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As many countries are hit by the social economic impact of heroin addiction, there is an urgent need to have an effective awareness program that focuses on educating the population on the danger of heroin addiction and helping the heroin‐users quit. This paper aims to study the effect of the awareness program on the spread of heroin dependence using a mathematical model with distributed delay. First, we show the existence threshold parameter R0$$ {\mathfrak{R}}_0 $$, which we call the basic reproduction number of the spread of heroin use. We prove, via the Lyapunov direct method, that the drug‐free equilibrium is globally asymptotically stable if R0<1$$ {\mathfrak{R}}_0<1 $$. If R0>1$$ {\mathfrak{R}}_0>1 $$, the drug dependence persists, and the drug equilibrium is globally asymptotically stable. We give three possible scenarios to find the optimal awareness program strategy that puts the heroin epidemic under control. These scenarios take into consideration the reachability of the population, the immunity against heroin addiction, and the effectiveness of the program to contain the use of heroin in the population and bring R0$$ {\mathfrak{R}}_0 $$ below one. [ABSTRACT FROM AUTHOR]

We introduce, in this paper, a delayed hybrid epidemic model describing the evolution of heroin addiction in a given population. The main objective of our mathematical analysis will be to provide the global behavior of the solutions. To the best of our knowledge, there are no results about the global dynamics of such a model containing both age structure and distributed time delay. We will prove that our model has threshold dynamics in terms of the basic reproduction number R0$$ {R}_0 $$. It will be established that for R0≤1$$ {R}_0\le 1 $$, the addiction‐free equilibrium is globally stable and for R0>1$$ {R}_0&gt;1 $$, the endemic equilibrium is globally stable. The theoretical results are checked numerically. Especially, we focus on the influence of the addiction rate on the population dynamic. [ABSTRACT FROM AUTHOR]

This research is concerned to study the global threshold dynamics of a hybrid viral infection model, with the assumption that all parameters are space dependent. The considered hybrid model contains three principal equations, the first is an ordinary differential equation that represents the uninfected cells, an age-structured equation modeled by the transport equation for infected-cells, and a diffusive equation that studies the evolution of the virus. The challenging mathematical aspect of this research lies in the fact that the model is partially degenerate and the solution map is not compact. Also, understanding the global threshold dynamics in terms of this partial degeneracy, mostly with age-structured equation and nonlocal diffusion is the ultimate challenging point of this research. The existence and the uniqueness of a global solution are proved. Further, the existence of a global compact attractor is shown. Moreover, the basic reproduction number $ \mathfrak{R}_0 $ is identified for the model with its threshold role: If $ \mathfrak{R}_0<1 $, the infection-free-steady state is globally asymptotically stable; for $ \mathfrak{R}_0>1 $, the solution of the studied model is uniformly persistent and the pathogen persists in a unique positive steady state, which it is shown using the Lyapunov approach. The results are supported by some graphical representations for illustrating the finding. [ABSTRACT FROM AUTHOR]

We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue λ0$$ {\lambda}_0 $$ corresponding to strictly positive eigenfunction which is expressed as λ0=R0−1$$ {\lambda}_0={R}_0-1 $$. By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for R0<1$$ {R}_0<1 $$. For R0>1$$ {R}_0>1 $$, and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function. [ABSTRACT FROM AUTHOR]

This article is an editorial that discusses the development of new modeling frameworks for ecological and epidemiological systems. The authors highlight ten research articles that focus on predator-prey dynamics, disease modeling, and other biological processes. The articles cover topics such as harvesting, cannibalism, dengue dynamics, and controlling the spread of COVID-19. The authors conclude that the contributions provide valuable insights into ecological dynamics and suggest effective disease control strategies. [Extracted from the article]

Immigration is the responsible for transmitting disease from one country to another. In this research, we investigate the global proprieties of an SIR model with nonlocal dispersal and immigration. The main purpose is to show that immigration will eliminate the threshold dynamics known for SIR epidemic models, and leads to the persistence of the disease independently of the parameters. Based on Lipschitz continuity of parameters, it has been shown that the strictly positive equilibrium state exists and it is unique. Using the infected density equilibrium as the integral kernel of a Lyapunov function, we also proved the global attractiveness of the endemic equilibrium state, the mathematical findings are checked numerically. • Global proprieties of an SIR model with nonlocal disperse and immigration effect. • Persistence and extinction of the disease. • Existence of the positive equilibrium state and its uniqueness. • Global stability analysis of the model. [ABSTRACT FROM AUTHOR]

In this research, we investigate the global properties of the heroin epidemic model with time distributed delay and nonlinear incidence function. We show that the system has threshold dynamics in terms of R 0 , and we prove, a Lyapunov function, that for R 0 < 1 the drug-free equilibrium is globally asymptotically stable. For R 0 > 1 , we give the persistence result of the heroin consumption. We also show the global stability of the endemic equilibrium for R 0 > 1 using a suitable Lyapunov function. The mathematical results are illustrated by numerically simulations. [ABSTRACT FROM AUTHOR]

In this paper, the global dynamics of an S E I epidemic model with constant immigration and general nonlinear incidence function is investigated. It is shown that there is neither a disease free equilibrium nor a basic reproduction number for this kind of models containing immigration terms. Moreover, the existence of a unique endemic equilibrium is proved. Using second Lyapunov method, we establish the global stability of the positive equilibrium. For a specific type of incidence function, some numerical simulations are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]