Xue-Zhi Li

Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment

The dynamical behaviors of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.

Global analysis of a vector-host epidemic model with nonlinear incidences☆

Abstract In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R 0 . It is shown that if R 0 is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R 0 is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.

Linking immunological and epidemiological dynamics of HIV: the case of super-infection.

In this paper, a two-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the two strains eliminate each other with the strain with the larger immunological reproduction persisting. However, on the population scale superinfection is possible, with the strain with larger immunological reproduction number super-infecting the strain with the smaller immunological reproduction number. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition, the between-host transmission and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers, the epidemiological reproduction numbers and invasion reproduction numbers are computed. Besides the disease-free equilibrium, there are two population-level strain one and strain two isolated equilibria, as well as a population-level coexistence equilibrium when both invasion reproduction numbers are greater than one. The single-strain population-level equilibria are locally asymptotically stable suggesting that in the absence of superinfection oscillations do not occur, a result contrasting previous studies of HIV age-since-infection structured models. Simulations suggest that the epidemiological reproduction number and HIV population prevalence are monotone functions of the within-host parameters with reciprocal trends. In particular, HIV medications that decrease within-host viral load also increase overall population prevalence. The effect of the immunological parameters on the population reproduction number and prevalence is more pronounced when the initial viral load is lower.

Presentation of Malaria Epidemics Using Multiple Optimal Controls

An existing model is extended to assess the impact of some antimalaria control measures, by reformulating the model as an optimal control problem. This paper investigates the fundamental role of three type of controls, personal protection, treatment, and mosquito reduction strategies in controlling the malaria. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. The optimality system is solved numerically by a competitive Gauss-Seidel-like implicit difference method. Finally, numerical simulations of the optimal control problem, using a set of reasonable parameter values, are carried out to investigate the effectiveness of the proposed control measures.

Global stability of an epidemic model for vector-borne disease

This paper considers an epidemic model of a vector-borne disease which has the vectormediated transmission only. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R0. If R0 ≤ 1, the diseasefree equilibrium is globally stable and the disease dies out. If R0 > 1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium. Numerical simulations are presented to illustrate the results.

An age-structured two-strain epidemic model with super-infection.

This article focuses on the study of an age-structured two-strain model with super-infection. The explicit expression of basic reproduction numbers and the invasion reproduction numbers corresponding to strain one and strain two are obtained. It is shown that the infection-free steady state is globally stable if the basic reproductive number R(0) is below one. Existence of strain one and strain two exclusive equilibria is established. Conditions for local stability or instability of the exclusive equilibria of the strain one and strain two are established. Existence of coexistence equilibrium is also obtained under the condition that both invasion reproduction numbers are larger than one.

Competitive exclusion in a vector–host epidemic model with distributed delay†

A multi-strain model of a vector-borne disease with distributed delay in the vector and the host is investigated. It is shown that if the reproduction number of the model ℛ0<1, the unique disease-free equilibrium is globally asymptotically stable. Without loss of generality, strain one is assumed to have the largest reproduction number. In this case, the dominance equilibrium of strain one is shown to be locally stable. The basic reproduction number for a strain i () is written as a product of the reproduction number of the vector and the reproduction number of the host , i.e. . The competitive exclusion principle is derived under the somewhat stronger condition that if strain one maximizes both the reproduction number of the host and the reproduction number of the vector , strain one dominance equilibrium is globally asymptotically stable.

Global stability for a delayed HIV-1 infection model with nonlinear incidence of infection

Abstract In this paper, a delayed HIV-1 infection model with nonlinear incidence of infection is reinvestigated. It is shown that if the reproduction number R > 1 , then the system is permanent, and the infective equilibrium of the system is globally asymptotically stable. Thus, the global dynamics of the system is completely determined by the reproduction number R . The results obtained enrich and improve the corresponding results given by Wang et al. [X. Wang, Y. Tao, X. Song, A delayed HIV-1 infection model with Beddington–DeAngelis functional response, Nonlinear Dynamics 62 (2010) 67–72]. The conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper [D. Li, W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007) 683–691].

Optimal control of a malaria model with asymptomatic class and superinfection

In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R0, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R0<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.

Competitive exclusion in a multi-strain immuno-epidemiological influenza model with environmental transmission

ABSTRACT In this paper, a multi-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the n strains eliminate each other with the strain having the largest immunological reproduction number persisting. However, on the population scale, we extend the competitive exclusion principle to a multi-strain model of SI-type for the dynamics of highly pathogenic flu in poultry that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition the between-host transmission rate, the shedding rate of individuals infected by strain j and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers and the epidemiological reproduction numbers are computed. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained if all reproduction numbers are smaller or equal to one. If , the reproduction number of strain j is larger than one, then a single-strain equilibrium, corresponding to strain j exists. This single-strain equilibrium is globally stable whenever and is the unique maximal reproduction number and all of the reproduction numbers are distinct. That is, the strain with the maximal basic reproduction number competitively excludes all other strains.