. \end{gathered}$$ (5.11)

Integrating by parts for $v^{*}(a)g\left(\frac{v(a,t)}{v^{*}(a)}\right)$, we have

$$\begin{gathered} {\int }_0^{+\infty }{v}^{*}(a)\frac{\partial }{\partial a}g(\frac{v(a,t)}{v^{*}(a)})da \\ ={v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_0^{+\infty }-{\int }_0^{+\infty }g(\frac{v(a,t)}{v^{*}(a)})\frac{\partial }{\partial a}{v}^{*}(a)da \\ ={v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_{a=+\infty }-v^{*}(0)g(\frac{v(0,t)}{v^{*}(0)})-{\int }_0^{+\infty }g(\frac{v(a,t)}{v^{*}(a)})\frac{\partial }{\partial a}{v}^{*}(a)da. \end{gathered}$$ (5.12)

Noting that

$$v^0(0)=p\mathrm{\Lambda },v(0,t)=p\mathrm{\Lambda },{\partial }_a{v}^{*}(a)=-(\mu +\alpha (a))v^{*}(a)-(1-\sigma (a))\beta v^{*}(a)B^{*}.$$

Thus, we obtain

$$\begin{gathered} {\int }_0^{+\infty }{v}^{*}(a)\frac{\partial }{\partial a}g(\frac{v(a,t)}{v^{*}(a)})da \\ ={v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_{a=+\infty }+{\int }_0^{+\infty }(\mu +\alpha (a)+(1-\sigma (a)\beta B^{*}))v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})da. \end{gathered}$$ (5.13)

Substituting Eq. (5.13) into Eq. (5.11) gives

$$\begin{gathered} \frac{d{G}_5(t)}{dt}=-{v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_{a=+\infty }-{\int }_0^{+\infty }(\mu +\alpha (a))v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})da \\ +\beta B^{*}{\int }_0^{+\infty }(1-\sigma (a))v^{*}(a)(ln\frac{v(a,t)}{v^{*}(a)}-\frac{v(a,t)B(t)}{v^{*}(a)B^{*}}+\frac{B(t)}{B^{*}})da \\ =-{v^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_{a=+\infty }-\mu {\int }_0^{+\infty }{v}^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})da-\alpha (a){\int }_0^{+\infty }{v}^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})da \\ +\beta B^{*}{\int }_0^{+\infty }(1-\sigma (a))v^{*}(a)[-g(\frac{v(a,t)B(t)}{v^{*}(a)B^{*}})+g(\frac{B(t)}{B^{*}})]da. \end{gathered}$$ (5.14)

To proceed, we assume that η₁ = η₂ and θ = 1. That is, we assume that the asymptomatic infected and infected individuals have the same recovery rates, and that all the initially infected individuals are asymptomatic when they first contract the disease (though, some of them will become symptomatic later on). Combining (5.6) and (5.14), we have

$$\begin{gathered} \frac{dG(t)}{dt}=b_1\frac{d{G}_1(t)}{dt}+b_2\frac{d{G}_2(t)}{dt}+b_3\frac{d{G}_3(t)}{dt}+b_2\frac{d{G}_4(t)}{dt}+b_2\frac{d{G}_5(t)}{dt} \\ =-b_1(1-p)\mathrm{\Lambda }g(\frac{S^{*}}{S(t)})-b_1\mu S^{*}g(\frac{S(t)}{S^{*}})-b_2\theta \beta B^{*}{S}^{*}g(\frac{B(t)S(t)A^{*}}{B^{*}{S}^{*}A(t)}) \\ -b_2{\int }_0^{\infty }(1-\sigma (a))\beta v^{*}(a)B^{*}dag(\frac{B(t)v(a,t)A^{*}}{v^{*}(a)B^{*}A(t)})-b_3(1-\theta )\beta B^{*}{S}^{*}g(\frac{B(t)S(t)I^{*}}{B^{*}{S}^{*}I(t)}) \\ -b_3\gamma A^{*}g(\frac{I^{*}A(t)}{I(t)A^{*}}) \\ -{\eta }_1{A}^{*}g(\frac{B^{*}A(t)}{B(t)A^{*}})-{\eta }_2{I}^{*}g(\frac{B^{*}I(t)}{B(t)I^{*}}) \\ -b_1{\int }_0^{+\infty }\alpha (a)v^{*}(a)dag(\frac{v(a,t)S^{*}}{v^{*}(a)S(t)})-{b_2{v}^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})|}_{a=+\infty } \\ -b_2\mu {\int }_0^{+\infty }{v}^{*}(a)g(\frac{v(a,t)}{v^{*}(a)})da. \end{gathered}$$

Therefore, we obtain $\frac{dG(t)}{dt}\le 0$ and the largest invariant set of $\left\{\frac{dG(t)}{dt}\le 0\right\}$ is {E*}. Hence by LaSalle’s invariance principle, the endemic equilibrium E* of system (2.2) is globally asymptotically stable. □

We additionally comment that the conditions η₁ = η₂ and θ = 1 in Theorem 5.1 are introduced to facilitate our stability analysis of the endemic equilibrium for R_v > 1. These conditions are sufficient, but may not be necessary, to ensure the global asymptotic stability of the endemic equilibrium. Nevertheless, in a practical sense, the ration of asymptomatic to symptomatic cholera infections is generally very high. It is found that the ratio of asymptomatic to symptomatic infections ranges from 3 to 100 for cholera [26], which corresponds to a range between 75% to 99% for θ in our model.

6. Numerical results

To verify our analytical prediction, we have conducted numerical simulation to system (2.2). A typical scenario for R_v < 1 is shown in Fig. 3. We observe that the numbers of the asymptomatic infected and infected individuals, and the concentration of the environmental bacteria, all approach 0 over time, indicating the asymptotic stability of the disease-free equilibrium. A typical scenario for R_v > 1 is shown in Fig. 4, where the asymptomatic infected and infected populations and the bacterial concentration all converge to a positive steady state, indicating the asymptotic stability of the endemic equilibrium. Meanwhile, Fig. 5 displays the variation of the vaccinated population density with respect to time and vaccination age under these two settings. These numerical findings are consistent with the results from Theorems 3.2 and 5.1. The parameter values used in these numerical simulations are: p = 0.3, Λ = 10 persons per day, β = 0.6 * 10⁻⁸ per person per day, μ = 6.3 * 10⁻⁵ per day, θ = 0.8, γ = 0.05 per day, η₁ = 0.3 per person per day, η₂ = 0.1 per person per day, and m₀ = 0.05 per day (for R_v < 1) or m₀ = 0.005 per day (for R_v > 1).

Fig. 3.

Typical numerical results illustrating the stability of the disease-free equilibrium E₀ when R_v < 1 : (a) asymptomatic infected and infected populations with respect to time; (b) bacterial concentration with respect to time.

Fig. 4.

Typical numerical results illustrating the stability of the endemic equilibrium E* when R_v > 1 : (a) asymptomatic infected and infected populations with respect to time; (b) bacterial concentration with respect to time.

Fig. 5.

Typical scenarios for the vaccinated population V(a, t) with respect to time and vaccination age: (a) R_v < 1; (b) R_v > 1.

7. Conclusion

A goal of this paper is to study the impact of vaccination age on the transmission dynamics of cholera. To that end, we have proposed a new mathematical model that couples an age-structured PDE (for the vaccinated individuals) with ODEs and integral equations (for the bacterial concentration and the susceptible, asymptomatic infected, and infected, individuals). We have carefully analyzed the equilibria of this model and derived the basic reproduction number R_v, based on which we have established threshold-type results for cholera transmission dynamics: when R_v < 1, the disease will be eliminated; when R_v > 1, the disease will persist.

The basic reproduction number R_v of our age-structured model represents the overall risk of cholera in a host population where vaccination is implemented. Compared to the basic reproduction number ${\mathcal{R}}_0$ associated with the simplified ODE model, R_v shows strong dependence on the vaccination age, through the age-dependent immunoprotective rate and immunity waning rate, and through the age-related disease-free state of the susceptible and vaccinated populations. Our analysis of both models (see Section 3, in particular) indicates that without incorporating the vaccination age, we might obtain an inaccurate estimate of the risk for cholera infection.

Our modeling framework incorporates a number of important factors associated with cholera vaccination, such as the vaccination age, the imperfect vaccination, the asymptomatic infection for vaccinated hosts, the age-dependent immuno-protective effect, and the age-dependent waning rate of the vaccine-induced immunity. The interplay of these different factors leads to complex transmission dynamics of cholera. Our current study emphasizes such interactions and tries to resolve the complex dynamics through rigorous and nontrivial mathematical analysis. The findings help to better understand the effects of vaccination, especially the impact of vaccination age, on cholera epidemic. Finally, our current model is based on the assumption that vaccines are administered for a portion of individuals that enter the host population, and this may be different from some common practice of cholera vaccination where people of different chronological age groups are vaccinated. We hope to relax this assumption in our future research and study the interaction between vaccination age and chronological age, and their collective impact on cholera dynamics.

Fig. 1.

Flow diagram of the cholera model (2.2).

Appendix A.

Appendix: ODE model

Here we consider a simplification of system (2.2). Let α(a) and σ(a) both be constants; i.e., α(a) ≡α and σ(a) ≡σ. Meanwhile, let $\tilde{V}(t)={\int }_0^{\infty }V(a,t)da$. Then system (2.2) is reduced to the following ODE system

$$\begin{gathered} \frac{dS}{dt}=(1-p)\mathrm{\Lambda }+\alpha V(t)-\beta B(t)S(t)-\mu S(t), \\ \frac{dA}{dt}=\theta \beta B(t)S(t)+(1-\sigma )\beta V(t)B(t)-\left(\gamma +\mu +{\eta }_1\right)A(t), \\ \frac{dI}{dt}=(1-\theta )\beta B(t)S(t)+\gamma A(t)-\left(\mu +{\eta }_2\right)I(t), \\ \frac{dV}{dt}=p\mathrm{\Lambda }-(\mu +\alpha )V(t)-(1-\sigma )\beta V(t)B(t), \\ \frac{dB}{dt}={\eta }_{1}A(t)+{\eta }_{2}I(t)-m_{0}B(t), \end{gathered}$$ (7.1)

where, for convenience of notation, we have changed $\tilde{V}(t)$ back to V(t).

System (7.1) has a unique disease-free equilibrium (DFE) at

$${\mathcal{E}}_0=\left(S_0,0,0,V_0,0\right)=\left(\frac{\mathrm{\Lambda }}{\mu }\left(1-\frac{p\mu }{\mu +\alpha }\right),0,0,\frac{p\mathrm{\Lambda }}{\mu +\alpha },0\right).$$ (7.2)

The new infection matrix $\mathcal{F}$ and the transition matrix $\mathcal{V}$ are given by

$$\mathcal{F}=\left[\begin{array}{lll}0\hfill & 0\hfill & {\beta }_1\hfill \\ 0\hfill & 0\hfill & {\beta }_2\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right]\text{and}\mathcal{V}=\left[\begin{array}{ccc}\gamma +\mu +{\eta }_1& 0& 0\\ -\gamma & \mu +{\eta }_2& 0\\ -{\eta }_1& -{\eta }_2& m_0\end{array}\right],$$ (7.3)

where

$${\beta }_1\triangleq \theta \beta S_0+(1-\sigma )\beta V_0,{\beta }_2\triangleq (1-\theta )\beta S_0.$$

It follows that the next-generation matrix is given by

$$\mathcal{F}{\mathcal{V}}^{-1}=\left[\begin{array}{ccc}\frac{{\beta }_1}{m_0\left(\gamma +\mu +{\eta }_1\right)}\left({\eta }_1+\frac{{\eta }_2\gamma }{\mu +{\eta }_2}\right)& \frac{{\beta }_1{\eta }_2}{m_0\left(\mu +{\eta }_2\right)}& \frac{{\beta }_1}{m_0}\\ \frac{{\beta }_2}{m_0\left(\gamma +\mu +{\eta }_1\right)}\left({\eta }_1+\frac{{\eta }_2\gamma }{\mu +{\eta }_2}\right)& \frac{{\beta }_2{\eta }_2}{m_0\left(\mu +{\eta }_2\right)}& \frac{{\beta }_2}{m_0}\\ 0& 0& 0\end{array}\right].$$ (7.4)

The basic reproduction number of model (7.1) is then defined as the spectral radius of the matrix $\mathcal{F}{\mathcal{V}}^{-1}$ i.e.,

$${\mathcal{R}}_0=\rho \left(\mathcal{F}{\mathcal{V}}^{-1}\right)=\frac{{\eta }_1{\beta }_1}{m_0\left(\gamma +\mu +{\eta }_1\right)}+\frac{\gamma {\eta }_2{\beta }_1}{m_0\left(\gamma +\mu +{\eta }_1\right)\left(\mu +{\eta }_2\right)}+\frac{{\eta }_2{\beta }_2}{m_0\left(\mu +{\eta }_2\right)}.$$ (7.5)

There are three parts in the expression of ${\mathcal{R}}_0$ in Eq. (7.5). The first part describes the contribution from the asymptomatic infection, the third part describes the contribution from the symptomatic infection, and the second part represents the contribution from the interaction between the two. Together, these three factors shape the overall disease risk for the simplified cholera model given by the ODE system (7.1).

In addition, since $\frac{dV}{dt}\le p\mathrm{\Lambda }-(\mu +\alpha )V(t)$ leads to $\underset{t\to \infty }{\text{lim}\text{sup}}V(t)\le V_0$, we obtain the following biologically feasible domain for system (7.1):

$$\Gamma =\left\{(S,A,I,V,B)\in {ℝ}_{+}^5:S+A+I+V\le \frac{\mathrm{\Lambda }}{\mu },S\le S_0,V\le V_0,B\le \frac{\mathrm{\Lambda }\left({\eta }_1+{\eta }_2\right)}{\mu m_0}\right\}.$$

The endemic equilibrium (S*, A*, I*, V*, B*) of system (7.1) satisfies the following equations

$$(1-p)\mathrm{\Lambda }+\alpha V^{*}-\beta B^{*}{S}^{*}-\mu S^{*}=0,$$ (7.6)

$$\theta \beta B^{*}{S}^{*}+(1-\sigma )\beta V^{*}{B}^{*}-\left(\gamma +\mu +{\eta }_1\right)A^{*}=0,$$ (7.7)

$$(1-\theta )\beta B^{*}{S}^{*}+\gamma A^{*}-\left(\mu +{\eta }_2\right)I^{*}=0,$$ (7.8)

$$p\mathrm{\Lambda }-(\mu +\alpha )V^{*}-(1-\sigma )\beta V^{*}{B}^{*}=0,$$ (7.9)

$${\eta }_1{A}^{*}+{\eta }_2{I}^{*}-m_0{B}^{*}=0.$$ (7.10)

Using Eqs. (7.6)–(7.9), we easily obtain

$$V^{*}\left(B^{*}\right)=\frac{p\mathrm{\Lambda }}{\mu +\alpha +(1-\sigma )\beta B^{*}},$$ (7.11)

$$S^{*}\left(B^{*}\right)=\frac{(1-p)\mathrm{\Lambda }+\alpha V^{*}\left(B^{*}\right)}{\beta B^{*}+\mu },$$ (7.12)

$$A^{*}\left(B^{*}\right)=\frac{\beta B^{*}}{\gamma +\mu +{\eta }_1}\left(\theta S^{*}\left(B^{*}\right)+(1-\sigma )V^{*}\left(B^{*}\right)\right),$$ (7.13)

$$I^{*}\left(B^{*}\right)=\frac{(1-\theta )\beta S^{*}\left(B^{*}\right)B^{*}+\gamma A^{*}\left(B^{*}\right)}{\mu +{\eta }_2},$$ (7.14)

one after another. If we denote $f\left(B^{*}\right)=\frac{{\eta }_1{A}^{*}\left(B^{*}\right)+{\eta }_2{I}^{*}\left(B^{*}\right)}{m_0{B}^{*}}$, then f(B*) is a decreasing function with f(+∞) = 0 and $f(0+)={\mathcal{R}}_0$. Hence, Eq. (7.10) has no positive solution when ${\mathcal{R}}_0\le 1$ and has a unique positive solution when ${\mathcal{R}}_0>1$. Thus, system (7.1) has a unique endemic equilibrium ${\mathcal{E}}_1=\left(S^{*},A^{*},I^{*},V^{*},B^{*}\right)$ if and only if ${\mathcal{R}}_0>1$. We restate this result and establish the global stability of the DFE ${\mathcal{E}}_0$ in the following theorem.

Theorem 7.1.

If ${\mathcal{R}}_0\le 1$, then system (7.1) has a unique equilibrium, the DFE ${\mathcal{E}}_0$, and it is globally asymptotically stable Г. If ${\mathcal{R}}_0>1$, ${\mathcal{E}}_0$ becomes unstable and system (7.1) admits a unique endemic equilibrium ${\mathcal{E}}_1$.

Proof.

Let y = (A, I, B)^T. One can verify that $\frac{dy}{dt}={\left(\frac{dA}{dt},\frac{dI}{dt},\frac{dB}{dt}\right)}^T\le (\mathcal{F}-\mathcal{V})y$. We define a Lyapunov function as follows

$$\mathcal{L}=w^T{\mathcal{V}}^{-1}y,$$

where w^T = (β₁, β₂, 0) is a left eigenvector of ${\mathcal{V}}^{-1}\mathcal{F}$ associated with the eigenvalue ${\mathcal{R}}_0$.

Hence,

$$\begin{gathered} \frac{d\mathcal{L}}{dt}=w^T{\mathcal{V}}^{-1}\frac{dy}{dt}\le w^T{\mathcal{V}}^{-1}(\mathcal{F}-\mathcal{V})y \\ =\left({\mathcal{R}}_0-1\right)w^{T}y\le 0,\text{if}{\mathcal{R}}_0\le 1. \end{gathered}$$

It is easy to see that $\frac{d\mathcal{L}}{dt}=0$ contains only the singleton $\left\{{\mathcal{E}}_0\right\}$. Therefore, by LaSalle’s invariance principle, ${\mathcal{E}}_0$ is globally asymptotically stable in Г when ${\mathcal{R}}_0\le 1$.

In contrast, if ${\mathcal{R}}_0>1$, then $\frac{d\mathcal{L}}{dt}>0$ in a neighborhood of ${\mathcal{E}}_0$ in Г. Solutions in Г sufficiently close to ${\mathcal{E}}_0$ will move away from x₀, which implies the instability of ${\mathcal{E}}_0$. □

Regarding the stability of the endemic equilibrium, we have the following result.

Theorem 7.2.

If ${\mathcal{R}}_0>1$, the endemic equilibrium ${\mathcal{E}}_1$ of system (7.1) is globally asymptotically stable in Г.

The proof of Theorem 7.2 is similar to that of Theorem 5.1. We outline the proof here without stating the details. Construct the following Lyapunov function:

$$G(t)=b{S}^{*}g(\frac{S(t)}{S^{*}})+b{A}^{*}g(\frac{A(t)}{A^{*}})+b{I}^{*}g(\frac{I(t)}{I^{*}})+b{V}^{*}g(\frac{V(t)}{V^{*}})+B^{*}g(\frac{B(t)}{B^{*}})$$ (7.15)

where $b=\frac{{\eta }_1}{{\eta }_1+\mu }$ Through similar algebraic manipulations as in Eqs. (5.11)–(5.24), we obtain

$$\begin{gathered} \frac{dG(t)}{dt}=-b(1-p)\mathrm{\Lambda }g(\frac{S^{*}}{S(t)})-b\mu S^{*}g(\frac{S(t)}{S^{*}})-b\beta B^{*}{S}^{*}g(\frac{B(t)S(t)A^{*}}{B^{*}{S}^{*}A(t)}) \\ -b(1-\sigma )\beta V^{*}{B}^{*}g(\frac{B(t)V(t)A^{*}}{V^{*}{B}^{*}A(t)})-b\gamma A^{*}g(\frac{I^{*}A(t)}{I(t)A^{*}}) \\ -{\eta }_1{A}^{*}g(\frac{B^{*}A(t)}{B(t)A^{*}})-{\eta }_1{I}^{*}g(\frac{B^{*}I(t)}{B(t)I^{*}})-b\alpha V^{*}g(\frac{V(t)S^{*}}{V^{*}S(t)})-b\mu V^{*}g(\frac{V(t)}{V^{*}}). \end{gathered}$$ (7.16)

Thus $\frac{dG(t)}{dt}\le 0$, and the largest invariant set of $\left\{\frac{dG(t)}{dt}=0\right\}$ is clearly the singleton $\left\{{\mathcal{E}}_1\right\}$. LaSalle’s invariance principle then implies that the endemic equilibrium ${\mathcal{E}}_1$ is globally asymptotically stable.