Abstract
A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.
1. Introduction
Mathematical model that reflects the characteristics of an epidemic to some extent can help us to understand better how the disease spreads in the community and can investigate how changes in the various assumptions and parameter values affect the course of epidemic. In [1], Hyman et al. proposed a differential infectivity model that accounted for differences in infectiousness between individuals during the chronic stages and the correlation between viral loads and rates of developing AIDS. They assumed that the susceptible population was homogeneous and neglected variations in susceptibility, risk behavior, and many other factors associated with the dynamics of HIV spread. Ma et al. [2] presented several differential infectivity epidemic models under different assumptions.
In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea, and bacterial infections, may have a different ability to transmit the infections in different ages. For example, measles and varicella always occur in juveniles, while it is reasonable to consider the disease transmission in adult population such as typhus and diphtheria. In recent years, epidemic models with stage structure have been studied in many papers [3–10].
In this paper, we formulate a differential infectivity epidemic model with stage structure. The proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach [11–22] to the method of Lyapunov functions. Let and denote the immature susceptible and mature susceptible populations, respectively. The infectious population was subdivided into subgroups . and denote the probabilities of an immature infectious individual and a mature infectious individual enter subgroup , respectively, where . The disease incidence in the th subgroup can be calculated as , where is the transmission coefficient between compartments and . includes some special incidence functions in the literature. For instance, (saturation effect). Since we do not assume that recovered individuals return into the susceptible class, the recovered class does not need to be explicitly modeled. Then, we obtain the following model:where with being the recruitment constant and being the natural death rate. is the conversion rate from an immature individual to a mature individual. is the natural death rate of the mature susceptible class. , where is the death rate of population in subgroup and is the recovery rate in the th subgroup. All parameter values are assumed to be nonnegative and , , , , .
The organization of this paper is as follows. In Section 2, we prove some preliminary results for system (1). In Section 3, the main theorem of this paper is stated and proved. In Section 4, numerical simulations which support our theoretical analysis are given.
2. Preliminaries
We assume the following.
(A1) is continuous and Lipschitz on , is nonincreasing on , and
From our assumptions, it is clear that system (1) has a unique solution for any given initial data with , , and for and the solution remains nonnegative. We see that system (1) exits in a disease-free equilibrium , where . Let . Then, we derive from (1) that the region, is a forward invariant compact absorbing set with respect to (1). Also let denote the interior of . The next generation matrix for system (1) isThen, we define the basic reproduction number as the spectral radius of , . A square matrix is said to be reducible, if there is a permutation matrix , such that is a block upper triangular matrix; otherwise it is irreducible.
3. Main Results
In the section, we will study the global asymptotical stability of equilibria of system (1).
Theorem 1. Assume that (A1) holds and is irreducible. (1)If , then is globally asymptotically stable in .(2)If , then is unstable and system (1) admits at least one endemic equilibrium in .
Proof. Let , , , and . Notice that is irreducible, and then is also irreducible. Hence, there exists , , such thatDefine . Then We see that the only compact invariant subset of the set where is the singleton . By LaSalle’s Invariance Principle, is globally asymptotically stable in if .
If , by continuity, we obtain that in a neighborhood of in . This implies that is unstable. From a uniform persistence result of [23] and a similar argument as in the proof of Proposition of [24], we can deduce that the instability of implies the uniform persistence of system (1) in . This together with the uniform boundedness of solutions of system (1) in implies that system (1) has an endemic equilibrium in (see Theorem of [25] or Theorem of [26]). The proof is completed.
By Theorem 1, we have the idea that if is irreducible, (A1) holds and , and then system (1) exists in endemic equilibrium in . Let , and then the components of satisfySince is strictly decreasing on , we have
For convenience of notations, setThen, is also irreducible. It follows from Lemma of [11] that the solution space of linear system,has dimension 1, with a basiswhere denotes the cofactor of the th diagonal entry of . Note that from (12) we haveFrom (14), we have
We further make the following assumption.
(A2) is strictly increasing on , andwhere is chosen in an arbitrary way and equality holds if .
Theorem 2. Assume that (A1) and (A2) hold, , and is irreducible. If , then is globally asymptotically stable in and thus is the unique endemic equilibrium.
Proof. Consider a Lyapunov functionalDifferentiating along the solution of system (1), we obtain From (7), we know thatIt follows from (8), (9), and (19) that From (10), (15), and (16), we obtainBy and the arithmetic-geometric mean, we easily see thatWe can rewrite as Using the fact that , where equality holds if only if , we obtainIn the following, we will show thatWe first give the proof of (25) for , which would give a reader the basic yet clear ideas without being hidden by the complexity of terms caused by larger values of . When , we have Formula (13) gives and in this case. Expanding yields For more general , by a similar argument as in the proof of in [16], we obtain
From (21), (22), and (24), we see that if , thenIf (27) holds, it follows from (1) thatThen, we obtain that This implies thatBy the characteristics of , we obtain the idea that the largest invariant subset of the set where is the singleton . By LaSalle’s Invariance Principle, is globally asymptotically stable for . This completes the proof.
4. Numerical Examples
In the section, numerical simulations are presented to support and complement the theoretical findings. We consider the following model:where . Clearly, (A1) and (A2) hold. We fix the parameters as follows:Then, we have .
Case 1. If , then we obtain . By Theorem 1, the disease dies out in both subgroups. Numerical simulation illustrates this fact (see Figure 1).
Case 2. If , then we have , , , , and . By Theorem 2, the disease persists in both subgroups. Numerical simulation illustrates this fact (see Figure 2).
5. Conclusions
A differential infectivity epidemic model with stage structure has been used to describe the spreading of such a disease. We have focused on the theoretical analysis of the equilibriums. Using a graph-theoretic approach to the method of Lyapunov functions, we have proved the global stability of the endemic equilibrium. We have established uniform persistence and the sharp threshold. The work has potential extensions and improvements, which remains to be discussed in the future.
Competing Interests
The author declares that there are no competing interests.
Acknowledgments
This work was supported by Key Laboratory of Statistical Information Technology and Data Mining, State Statistics Bureau (SDL201601).