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In this paper, we study an SI epidemic model with nonlinear infection rate and stage structure. The conditions for the existence of various equilibriums are established. The threshold that the epidemic will die out or become local epidemic is found.

This paper is concerned with a stochastic single-species system with Lévy jumps in a polluted environment. Some sufficient conditions on extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean, stability in the mean and stochastic permanence are obtained. The threshold between extinction and weak persistence in the mean is established. At the same time, under a simple condition, it is proved that this threshold also is the threshold between extinction and stability in the mean. The results reveal that Lévy jumps have significant effects to the persistence and extinction results.

We study an SEIRS epidemic model with an isolation and nonlinear incidence rate function. We have obtained a threshold value $R_0$ and shown that there is only a disease-free equilibrium point, when $R_0<1$ and an endemic equilibrium point if $R_0>1$. We have shown that both disease-free and endemic equilibrium point are globally stable.

Understanding the dynamical mechanism of the host–parasite interactions is one of important issues on host–parasite association. In this paper, we formulate a three-dimensional host–macroparasite system to describe the host–parasite interactions, which includes the logistic growth rate of host population, the important free-living stage and the host fecundity reduction due to parasite infection. The purpose of the paper is to investigate the asymptotical behavior of the system. By using the properties of the solution to non-autonomous linear system, the basic production number ${{ℛ}}_0$ is proved to be a threshold which determines the outcome of the parasites. If ${{ℛ}}_0<1$, the parasite will eventually die out, and if ${{ℛ}}_0>1$ the parasite will be uniformly persistent. Hopf bifurcation of the system is further studied, and sufficient conditions for the Hopf bifurcation are obtained. By using the singular perturbation techniques, the system is separated into two time scales with a faster time scale for the free-living infective particles and a slower time scale for the population dynamics of host and parasite, and then a complete analysis of the dynamics on the slow manifold is conducted. The theoretical results show that the level of aggregation of parasites within host may influence the persistence and stability of the system.

In the interaction between plants and herbivores that live in the same ecosystem, understanding the conditions in which co-existence equilibrium occurs answers a major question in Ecology. In this interaction, plants serve as food for herbivores on the food chain. Then the livelihood of herbivores highly depends on the availability of food, in this case the availability of plants. Moreover, the abundance of the plant density alone does not guarantee the non-extinction of the herbivore population as they are assumed to reproduce sexually. With this motivation, in this paper a predator–prey mathematical model is reformulated such that the death rate of the herbivore population is dependent on the plant density and their emergence is also governed by the Allee effect. Using the mathematical theory of dynamical system, threshold conditions are obtained for the non-extinction of the herbivore population and a trapping region is obtained to ensure co-existence of the population. Moreover, it has been shown that the dynamics of the population is significantly sensitive to the feeding rate and the harvest rate of the herbivore population.

This paper formulates and studies a delayed chemostat with Lévy noises. Existence of the globally positive solution is proved first by establishing suitable Lyapunov functions, and a further result on exact Lyapunov exponent shows the growth of the total concentration in the chemostat. Then, we prove existence of the uniquely ergodic stationary distribution for a subsystem of the nutrient, based on this, a unique threshold is identified, which completely determines persistence or not of the microorganism in the chemostat. Besides, recurrence is studied under special conditions in case that the microorganism persists. Results indicate that all the noises have negative effects on persistence of the microorganism, and the time delay has almost no effects on the sample Lyapunov exponent and the threshold value of the chemostat.

This paper deals with stochastic Chikungunya (CHIKV) virus model along with saturated incidence rate. We show that there exists a unique global positive solution and also we obtain the conditions for the disease to be extinct. We also discuss about the existence of a unique ergodic stationary distribution of the model, through a suitable Lyapunov function. The stationary distribution validates the occurrence of disease; through that, we find the threshold value for prevail and disappear of disease within host. With the help of numerical simulations, we validate the stochastic reproduction number $R_0^S$ as stated in our theoretical findings.

Acquired immunodeficiency syndrome (AIDS) has a serious impact on human health and life safety. In order to study its related factors, this paper establishes an HIV/AIDS model with treatment individuals based on heterosexual contact and male-to-male sexual contact. Using the method of next generation matrix, the threshold $R_0$ of the model is given. When $R_0<1$, it proves the global stability of the disease-free equilibrium. When $R_0>1$, it studies the dynamics of the boundary equilibrium and the endemic equilibrium under different conditions. Finally, through numerical simulations, the correctness of the theoretical results is verified. The key parameters affecting the spread of HIV are found through parameter sensitivity analysis, which provides a theoretical basis for effective control of the spread of HIV.