An Epidemic Model for Tick-Borne Disease with Two Delays

Abstract

We have considered an epidemic model of a tick-borne infection which has nonviraemic transmission in addition to the viremic transmission. The basic reproduction number ℜ₀, which is a threshold quantity for stability of equilibria, is calculated. If ℜ₀≤1, then the infection-free equilibrium is globally asymptotically stable, and this is the only equilibrium. On the contrary, if ℜ₀>1, then an infection equilibrium appears which is globally asymptotically stable, when one time delay is absent. By applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when ℜ₀>1.

1. Introduction

Vector-borne diseases are infectious diseases caused by viruses, bacteria, protozoa, or rickettsia which are primarily transmitted by disease-transmitting biological agents, called vectors, who carry the disease without getting it themselves. Vector-borne disease, in particular, tick-borne disease, such as Lyme disease, transmitted between humans by blood-feeding ticks, have been big concerns for the public health in the world.

The dynamics of the tick vector are based on the biology of the tick Ixodes ricinus that transmits the virus that causes Louping ill. Louping ill is a tick-borne, zoonotic, viral disease that is most important in sheep and red grouse. This tick has a life cycle that develops from the egg through two immature stages (larva and nymph) to the adult stage. Each immature stage requires a blood meal from a suitable vertebrate host. The adult female requires a meal before producing eggs once and then dying. Adult females can only obtain a feed from large mammals, that is, hares and not grouse, whilst the immature stages will also feed on smaller warm-blooded vertebrates such as grouse chicks. Once a questing tick locates a host it generally feeds for several days. The tick life cycle usually takes 3 to 4 years but can be longer depending on host availability and climatic conditions; ticks need high relative humidity to survive [1].

Mathematical models have made considerable contributions to our understanding of tick-borne infections [1–8] or tick population dynamics [9–11]. Transmission from host to tick is generally believed to occur when ticks bite and feed on the blood of a viraemic host. However, in recent years it has become clear that pathogen transmission can occur through a number of other routes. For example, transmission between infected and uninfected ticks that cofeed on a host can occur in the absence of systemic infection [1]. Another route of infection for some tick-borne pathogens is through nonviraemic transmission. This occurs via the skin aided by tick saliva when a susceptible tick feeds close to an infected tick on a host that is not displaying high levels of viraemia in the blood [12]. Grouse chicks are known to eat ticks as part of their diet of invertebrates during the first three weeks of life. Gilbert et al. [13] suggest that the ingestion of infected ticks may be the main route of infection in wild grouse chicks; the authors consider the ingestion model; in this case, the virus persists over a greater range of grouse densities when ingestion is included. This is because it is now easier for the grouse to become infected with the inclusion of an extra route of infection via the oral route. In this paper, we consider the nonseasonal model; that is, the behavior of grouse and ticks was assumed constant over the course of a year. The non seasonal model can be analyzed analytically unlike the seasonal models. To investigate whether the incorporation of the seasonal biology of grouse alters the model predictions dramatically from the non seasonal model. In [13], they present a more realistic model which describes the chicks and adult grouse separately with chicks hatching only at a given time of the year. In this paper we build on the model by Norman et al. [1], which have proposed an epidemic dynamics of Louping ill virus in red grouse (the viremic host) and hares (the nonviremic host) in the following form: 𝐺̇_𝑠(𝑡)=𝑎_𝑔−𝑠_𝑔𝐺_𝑠(𝑡)𝐺_𝑠(𝑡)−𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−𝑏_𝑔𝐺_𝑠(𝑡),𝐺̇_𝑖(𝑡)=𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−Γ𝐺_𝑖(𝑡),𝐺̇_𝑧(𝑡)=𝛾𝐺_𝑖(𝑡)−𝑏_𝑔𝐺_𝑧(𝑡),𝑇̇_𝑠(𝑡)=𝑎_𝑇−𝑠_𝑇𝑇_𝑠(𝑡)𝑇_𝑠(𝑡)−𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)−𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑠(𝑡),𝑇̇_𝑖(𝑡)=𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)+𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡). (1) The model considers two hosts, a viremic host that does not support adult ticks and a non-viremic host that supports all stages. The total viremic host population size at time 𝑡, given by 𝐺(𝑡), is partitioned into subclasses of individuals who are susceptible, infectious, and recovered, with sizes denoted by 𝐺_𝑠(𝑡), 𝐺_𝑖(𝑡), and 𝐺_𝑧(𝑡), respectively. Hares are assumed to be at a constant density 𝐻. 𝑇_𝑠(𝑡) and 𝑇_𝑖(𝑡) represent population density of susceptible ticks and infected ticks at any time 𝑡, with the total tick population, 𝑇(𝑡)=𝑇_𝑠(𝑡)+𝑇_𝑖(𝑡).

The rate of non-viremic transmission is assumed to be proportional to the number of hares present, the number of infected ticks present and the number of susceptible ticks present, and therefore takes the form 𝜃𝐻𝑇_𝑖𝑇_𝑠, where 𝜃 is a measure of the probability of non-viremic transmission occurring. Furthermore, the viremic host population dies at a natural death rate 𝑏_𝑔. The vertical transmission in the viremic host does not occur so that all newly recruited individuals are susceptible. The rate at which a grouse is bitten and infected by a tick 𝛽₁ can only involve the nymphal stage as adults do not generally bite grouse and the larvae are not infected. The rate at which nymphs bite grouse was estimated in [14] from tick counts of nymphs on deer and grouse as well as from model predictions; 𝛾 is the rate at which infectious grouse become immune, and Γ=𝛼+𝑏_𝑔+𝛾 is the rate at which infectious grouse are lost from the system and is the sum of the death rate, due to the disease, the natural death rate, and the rate at which individuals become immune. For the ticks, 𝑏_𝑇 is the per capita natural death rate. 𝛽₂ is the probability of an average tick biting a grouse and becoming infected per unit time, 𝛽₃ is the probability of an adult tick being female, biting a hare, and then going on to reproduce per unit time before dying and being lost from the system.

Norman et al. [1] consider the local stability of six biological relevant equilibria; this is done by determining the eigenvalues of the Jacobian evaluated at each equilibrium. The author chose the logistic growth function to describe the growth of grouse and ticks, obviously, it is more reasonable, but it increases the complexity of the algebraic analysis. To obtain global properties for model (1), we suppose that the host and tick population are recruited at rate that𝜆and𝑠. Note that the variables 𝐺_𝑧 do not appear in the other equations of the above system. This allows us to consider the following fours dimensional system: 𝐺̇_𝑠(𝑡)=𝜆−𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−𝑏_𝑔𝐺_𝑠(𝑡),𝐺̇_𝑖(𝑡)=𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−Γ𝐺_𝑖(𝑡),𝑇̇_𝑠(𝑡)=𝑠−𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)−𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑠(𝑡),𝑇̇_𝑖(𝑡)=𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)+𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡). (2) The incubation period for Louping ill is 6 to 18 days in sheep. Parenterally inoculated red grouse develop clinical signs in 2 to 8 days; the dynamic of tick feeding the incubation is effectively 12 months. We introduce two types of the delays that correspond to (i) the latent period in an infected host and (ii) the latency in a vector. We know that the number of infected hosts at time 𝑡 may depend on the number of susceptible hosts which is infected by the infected ticks at time 𝑡−𝜏, for a time delay 𝜏>0, and 𝑒^{−𝑏𝑔𝜏} is the survival rate. The number of infected ticks at time 𝑡 may depend on the number of susceptible ticks which is feeding on the infected hosts at time 𝑡−𝜔, for a time delay 𝜔>0, and 𝑒^{−𝑏𝑇𝜔} is the survival rate. Thus, the following model is proposed: 𝐺̇_𝑠(𝑡)=𝜆−𝑒^{−𝑏𝑔𝜏}𝛽₁𝑇_𝑖(𝑡−𝜏)𝐺_𝑠(𝑡)−𝑏_𝑔𝐺_𝑠(𝑡),𝐺̇_𝑖(𝑡)=𝑒^{−𝑏𝑔𝜏}𝛽₁𝑇_𝑖(𝑡−𝜏)𝐺_𝑠(𝑡)−Γ𝐺_𝑖(𝑡),𝑇̇_𝑠(𝑡)=𝑠−𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)−𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑠(𝑡),𝑇̇_𝑖(𝑡)=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)+𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡). (3) Although the models incorporation of time delay have more real biological meaning, they are difficult to deal with it mathematically. However, the Lyapunov functional and the LaSalle-type theorem in [15] can provide a direct and effective method to establish global dynamical properties for the system of those nonlinear functional differential equations. Recently, Korobeinikov [16–18], and Huang et al. [19] have investigated the global dynamical properties of epidemiological models with and without delay by construc suitable Lyapunov functional methods. Inspired by the work in McCluskey [20], we construct a global Lyapunov functional and show that the disease-free equilibrium of system (3) is globally asymptotically stable if ℜ₀≤1, and if ℜ₀>1 the infection equilibrium is globally asymptotically stable when one delay is absent. Furthermore, by applying a permanence theorem by Smith and Zhao [21] for infinite dimensional systems (also see [22]), we obtain that the disease is always present when ℜ₀>1. For its applications, we refer the reader to [10, 11, 21–23].

The paper is organized as follows. In the next section, we analyze some basic mathematical properties (positivity and boundedness of solutions). In Section 3, we give a complete global stability analysis of the disease-free equilibrium. In Section 4, we are concerned with the global stability of the infection equilibrium with one delay. In Section 5, we show that the disease is endemic in the sense of permanence whenever ℜ₀>1. A discussion of the mathematical results and of their biological implications is presented in Section 6.

2. Nonnegativity and Boundedness of Solutions

Let 𝐶=𝐶([−𝜎,0];𝑅⁴) be the Banach space of continuous functions from [−𝜎,0] to 𝑅⁴ equipped with the sup-norm, where 𝜎=max{𝜏,𝜔}. The initial condition of (3) is given as 𝐺_𝑠(𝜃)=𝜑₁(𝜃),𝐺_𝑖(𝜃)=𝜑₂(𝜃),𝑇_𝑠(𝜃)=𝜑₃(𝜃),𝑇_𝑖(𝜃)=𝜑₄(𝜃)𝜃∈[−𝜎,0], (4) where 𝜑=(𝜑₁,𝜑₂,𝜑₃,𝜑₄)∈𝐶 such that 𝜑_𝑗(𝜃)≥0(𝜎≤𝜃≤0,𝑗=1,2,3,4).

The following result establishes the feasible region of the model and shows that the model is wellposed.

Theorem 1. Under initial conditions in (4), all solutions of system (3) are nonnegative on [0,+∞) and ultimately bounded.

Proof. First, we prove that 𝐺_𝑠(𝑡) is positive for 𝑡≥0. Assuming the contrary and letting 𝑡₁>0 be the first time such that 𝐺_𝑠(𝑡₁)=0, by the first equation of system (3), we have 𝐺^′_𝑠(𝑡₁)=𝜆>0, and hence 𝐺_𝑠(𝑡)<0 for 𝑡∈(𝑡₁−𝜖,𝑡₁), where 𝜖 is an arbitrarily small positive constant. This leads to a contradiction. It follows that 𝐺_𝑠(𝑡) is always positive. Using the same method, we have that 𝑇_𝑠(𝑡) is always positive. Further, from the second and the forth equations in (3), we have, respectively, 𝐺_𝑖(𝑡)=𝐺_𝑖(0)𝑒^−Γ𝑡+𝑒^{−𝑏𝑔𝜏}𝛽₁^𝑡₀𝑇_𝑖(𝜉−𝜏)𝐺_𝑠(𝜉)𝑒^{−Γ(𝑡−𝜉)}𝑇𝑑𝜉,_𝑖(𝑡)=𝑇_𝑖(0)𝑒^{−∫𝑡0(𝜃𝐻𝑇𝑠(𝛾)−(𝛽3𝐻+𝑏𝑇))𝑑𝛾}+𝑒^{−𝑏𝑇𝜔}𝛽₂×^𝑡₀𝑇_𝑠(𝜉)𝐺_𝑖(𝜉−𝜔)𝑒^{−∫𝑡𝜉(𝜃𝐻𝑇𝑠(𝛾)−(𝛽3𝐻+𝑏𝑇))𝑑𝛾}𝑑𝜉. (5) Using (4) we know that the 𝐺_𝑖(𝑡) and 𝑇_𝑖(𝑡) of (3) are positive for all 𝑡≥0.
Next we show that positive solutions of (3) are ultimately bounded for 𝑡≥0.
Define 𝐹(𝑡)=𝐺_𝑠(𝑡)+𝐺_𝑖(𝑡)+𝑇_𝑠(𝑡)+𝑇_𝑖(𝑡) and 𝑑=min{𝑏_𝑔,𝛽₃𝐻+𝑏_𝑇}. By non-negativity of the solution, it follows that 𝐹^′(𝑡)=𝜆+𝑠−𝑏_𝑔𝐺_𝑠+Γ𝐺_𝑖+𝛽₃𝐻+𝑏_𝑇×𝑇_𝑠+𝑇_𝑖≤𝜆+𝑠−𝑑𝐹(𝑡). (6) This implies that 𝐹(𝑡) is ultimately bounded, and so are 𝐺_𝑠(𝑡), 𝐺_𝑖(𝑡), 𝑇_𝑠(𝑡), and 𝑇_𝑖(𝑡).
This completes the proof of Theorem 1.

The basic reproduction number [24] for system (3) is ℜ₀=𝑒^{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝜆𝑠𝛽₁𝛽₂Γ𝑏_𝑔𝛽₃𝐻+𝑏_𝑇²+𝑠𝜃𝐻𝛽₃𝐻+𝑏_𝑇². (7)ℜ₀ represents the average number of secondary infections that single infections host can generate in a totally susceptible population of hosts and ticks. The basic reproduction number consists of two terms representing two routes of infection. The tick would live for 1/(𝛽₃𝐻+𝑏_𝑇) units of time and produces (𝑒^{−𝑏𝑔𝜏}𝜆𝛽₁)/𝑏_𝑔(𝛽₃𝐻+𝑏_𝑇) infected grouse through feeding and 𝜃𝑠𝐻/(𝛽₃𝐻+𝑏_𝑇)² infected ticks through cofeeding. Each infected grouse lives for 1/Γ units of time and produces 𝑒^{−𝑏𝑇𝜔}𝛽₂𝑠/Γ(𝛽₃𝐻+𝑏_𝑇) infected ticks.

System (3) has always an infection-free equilibrium 𝐸₀=(𝐺⁰_𝑠,0,𝑇⁰_𝑠,0), that is, grouse and ticks both at their carrying capacity with no disease present, where 𝐺⁰_𝑠=𝜆/𝑏_𝑔, 𝑇⁰_𝑠=𝑠/(𝛽₃𝐻+𝑏_𝑇), and an infection equilibrium 𝐸^∗=(𝐺^∗_𝑠,𝐺^∗_𝑖,𝑇^∗_𝑠,𝑇^∗_𝑖), when ℜ₀>1, that is, ticks, grouse, and disease coexisting together, where 𝑇^∗_𝑖 is given by the following cubic: 𝑓𝑇^∗_𝑖=𝐴𝑇_𝑖^∗2+𝐵𝑇^∗_𝑖+𝐶, (8) where 𝐴=𝑒^{−𝑏𝑔𝜏}Γ𝜃𝐻𝛽₁𝛽₃𝐻+𝑏_𝑇,𝐵=−𝑠𝜃𝐻Γ𝛽₁𝑒^{−𝑏𝑔𝜏}+𝑒^{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝜆𝛽₁𝛽₂𝛽₃𝐻+𝑏_𝑇+Γ𝛽₃𝐻+𝑏_𝑇𝜃𝐻𝑏_𝑔+𝑒^{−𝑏𝑔𝜏}𝛽₁𝛽₃𝐻+𝑏_𝑇,𝐶=−𝑒^{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝜆𝑠𝛽₁𝛽₂−𝑠Γ𝜃𝐻𝑏_𝑔+Γ𝑏_𝑔𝛽₃𝐻+𝑏_𝑇². (9) Hence, we can write all of the densities in terms of the infectious tick density, 𝑇^∗_𝑖, 𝐺^∗_𝑠=𝜆𝑒^{−𝑏𝑔𝜏}𝛽₁𝑇^∗_𝑖+𝑏_𝑔,𝐺^∗_𝑖=𝑒^{−𝑏𝑔𝜏}𝜆𝛽₁𝑇^∗_𝑖Γ𝑒^{−𝑏𝑔𝜏}𝛽₁𝑇^∗_𝑖+𝑏_𝑔,𝑇^∗_𝑠=𝑠𝛽₃𝐻+𝑏_𝑇−𝑇^∗_𝑖. (10)

3. Global Stability of the Infection-Free Equilibrium of Model (3)

In this section, we would consider the global stability of the infection-free equilibrium. Next, we show that ℜ₀ determines the stability of the infection-free equilibrium.

Theorem 2. If ℜ₀≤1, then the infection-free equilibrium 𝐸₀ is globally asymptotically stable for any time delay 𝜏≥0 and 𝜔≥0.

Proof. Consider a Lyapunov functional: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (11) with 𝑉₁(𝑡)=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠Γ×𝐺_𝑠(𝑡)−𝐺⁰_𝑠−𝐺⁰_𝑠𝐺ln^𝑠(𝑡)𝐺⁰_𝑠+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠Γ𝐺_𝑖(𝑡)+𝑇_𝑠(𝑡)−𝑇⁰_𝑠−𝑇⁰_𝑠𝑇ln^𝑠(𝑡)𝑡⁰_𝑠+𝑇_𝑖(𝑡),𝑉₂(𝑡)=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠^𝑡_𝑡−𝜔𝐺_𝑖(𝜉)𝑉𝑑𝜉,₃(𝑡)=𝑒^{−(𝑏𝑔𝜏+𝑏𝑇𝜔)}𝛽₁𝛽₂𝐺⁰_𝑠𝑇⁰_𝑠Γ^𝑡_𝑡−𝜏𝑇_𝑖(𝜉)𝑑𝜉. (12) Calculate the time derivative of 𝑉: 𝑑𝑉=𝑒𝑑𝑡^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠Γ𝐺1−^0𝑠𝐺_𝑠(𝑡)𝐺^′_𝑠(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠Γ𝐺^′_𝑖(𝑡)+𝑇1−^0𝑠𝑇_𝑠(𝑡)𝑇^′_𝑠(𝑡)+𝑇^′_𝑖(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠𝐺_𝑖(𝑡)−𝐺_𝑖(+𝑒𝑡−𝜔)_{−(𝑏𝑔𝜏+𝑏𝑇𝜔)}𝛽₁𝛽₂𝐺⁰_𝑠𝑇⁰_𝑠Γ𝑇_𝑖(𝑡)−𝑇_𝑖(𝑡−𝜏). (13) Since 𝜆=𝑏_𝑔𝐺⁰_𝑠, 𝑠=(𝛽₃𝐻+𝑏_𝑇)𝑇⁰_𝑠, it follows that 𝑑𝑉=𝛽𝑑𝑡²𝑏_𝑔𝐺⁰_𝑠𝑇⁰_𝑠Γ𝐺2−^0𝑠𝐺_𝑠(𝑡)−𝐺_𝑠(𝑡)𝐺⁰_𝑠+𝛽₃𝐻+𝑏_𝑇𝑇⁰_𝑠𝑇2−^0𝑠𝑇_𝑠(𝑡)−𝑇_𝑠(𝑡)𝑇⁰_𝑠+𝛽₃𝐻+𝑏_𝑇ℜ₀−1𝑇_𝑖(𝑡). (14) Obviously ℜ₀≤1 ensures that 𝑑𝑉/𝑑𝑡≤0. It is clear that 𝑉≥0 and 𝑉=0 if and only if 𝐺_𝑠(𝑡)=𝐺⁰_𝑠, 𝐺_𝑖(𝑡)=0, 𝑇_𝑠(𝑡)=𝑇⁰_𝑠, and 𝑇_𝑖(𝑡)=0. Hence, it follows from the Lyapunov-LaSalle invariance principle that the infection-free equilibrium 𝐸₀ is globally asymptotically stable when ℜ₀≤1.
This completes the proof of Theorem 2.

4. Dynamical Analysis for the Infection Equilibrium

System (3) has an infection equilibrium 𝐸^∗ when ℜ₀>1. We consider two situations: (a) 𝜏=0, 𝜔≥0 and (b) 𝜏≥0, 𝜔=0.

4.1. Global Stability of the Infection Equilibrium When 𝜏=0 and ss𝜔≥0

In this subsection, the situations 𝜏=0 and 𝜔≥0 are considered.

Theorem 3. If ℜ₀>1, then the infection equilibrium 𝐸^∗ is globally asymptotically stable for 𝜏=0 and 𝜔≥0.

Proof. Define a Lyapunov functional: 𝑈₁(𝑡)=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺_𝑠(𝑡)−𝐺^∗_𝑠−𝐺^∗_𝑠𝐺ln^𝑠(𝑡)𝐺^∗_𝑠+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺_𝑖(𝑡)−𝐺^∗_𝑖−𝐺^∗_𝑖𝐺ln^𝑖(𝑡)𝐺^∗_𝑖+𝑇_𝑠(𝑡)−𝑇^∗_𝑠−𝑇^∗_𝑠𝑇ln^𝑠(𝑡)𝑇^∗_𝑠+𝑇_𝑖(𝑡)−𝑇^∗_𝑖−𝑇^∗_𝑖𝑇ln^𝑖(𝑡)𝑇^∗_𝑖+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝑈₊(𝑡), (15) where 𝑈₊(𝑡)=^𝜔₀𝐺_𝑖(𝑡−𝜃)𝐺^∗_𝑖𝐺−1−ln^𝑖(𝑡−𝜃)𝐺^∗_𝑖𝑑𝜃. (16) The derivatives of 𝑈₊(𝑡) are given by 𝑑𝑈₊(𝑡)=𝑑𝑑𝑡𝑑𝑡^𝜔₀𝐺_𝑖(𝑡−𝜃)𝐺^∗_𝑖𝐺−1−ln^𝑖(𝑡−𝜃)𝐺^∗_𝑖=𝑑𝜃_𝜔0𝑑𝐺𝑑𝑡^𝑖(𝑡−𝜃)𝐺^∗_𝑖𝐺−1−ln^𝑖(𝑡−𝜃)𝐺^∗_𝑖𝑑𝜃=−^𝜔₀𝑑𝐺𝑑𝜃^𝑖(𝑡−𝜃)𝐺^∗_𝑖𝐺−1−ln^𝑖(𝑡−𝜃)𝐺^∗_𝑖=𝐺𝑑𝜃_𝑖(𝑡)𝐺^∗_𝑖−𝐺_𝑖(𝑡−𝜔)𝐺^∗_𝑖𝐺+ln^𝑖(𝑡−𝜔)𝐺_𝑖(𝑡). (17) Hence, we obtain 𝑑𝑈₁=𝑒𝑑𝑡^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺1−^∗𝑠𝐺_𝑠(𝑡)𝐺^′_𝑠(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝑆^∗𝐺1−^∗𝑖𝐺_𝑖(𝑡)𝐺^′_𝑖(𝑡)+𝑇1−^∗𝑠𝑇_𝑠(𝑡)𝑇^′_𝑠(𝑡)+𝑇1−^∗𝑖𝑇_𝑖(𝑡)𝑇^′_𝑖(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝑑𝑈₊(𝑡)=𝑒𝑑𝑡_{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝜆−𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−𝑏_𝑔𝐺_𝑠(𝑡)+𝐺^∗_𝑠𝐺_𝑠(𝑡)𝜆−𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−𝑏_𝑔𝐺_𝑠(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−Γ𝐺_𝑖(𝑡)+𝐺^∗_𝑖𝐺_𝑖(𝑡)𝛽₁𝑇_𝑖(𝑡)𝐺_𝑠(𝑡)−Γ𝐺_𝑖(𝑡)+𝑠−𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)−𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑠(𝑡)+𝑇^∗_𝑠𝑇_𝑠(𝑡)𝑠−𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)−𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑠(𝑡+𝑒_{−𝑏𝑇𝜔})𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)+𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡)+𝑇^∗_𝑖𝑇_𝑖(𝑡)𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)+𝜃𝐻𝑇_𝑠(𝑡)𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝐺_𝑖(𝑡)𝐺^∗_𝑖−𝐺_𝑖(𝑡−𝜔)𝐺^∗_𝑖𝐺+ln^𝑖(𝑡−𝜔)𝐺_𝑖(𝑡). (18) Since 𝜆=𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠+𝑏_𝑔𝐺^∗_𝑠, 𝑠=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖−𝜃𝐻𝑇^∗_𝑠𝑇^∗_𝑖−(𝛽₃𝐻+𝑏_𝑇)𝑇^∗_𝑠, and Γ𝐺^∗_𝑖=𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠,𝛽₃𝐻+𝑏_𝑇𝑇^∗_𝑖=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖+𝜃𝐻𝑇^∗_𝑠𝑇^∗_𝑖, (19) it follows that 𝑑𝑈₁=𝑒𝑑𝑡^{−𝑏𝑇𝜔}𝛽₂𝑏_𝑔𝐺^∗_𝑠𝑇^∗_𝑠𝛽₁𝑇^∗_𝑖×𝐺2−^∗𝑠𝐺_𝑠(𝑡)−𝐺_𝑠(𝑡)𝐺^∗_𝑠+𝜃𝐻𝑇^∗_𝑖𝑇^∗_𝑠𝑇2−^∗𝑠𝑇_𝑠(𝑡)−𝑇_𝑠(𝑡)𝑇^∗_𝑠+𝛽₃𝐻+𝑏_𝑇𝑇^∗_𝑠𝑇2−^∗𝑠𝑇_𝑠(𝑡)−𝑇_𝑠(𝑡)𝑇^∗_𝑠+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖×𝐺4−^∗𝑠𝐺_𝑠(𝑡)−𝐺^∗_𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡)−𝑇^∗_𝑠𝑇_𝑠(𝑡)−𝑇^∗_𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡). (20) Using equality 𝐺ln^𝑖(𝑡−𝜔)𝐺_𝑖(𝑡)𝐺=ln^∗𝑠𝐺_𝑠(𝑡)𝑇+ln^∗𝑠𝑇_𝑠(𝑡)𝑇+ln^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡)𝐺+ln^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡), (21) we obtain 𝑑𝑈₁𝑑𝑡=−𝑏_𝑔𝐺^∗_𝑠1𝐺_𝑠(𝑡)𝐺_𝑠(𝑡)−𝐺^∗_𝑠²−𝜃𝐻𝑇^∗_𝑖𝑇^∗_𝑠1𝑇_𝑠(𝑡)𝑇_𝑠(𝑡)−𝑇^∗_𝑠²−𝛽₃𝐻+𝑏_𝑇𝑇^∗_𝑠1𝑇_𝑠(𝑡)𝑇_𝑠(𝑡)−𝑇^∗_𝑠²+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖Ω₁+Ω₂+Ω₃+Ω₄, (22) where Ω₁𝐺=1−^∗𝑠𝐺_𝑠(𝑡)𝐺+ln^∗𝑠𝐺_𝑠(𝑡),Ω₂𝐺=1−^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡)𝐺+ln^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡),Ω₃𝑇=1−^∗𝑠𝑇_𝑠(𝑡)𝑇+ln^∗𝑠𝑇_𝑠(𝑡),Ω₄𝑇=1−^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡)𝑇+ln^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡−𝜔)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡). (23) Since the function 𝑔(𝑡)=1−𝑓(𝑡)+ln𝑓(𝑡) (24) is always nonpositive for any function 𝑓(𝑡)>0, and 𝑔(𝑡)=0 if and only if 𝑓(𝑡)=1. Therefore, we have Ω₁,Ω₂,Ω₃,Ω₄≤0.
It is easy to see that 𝑑𝑈₁/𝑑𝑡≤0. It is clear that 𝑑𝑈₁/𝑑𝑡=0 if and only if 𝐺_𝑠(𝑡)=𝐺^∗_𝑠, 𝐺_𝑖(𝑡)=𝐺^∗_𝑖, 𝑇_𝑠(𝑡)=𝑇^∗_𝑠, 𝑇_𝑖(𝑡)=𝑇^∗_𝑖. Hence, it is also known from stability theorem [25] that the infection equilibrium 𝐸^∗ is globally asymptotically stable for any time delay 𝜔≥0 under the condition ℜ₀>1.
This completes the proof of Theorem 3.

4.2. Global Stability of the Infection Equilibrium When 𝜏≥0 and 𝜔=0

In this subsection, the situations 𝜏≥0 and 𝜔=0 are considered.

Theorem 4. If ℜ₀>1, then the infection equilibrium 𝐸^∗ is globally asymptotically stable for 𝜏≥0 and 𝜔=0.

Proof. Define a Lyapunov functional: 𝑈₂(𝑡)=𝑒^𝑏𝑔𝜏𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺_𝑠(𝑡)−𝐺^∗_𝑠−𝐺^∗_𝑠𝐺ln^𝑠(𝑡)𝐺^∗_𝑠+𝑒^𝑏𝑔𝜏𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺_𝑖(𝑡)−𝐺^∗_𝑖−𝐺^∗_𝑖𝐺ln^𝑖(𝑡)𝐺^∗_𝑖+𝑇_𝑠(𝑡)−𝑇^∗_𝑠−𝑇^∗_𝑠𝑇ln^𝑠(𝑡)𝑇^∗_𝑠+𝑇_𝑖(𝑡)−𝑇^∗_𝑖−𝑇^∗_𝑖𝑇ln^𝑖(𝑡)𝑇^∗_𝑖+𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝑈₋(𝑡), (25) where 𝑈₋(𝑡)=^𝜏₀𝑇_𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑇−1−ln^𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑑𝜃. (26) The derivatives of 𝑈₋(𝑡) are given by 𝑑𝑈₋(𝑡)=𝑑𝑑𝑡𝑑𝑡^𝜏₀𝑇_𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑇−1−ln^𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑑𝜃=^𝜏₀𝑑𝑇𝑑𝑡^𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑇−1−ln^𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑑𝜃=−^𝜏₀𝑑𝑇𝑑𝜃^𝑖(𝑡−𝜃)𝑇^∗_𝑖𝑇−1−ln^𝑖(𝑡−𝜃)𝑇^∗_𝑖=𝑇𝑑𝜃_𝑖(𝑡)𝑇^∗_𝑖−𝑇_𝑖(𝑡−𝜏)𝑇^∗_𝑖𝑇+ln^𝑖(𝑡−𝜏)𝑇_𝑖(𝑡). (27) Hence, we obtain 𝑑𝑈₂=𝑒𝑑𝑡^𝑏𝑔𝜏𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺^∗_𝑠𝐺1−^∗𝑠𝐺_𝑠(𝑡)𝐺^′_𝑠(𝑡)+𝑒^𝑏𝑔𝜏𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝑆^∗𝐺1−^∗𝑖𝐺_𝑖(𝑡)𝐺^′_𝑖(𝑡)+𝑇1−^∗𝑠𝑇_𝑠(𝑡)𝑇^′_𝑠(𝑡)+𝑇1−^∗𝑖𝑇_𝑖(𝑡)𝑇^′_𝑖(𝑡)+𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝑑𝑈₋(𝑡)=𝑏𝑑𝑡_𝑔𝛽₂𝐺^∗_𝑠𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖𝐺2−^∗𝑠𝐺_𝑠(𝑡)−𝐺_𝑠(𝑡)𝐺^∗_𝑠+𝜃𝐻𝑇^∗_𝑖𝑇^∗_𝑠𝑇2−^∗𝑠𝑇_𝑠(𝑡)−𝑇_𝑠(𝑡)𝑇^∗_𝑠+𝛽₃𝐻+𝑏_𝑇𝑇^∗_𝑠𝑇2−^∗𝑠𝑇_𝑠(𝑡)−𝑇_𝑠(𝑡)𝑇^∗_𝑠+𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝑇ln^𝑖(𝑡−𝜏)𝑇_𝑖(𝑡)+𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖𝐺4−^∗𝑠𝐺_𝑠(𝑡)−𝐺^∗_𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡−𝜏)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡)−𝑇^∗_𝑠𝑇_𝑠(𝑡)−𝑇^∗_𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡). (28) Using equality 𝑇ln^𝑖(𝑡−𝜏)𝑇_𝑖(𝑡)𝐺=ln^∗𝑠𝐺_𝑠(𝑡)𝑇+ln^∗𝑠𝑇_𝑠(𝑡)𝑇+ln^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡)𝐺+ln^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡−𝜏)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡), (29) we obtain 𝑑𝑈₂𝑏𝑑𝑡=−^𝑔𝛽₂𝐺^∗_𝑠𝑇^∗_𝑠𝐺^∗_𝑖𝛽₁𝑇^∗_𝑖1𝐺_𝑠(𝑡)𝐺_𝑠(𝑡)−𝐺^∗_𝑠²−𝜃𝐻𝑇^∗_𝑖𝑇^∗_𝑠1𝑇_𝑠(𝑡)𝑇_𝑠(𝑡)−𝑇^∗_𝑠²−𝛽₃𝐻+𝑏_𝑇𝑇^∗_𝑠1𝑇_𝑠(𝑡)𝑇_𝑠(𝑡)−𝑇^∗_𝑠²+𝛽₂𝑇^∗_𝑠𝐺^∗_𝑖Ω₁+Ω₅+Ω₃+Ω₆, (30) where Ω₅𝐺=1−^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡−𝜏)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡)𝐺+ln^∗𝑖𝐺_𝑠(𝑡)𝑇_𝑖(𝑡−𝜏)𝐺^∗_𝑠𝑇^∗_𝑖𝐺_𝑖(𝑡),Ω₆𝑇=1−^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡)𝑇+ln^∗𝑖𝑇_𝑠(𝑡)𝐺_𝑖(𝑡)𝑇^∗_𝑠𝐺^∗_𝑖𝑇_𝑖(𝑡). (31) Since the function 𝑔(𝑡)=1−𝑓(𝑡)+ln𝑓(𝑡) (32) is always non-positive for any function 𝑓(𝑡)>0, and 𝑔(𝑡)=0 if and only if 𝑓(𝑡)=1. Therefore, we have Ω₁,Ω₅,Ω₃,Ω₆≤0.
It is easy to see that 𝑑𝑈₂/𝑑𝑡≤0. It is clear that 𝑑𝑈₂/𝑑𝑡=0 if and only if 𝐺_𝑠(𝑡)=𝐺^∗_𝑠, 𝐺_𝑖(𝑡)=𝐺^∗_𝑖, 𝑇_𝑠(𝑡)=𝑇^∗_𝑠, 𝑇_𝑖(𝑡)=𝑇^∗_𝑖. Hence, it is also known from stability theorem [25] that the infection equilibrium 𝐸^∗ is globally asymptotically stable for any time delay 𝜏≥0 under the condition ℜ₀>1.
This completes the proof of Theorem 4.

5. Permanence

System (3) is said to be uniformly persistent if there exists a constant 𝜂>0 such that any solution (𝐺_𝑠(𝑡),𝐺_𝑖(𝑡),𝑇_𝑠(𝑡),𝑇_𝑖(𝑡)) of (3) satisfies liminf_𝑡→∞𝐺_𝑠(𝑡)≥𝜂,liminf_𝑡→∞𝐺_𝑖(𝑡)≥𝜂,liminf_𝑡→∞𝑇_𝑠(𝑡)≥𝜂,liminf_𝑡→∞𝑇_𝑖(𝑡)≥𝜂. (33)

Now we give a result on the uniform persistence of system (3). To proceed, we introduce the following notation and terminology. Denote by 𝑃(𝑡), 𝑡≥0 the family of solution operators corresponding to (3). The 𝜔-limit set 𝜔(𝑥) of 𝑥 consists of 𝑦∈𝑋 such that there exists a sequence 𝑡_𝑛→∞ as 𝑛→∞ with 𝑃(𝑡_𝑛)𝑥→𝑦 as 𝑛→∞.

Theorem 5. System (3) is uniformly persistent if it satisfies ℜ₀>1.

Proof . Let 𝑋⁰=𝜙∈𝑋∶𝜙₂(0)>0,𝜙₄(0)>0,𝜕𝑋=𝑋⧵𝑋⁰=𝜙∈𝑋∶𝜙₂(0)=0or𝜙₄(0)=0, (34) which is relatively closed in 𝑋.
Now we prove that 𝑋⁰ is positively invariant for 𝑃(𝑡). By the second and forth equations of (3), we have 𝑑𝐺_𝑖(𝑡)𝑑𝑡≥−Γ𝐺_𝑖(𝑡),𝑑𝑇_𝑖(𝑡)𝛽𝑑𝑡≥−₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡),∀𝑡≥0. (35) Since 𝐺_𝑖(0,𝜙)=𝜙₂(0)>0 and 𝑇_𝑖(0,𝜙)=𝜙₄(0)>0, it follows from (35) that 𝐺_𝑖𝑡,𝜙≥𝜙₂(0)𝑒^−Γ𝑡,𝑇_𝑖𝑡,𝜙≥𝜙₄(0)𝑒^{−(𝛽3𝐻+𝑏𝑇)𝑡},∀𝑡≥0. (36) Thus, 𝑋⁰ is positively invariant for 𝑃(𝑡).
We set 𝑀_𝜕=(𝐭)(3),𝐘(𝐭).𝜙∈𝑋∶𝐘𝜙satisfies𝜙∈𝜕𝑋,∀𝑡≥0 (37) We claim that 𝑀_𝜕=𝐺_𝑠,0,𝑇_𝑠,0. (38) Assuming 𝐘(𝐭)∈𝑀_𝜕, for all 𝑡≥0, it suffices to show that 𝐺_𝑖(𝑡)=𝑇_𝑖(𝑡)=0, for all 𝑡≥0. If it is not true, then there exists 𝑡₀>0 such that either (a) 𝐺_𝑖(𝑡₀)=0, 𝑇_𝑖(𝑡₀)>0 or (b) 𝐺_𝑖(𝑡₀)>0, 𝑇_𝑖(𝑡₀)=0.
For case (a), from the forth equation of (3), we have 𝑑𝑇_𝑖(𝑡)𝛽𝑑𝑡≥−₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡). (39) Since 𝑇_𝑖(𝑡₀)>0, we have 𝑇_𝑖(𝑡)≥𝑇_𝑖𝑡₀𝑒^{−(𝛽3𝐻+𝑏𝑇)(𝑡−𝑡0)}>0,∀𝑡≥𝑡₀. (40) By the invariant of 𝑀_𝜕, we have 𝐺_𝑖(𝑡)≡0, for all 𝑡≥0. Then from the second equation of (3), we have 𝑇_𝑖(𝑡−𝜏)𝐺_𝑠(𝑡)≡0, for all𝑡≥0.
From the first equation of (3), we obtain 𝐺̇_𝑠(𝑡)=𝜆−𝑏_𝑔𝐺_𝑠(𝑡), thus lim_𝑡→+∞𝐺_𝑠(𝑡)=𝜆/𝑏_𝑔. By the invariant of 𝑀_𝜕, we have 𝐺_𝑠(𝑡)=𝜆/𝑏_𝑔.
Then from the second equation of (3), we obtain 𝑇_𝑖(𝑡−𝜏)=0; by the invariant of 𝑀_𝜕, we have 𝑇_𝑖(𝑡)≡0, for all 𝑡≥0. From the first and third equations of (3), we obtain 𝑇̇_𝑠(𝑡)=𝑠−(𝛽₃𝐻+𝑏_𝑇)𝑇_𝑠(𝑡); thus lim_𝑡→+∞𝑇_𝑠(𝑡)=𝑠/(𝛽₃𝐻+𝑏_𝑇). By the invariant of 𝑀_𝜕, we have 𝑇_𝑠(𝑡)=𝑠/(𝛽₃𝐻+𝑏_𝑇), ∀𝑡≥0, which contradicts the assumption that (𝐺_𝑠(𝑡),𝐺_𝑖(𝑡),𝑇_𝑠(𝑡),𝑇_𝑖(𝑡))∈𝑀_𝜕, for all 𝑡≥0. Similarly, we can obtain a contradiction for case (b). This proves the claim (38).
Let 𝐴=⋂_{𝑥∈𝐴𝑏}𝜔(𝑥), where 𝐴_𝑏 is the global attractor of 𝑃(𝑡) restricted to 𝜕𝑋. We show that 𝐴=𝐸₀. In fact, from 𝐴∈𝑀_𝜕 and the first and third equations of (3), we have lim_𝑡→∞𝐺_𝑠(𝑡)=𝐺⁰_𝑠 and lim_𝑡→∞𝑇_𝑠(𝑡)=𝑇⁰_𝑠. Thus, 𝐸₀ is the isolated invariant set in 𝑋.
Next, we show that 𝑊^𝑠(𝐸₀)∩𝑋⁰=∅. If this is not true, then there exists a solution (𝐺_𝑠𝑡,𝐺_𝑖𝑡,𝑇_𝑠𝑡,𝑇_𝑖𝑡)∈𝑋⁰ such that lim_𝑡→∞𝐺_𝑠(𝑡)=𝐺⁰_𝑠,lim_𝑡→∞𝐺_𝑖(𝑡)=0,lim_𝑡→∞𝑇_𝑠(𝑡)=𝑇⁰_𝑠,lim_𝑡→∞𝑇_𝑖(𝑡)=0. (41) For any sufficiently small constant 𝜀>0, there exists a positive constant 𝑇₀=𝑇₀(𝜀) such that 𝐺_𝑠(𝑡)>𝐺⁰_𝑠−𝜀>0,∀𝑡≥𝑇₀,𝑇_𝑠(𝑡)>𝑇⁰_𝑠−𝜀>0,∀𝑡≥𝑇₀. (42) For the constant 𝜀 given above, it follows from the second and forth equations of (3) that 𝐺̇_𝑖(𝑡)≥𝑒^{−𝑏𝑔𝜏}𝛽₁𝑇_𝑖()×𝐺𝑡−𝜏_0𝑠−𝜀−Γ𝐺_𝑖(𝑡),𝑇̇_𝑖(𝑡)≥𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠−𝜀×𝐺_𝑖(𝑡−𝜔)+𝜃𝐻𝑇⁰_𝑠−𝜀𝑇_𝑖(𝑡)−𝛽₃𝐻+𝑏_𝑇𝑇_𝑖(𝑡). (43) If 𝐺_𝑖(𝑡),𝑇_𝑖(𝑡)→0, as 𝑡→∞, then by a standard comparison argument and the nonnegativity, the solution (𝑥(𝑡),𝑦(𝑡)) of the following monotone system 𝑥̇(𝑡)=𝑒^{−𝑏𝑔𝜏}𝛽₁𝑦(𝑡−𝜏)𝐺⁰_𝑠(𝑡),𝑦̇(𝑡)−𝜀−Γ𝑥=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠𝑥()𝑇−𝜀𝑡−𝜔+𝜃𝐻⁰_𝑠𝑦(𝑡)−𝛽−𝜀₃𝐻+𝑏_𝑇𝑦(𝑡) (44) with the initial condition 𝑥(𝑡)=𝐺_𝑖(𝑡), 𝑦(𝑡)=𝑇_𝑖(𝑡), for all 𝑡∈[𝑡₀,𝑡₀+𝑇₀] converges to (0,0) as well. Thus, lim_𝑡→∞𝑊(𝑡)=0, where 𝑊(𝑡)>0 is defined by 𝑊(𝑡)=𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠−𝜀Γ𝑥(𝑡)+𝑦(𝑡)+𝑒^{−𝑏𝑇𝜔}𝛽₂𝑇⁰_𝑠−𝜀^𝑡_𝑡−𝜔𝑥(𝜉)+𝑒𝑑𝜉_{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝛽₁𝛽₂𝐺⁰_𝑠−𝜀𝑇⁰_𝑠−𝜀Γ^𝑡_𝑡−𝜏𝑦(𝜉)𝑑𝜉. (45) Differentiating 𝑊(𝑡) with respect to time 𝑡 gives 𝑑𝑊(𝑡)𝑑𝑡₍₄₄₎=𝜃𝐻𝑇⁰_𝑠+𝑒−𝜀_{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝛽₁𝛽₂𝐺⁰_𝑠−𝜀𝑇⁰_𝑠−𝜀Γ−𝛽₃𝐻+𝑏_𝑇𝑦(𝑡). (46) Because ℜ₀>1, we have 𝜃𝐻(𝑇⁰_𝑠−𝜀)+𝑒^{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝛽₁𝛽₂(𝐺⁰_𝑠−𝜀)(𝑇⁰_𝑠−𝜀)/Γ−(𝛽₃𝐻+𝑏_𝑇)>0 for a sufficiently small 𝜀. Therefore, 𝑊(𝑡) goes to either infinity or a positive number 𝑡→∞, which leads to a contradiction with lim_𝑡→∞𝑊(𝑡)=0. Thus we have 𝑊^𝑠(𝐸₀)∩𝑋⁰=∅.
Define 𝑝∶𝑋→𝑅₊ by 𝑝𝜙=min𝜙₂(0),𝜙₄(0),∀𝜙∈𝑋. (47) It is clear that 𝑋⁰=𝑝⁻¹(0,∞) and 𝜕𝑋=𝑝⁻¹(0). Thus by [20, Theorem 3], we have liminf_𝑡→∞𝐺_𝑖(𝑡),𝑇_𝑖(𝑡)≥𝜂₀,𝜂₀ (48) for some constant 𝜂₀>0. Let 𝜂=min{𝜂₀,𝜀}, where 𝜀 is the constant such thatliminf_𝑡→∞𝐺_𝑠(𝑡)≥𝜀>0 and liminf_𝑡→∞𝑇_𝑠(𝑡)≥𝜀>0. We showed that liminf_𝑡→∞𝐺_𝑠(𝑡)≥𝜂, liminf_𝑡→∞𝐺_𝑖(𝑡)≥𝜂, liminf_𝑡→∞𝑇_𝑠(𝑡)≥𝜂, and liminf_𝑡→∞𝑇_𝑖(𝑡)≥𝜂.
This completes the proof of Theorem 5.

6. Numerical Simulations

In this section, by using suitable numerical methods, we simulate the systems to illustrate the analytical results obtained in Sections 3–5. Moreover, we would verify the performance as shown in Section 5 and find that the system (3) is globally asymptotically stable when two delays are present simultaneously.

Table 1 shows the literature reported values for parameters which we used for the simulations. Besides the parameter value shown in Table 1, we can find 𝜂 by using the equation 𝜂=𝛼+𝛾+𝑏_𝑔.

Table 1 

Parameter values used in the simulations shown in figures.

For system (3), we choose 𝜆=3, 𝑠=2, 𝜏=10, and 𝜔=15, and other values are shown in Table 1. By computation, we obtain 𝑅₀=0.0818<1, and the infection equilibrium does not exist. By the analysis of Section 2, we know that the infection-free equilibrium 𝐸₀=(34.4828,0,14.2146,0) is globally asymptotically stable (see Figure 1). Furthermore, we choose 𝜆=8, 𝑠=6, 𝜏=0, and 𝜔=15, and other values are shown in Table 1. In Figure 2, we know that under the condition indicated in the Theorem 3 with 𝑅₀=1.5645>1 the infection equilibrium 𝐸^∗=(63.7684,1.324,38.421,4.15) will be globally asymptotically stable. In Figure 3, we know that under the condition indicated in the Theorem 4 with 𝑅₀=1.8425>1, the infection equilibrium 𝐸^∗=(75.346,1.237,39.96,3.856) will be globally asymptotically stable with 𝜏=3 and 𝜔=0. In Figure 4, we know that under the condition indicated in Theorem 5 with 𝑅₀=1.1638>1, the infection equilibrium 𝐸^∗=(68.72,1.0472,39.456,4.385) will be globally asymptotically stable with 𝜏=5 and 𝜔=10.

(a)

(b)

Figure 1 

The infection-free equilibrium 𝐸0 of system (3) is globally asymptotically stable when 𝜏=10, 𝜔=15, and ℜ0≤1. The initial functions are (60,10,40, and 10).

(a)

(b)

Figure 2 

The infection equilibrium 𝐸∗ of system (3) is asymptotically stable when 𝜏=0, 𝜔=15, and ℜ0>1. The initial functions are (100,10,100, and 10).

(a)

(b)

Figure 3 

The infection equilibrium 𝐸∗ of system (3) is asymptotically stable when𝜏=3, 𝜔=0, and ℜ0>1. The initial functions are (100,10,100, and 10).

(a)

(b)

Figure 4 

The infection equilibrium 𝐸∗ of system (3) is asymptotically stable when 𝜏=5, 𝜔=10, and ℜ0>1. The initial functions are (100,10,100, and 10).

7. Conclusion

In the present paper, we investigate an epidemic model for tick-borne disease with non-viraemic transmission and two time delays. By constructing Lyapunov functionals, we prove that the infection-free equilibrium 𝐸₀ of system (3) is globally asymptotically stable when ℜ₀≤1. If ℜ₀>1 then the infection equilibrium 𝐸^∗ is globally asymptotically stable when one delay is absent. By applying the persistence theory for infinite dimensional systems, we obtain that the populations of system (3) can coexist permanently if ℜ₀>1.

The reproductive number plays a crucial role for epidemic dynamics. Considering the basic reproductive number ℜ₀=𝑒^{−𝑏𝑔𝜏−𝑏𝑇𝜔}𝜆𝑠𝛽₁𝛽₂/Γ𝑏_𝑔(𝛽₃𝐻+𝑏_𝑇)²+𝑠𝜃𝐻/(𝛽₃𝐻+𝑏_𝑇)² as a function of 𝜏 and 𝜔, we can find that it is decreasing in 𝜏 and 𝜔 and it tends to 0 if the time delay 𝜏 or 𝜔 tends to 1. Therefore, the basic reproductive number can be reduced by increasing the intracellular delay. This may provide an insight for developing vaccination and regular treatment with acaricides that try to increase the intracellular delay.

It is interesting to consider the global stability of the infection equilibrium 𝐸^∗ with two time delays; from the numerical simulations, it is shown that the infection equilibrium 𝐸^∗ is globally asymptotically stable when ℜ₀>1; we leave this as a future work.

The nonseasonal model can be analysed analytically unlike the seasonal models. The non seasonal model is also simpler and quicker to implement in Mathematica. However, the effect of the seasonal behavior of ticks will be explored before deciding whether any form of seasonality will be incorporated in future work.

Acknowledgment

The research was partially supported by the Funds of USTB and NNSF of China (11071013).