Jinliang Wang

Global asymptotic stability for HIV-1 dynamics with two distributed delays

Based on the drugs treatment to control HIV-1 infection and viral replication, we express the intracellular eclipse phase of virions in host cell as distributed delays because of pharmacological actions. In present paper, we investigate a class of HIV-1 infection dynamical model with two distributed delays. One of them describes the period between the time that HIV virion enters (infects) target cell and the time that the infected cell starts to produce new viral particles. The other describes the time for the virion maturation process. They are both allowed to tend to be infinite because of drugs resistent strains. By the Lyapunov direct method of and utilizing the technology of constructing Lyapunov functionals, we identify the basic reproduction number R(0) as a threshold quantity for the stability of equilibria. More precisely, if R(0) ≤ 1, the infection-free equilibrium is globally asymptotically stable; on the contrary, if R(0) > 1, then an infected equilibrium appears which is globally asymptotically stable. The dynamical results indicate that time delays have effect on the global stability of two equilibria through threshold value R(0), which is a decreasing function of delays. The biological meanings imply that any drugs that can prolong the time of viral reproduction through slowing down the reverse transcription of HIV in host and virus maturation process may also help control the HIV-1 infection and virus loads. Another way to increase the efficacy of the protease inhibitor and the reverse transcriptase inhibitor (i.e. increasing n(p) and n(rt)) is also desirable treatment strategies.

GLOBAL DYNAMICS OF A MULTI-GROUP EPIDEMIC MODEL WITH GENERAL RELAPSE DISTRIBUTION AND NONLINEAR INCIDENCE RATE

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R0. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.

A MULTI-GROUP SVEIR EPIDEMIC MODEL WITH DISTRIBUTED DELAY AND VACCINATION

In this paper, based on a class of multi-group epidemic models of SEIR type with bilinear incidences, we introduce a vaccination compartment, leading to multi-group SVEIR model. We establish that the global dynamics are completely determined by the basic reproduction number which is defined by the spectral radius of the next generation matrix. Our proofs of global stability of the equilibria utilize a graph-theoretical approach to the method of Lyapunov functionals. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccines to obtain immunity or the possibility for them to be infected before acquiring immunity is neglected in each group, this condition will be satisfied and the disease can always be eradicated by suitable vaccination strategies. This may lead to over evaluation for the effect of vaccination.

SVEIR EPIDEMIOLOGICAL MODEL WITH VARYING INFECTIVITY AND DISTRIBUTED DELAYS

In this paper, based on an SEIR epidemiological model with distributed delays to account for varying infectivity, we introduce a vaccination compartment, leading to an SVEIR model. By employing direct Lyapunov method and LaSalles invariance principle, we construct appropriate functionals that integrate over past states to establish global asymptotic stability conditions, which are completely determined by the basic reproduction number R( V) (0). More precisely, it is shown that, if R( V) (0) ≤ 1, then the disease free equilibrium is globally asymptotically stable; if R( V) (0) > 1, then there exists a unique endemic equilibrium which is globally asymptotically stable. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before acquiring immunity can be neglected, this condition would be satisfied and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.

Qualitative and bifurcation analysis using an SIR model with a saturated treatment function

a b s t r a c t In this paper, we introduce a saturated treatment function into the SIR epidemic model with a bilinear incidence rate and density-dependent demographics, where the treatment function is limited for increasing number of infected individuals. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected individuals will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure disease eradication where the number of infective individuals tends to zero for all initial positions except the interior equilibria; (iii) ‘‘oscillating’’ backward bifurcation where either the number of infective individuals oscillates persistently, if the initial position lies in a region covering the stable endemic equilibrium, or disease eradication, if the initial position lies outside this region; (iv) disease eradication for all initial positions if the basic reproduction number is less than a turning point value. Numerical simulations are presented to illustrate the conclusions.

Adaptive evolution of attack ability promotes the evolutionary branching of predator species.

In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate the evolutionary branching phenomenon of predator species. We assume that both the prey and predators are density-dependent and the predators attack ability can adaptively evolve, but this has a cost in terms of its death rate. First, we identify the general properties of trade-off relationships that allow for a continuously stable strategy and evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly concave near the singular strategy, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the predator strategy, if the trade-off curve is convex-concave-convex, the predator species will eventually evolve into two different types, which can stably coexist on the much longer evolutionary timescale and no further branching is possible.

THRESHOLD DYNAMICS IN A PERIODIC SVEIR EPIDEMIC MODEL

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

Evolutionary Diversification of Prey and Predator Species Facilitated by Asymmetric Interactions

We investigate the influence of asymmetric interactions on coevolutionary dynamics of a predator-prey system by using the theory of adaptive dynamics. We assume that the defense ability of prey and the attack ability of predators all can adaptively evolve, either caused by phenotypic plasticity or by behavioral choice, but there are certain costs in terms of their growth rate or death rate. The coevolutionary model is constructed from a deterministic approximation of random mutation-selection process. To sum up, if prey’s trade-off curve is globally weakly concave, then five outcomes of coevolution are demonstrated, which depend on the intensity and shape of asymmetric predator-prey interactions and predator’s trade-off shape. Firstly, we find that if there is a weakly decelerating cost and a weakly accelerating benefit for predator species, then evolutionary branching in the predator species may occur, but after branching further coevolution may lead to extinction of the predator species with a larger trait value. However, if there is a weakly accelerating cost and a weakly accelerating benefit for predator species, then evolutionary branching in the predator species is also possible and after branching the dimorphic predator can evolutionarily stably coexist with a monomorphic prey species. Secondly, if the asymmetric interactions become a little strong, then prey and predators will evolve to an evolutionarily stable equilibrium, at which they can stably coexist on a long-term timescale of evolution. Thirdly, if there is a weakly accelerating cost and a relatively strongly accelerating benefit for prey species, then evolutionary branching in the prey species is possible and the finally coevolutionary outcome contains a dimorphic prey and a monomorphic predator species. Fourthly, if the asymmetric interactions become more stronger, then predator-prey coevolution may lead to cycles in both traits and equilibrium population densities. The Red Queen dynamic is a possible outcome under asymmetric predator-prey interactions.

Adaptive Evolution of Defense Ability Leads to Diversification of Prey Species

In this paper, by using the adaptive dynamics approach, we investigate how the adaptive evolution of defense ability promotes the diversity of prey species in an initial one-prey–two-predator community. We assume that the prey species can evolve to a safer strategy such that it can reduce the predation risk, but a prey with a high defense ability for one predator may have a low defense ability for the other and vice versa. First, by using the method of critical function analysis, we find that if the trade-off is convex in the vicinity of the evolutionarily singular strategy, then this singular strategy is a continuously stable strategy. However, if the trade-off is weakly concave near the singular strategy and the competition between the two predators is relatively weak, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the prey strategy, if the trade-off curve is globally concave, then the prey species might eventually evolve into two specialists, each caught by only one predator species. However, if the trade-off curve is convex–concave–convex, the prey species might eventually branch into two partial specialists, each being caught by both of the two predators and they can stably coexist on the much longer evolutionary timescale.

COEVOLUTIONARY DYNAMICS OF PREDATOR-PREY INTERACTIONS

In this paper, with the method of adaptive dynamics, we investigate the coevolution of phenotypic traits of predator and prey species. The evolutionary model is constructed from a deterministic approximation of the underlying stochastic ecological processes. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that evolutionary branching in the prey phenotype will occur when the frequency dependence in the prey carrying capacity is not strong. Furthermore, it is found that if the two prey branches move far away enough, the evolutionary branching in the prey phenotype will induce the secondary branching in the predator phenotype. The final evolutionary outcome contains two prey and two predator species. Secondly, we show that under symmetric interactions the evolutionary model admits a supercritical Hopf bifurcation if the frequency dependence in the prey carrying capacity is very weak. Evolutionary cycle is a likely outcome of the mutation-selection processes. Finally, we find that frequency-dependent selection can drive the predator population to extinction under asymmetric interactions.