A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control.

In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincaré criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions.

This paper inquires the global properties of a vector borne dengue out break model with non-linear incidence rates and latent delays of both host and vector. The global stability properties of disease free equilibrium as well as endemic equilibrium are obtained by means of appropriate Liapunov functionals and LaSalle's invariance principle for delay differential equations. The global dynamics of equilibria of the model are totally concluded by the basic reproductive number (R₀). We proved that if R₀ ≤ 1, the disease-free equilibrium globally asymptotically stable and R₀ ≤ 1, infected equilibrium is globally asymptotically stable. Numerical simulations are given at the end of paper for a specified model.

This study investigated the impact of latent and maturation delay on the qualitative behaviour of a human immunodeficiency virus-1 (HIV-1) infection model with nonlinear functional response and absorption effect. Basic reproduction number (R 0), which is defined as the average number of infected cells produced by one infected cell after inserting it into a fully susceptible cell population is calculated for the proposed model. As R o (threshold) depends on the negatively exponential function of time delay, these parameters are responsible to predict the future propagation behaviour of the infection. Therefore, for smaller positive values of delay and larger positive values of infection rate, the infection becomes chronic. Besides, infection dies out with larger delays and lower infection rates. To make the model biologically more sensible, we used the functional form of response function that plays an important role rather than the bilinear response function. Existence of equilibria and stability behaviour of the proposed model totally depend on R o. Local stability properties of both infection free and chronic infection equilibria are established by utilising the characteristic equation. As it is crucially important to study the global behaviour at equilibria rather than the local behaviour, we used the method of Liapunov functional. By constructing suitable Liapunov functionals and applying LaSalle's invariance principle for delay differential equations, we established that infection free equilibrium is globally asymptotically stable if R 0 ≤ 1, which biologically means that infection dies out. Moreover, sufficient condition is derived for global stability of chronic infection equilibrium if R 0 > 1 , which biologically means that infection becomes chronic. Numerical simulations are given to illustrate the theoretical results.

COVID-19, which broke out globally in 2019, is an infectious disease caused by a novel strain of coronavirus, and its spread is highly contagious and concealed. Environmental vectors play an important role in viral infection and transmission, which brings new difficulties and challenges to disease prevention and control. In this paper, a type of differential equation model is constructed according to the spreading functions and characteristics of exposed individuals and environmental vectors during the virus infection process. In the proposed model, five compartments were considered, namely, susceptible individuals, exposed individuals, infected individuals, recovered individuals, and environmental vectors (contaminated with free virus particles). In particular, the re-positive factor was taken into account (i.e., recovered individuals who have lost sufficient immune protection may still return to the exposed class). With the basic reproduction number ${\mathcal{R}}_0$ of the model, the global stability of the disease-free equilibrium and uniform persistence of the model were completely analyzed. Furthermore, sufficient conditions for the global stability of the endemic equilibrium of the model were also given. Finally, the effective predictability of the model was tested by fitting COVID-19 data from Japan and Italy.