In this paper a mathematical model that describes the flow of infectious disease in a population is proposed and studied. It is assumed that the disease divided the population into four classes: susceptible individuals (S), vaccinated individuals (V), infected individuals (I) and recover individuals (R). The impact of immigrants, vaccine and external sources of disease, on the dynamics of SVIRS epidemic model is studied. The existence, uniqueness and boundedness of the solution of the model are discussed. The local and global stability of the model is studied. The occurrence of local bifurcation as well as Hopf bifurcation in the model is investigated. Finally the global dynamics of the proposed model is studied numerically.

There are many factors effect on the spread of infectious disease or control it,
some of these factors are (immigration and vaccination). The main objective of this
paper is to study the effect of those factors on the dynamical behavior of an SVIR
model. It is assumed that the disease is spread by contact between members of
populations individuals. While the recovered individuals gain permanent immunity
against the disease. The existence, uniqueness and boundedness of the solution of
this model are investigated. The local and global dynamical behaviors of the model
are studied. The local bifurcations and Hopf bifurcation of the model are
investigated. Finally, in order to confirm our obtained results and specify t

It is well known that the spread of cancer or tumor growth increases in polluted environments. In this paper, the dynamic behavior of the cancer model in the polluted environment is studied taking into consideration the delay in clearance of the environment from their contamination. The set of differential equations that simulates this epidemic model is formulated. The existence, uniqueness, and the bound of the solution are discussed. The local and global stability conditions of disease-free and endemic equilibrium points are investigated. The occurrence of the Hopf bifurcation around the endemic equilibrium point is proved. The stability and direction of the periodic dynamics are studied. Finally, the paper is ended with a numerical simul

     A partial temporary immunity SIR epidemic model involv nonlinear treatment rate is proposed and studied. The basic reproduction number  is determined. The local and global stability of all equilibria of the model are analyzed. The conditions for occurrence of local bifurcation in the proposed epidemic model are established. Finally, numerical simulation is used to confirm our obtained analytical results and specify the control set of parameters that affect the dynamics of the model.

In this paper a mathematical model that analytically as well as numerically
the flow of infection disease in a population is proposed and studied. It is
assumed that the disease divided the population into five classes: immature
susceptible individuals (S1) , mature individuals (S2 ) , infectious individual
(I ), removal individuals (R) and vaccine population (V) . The existence,
uniqueness and boundedness of the solution of the model are discussed. The
local and global stability of the model is studied. Finally the global dynamics of
the proposed model is studied numerically.

In this paper, we established a mathematical model of an SI1I2R epidemic disease with saturated incidence and general recovery functions of the first disease I1. Considering the basic reproduction number, we obtained conditions for both disease-free and co-existing cases. The equilibrium points local stability is verified by using the Routh-Hurwitz criterion, while for the global stability, we used a suitable Lyapunov function to analyze the endemic spread of the positive equilibrium point. Moreover, we carried out the local bifurcation around both equilibrium points (disease-free and co-existing), where we obtained that the disease-free equilibrium point undergoes a transcritical bifurcation. We conduct numerical simulations that suppo

   In this research, dynamical study of an SIR epidemical model with nonlinear direct incidence rate (Beddington-De Angelis ) type, and regress of treatment investigated .An  analytical study  to the model shows that there are two equilibrium points appear, the discussed successfully with sufficient condition, the existence of local bifurcation and Hopf bifurcation was analyzed, finally numerical simulations are done to explain the analytic studies.

In this paper a prey-predator-scavenger food web model is proposed and studied. It is assumed that the model considered the effect of harvesting and all the species are infected by some toxicants released by some other species. The stability analysis of all possible equilibrium points is discussed. The persistence conditions of the system are established. The occurrence of local bifurcation around the equilibrium points is investigated. Numerical simulation is used and the obtained solution curves are drawn to illustrate the results of the model. Finally, the nonexistence of periodic dynamics is discussed analytically as well as numerically.

The objective of this paper is to study the stability of SIS epidemic model involving treatment. Two types of such eco-epidemiological models are introduced and analyzed. Boundedness of the system is established. The local and global dynamical behaviors are performed. The conditions of persistence of the models are derived.

It is proposed and studied a prey-predator system with a Holling type II functional response that merges predation fear with a predator-dependent prey's refuge. Understanding the impact of fear and refuge on the system's dynamic behavior is one of the objectives. All conceivable steady-states are investigated for their stability. The persistence condition of the system has been established. Local bifurcation analysis is performed in the Sotomayor sense. Extensive numerical simulation with varied parameters was used to explore the system's global dynamics. A limit cycle and a point attractor are the two types of attractors in the system. It's also interesting to note that the system exhibits bi-stability between these 2 types of attractors.