Stability analysis and optimal control of an epidemic model with vaccination

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number （R0） of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0 when R0 〈1. When R0 〉1 endemic equilibrium E1 exists and the system becomes locally asymptotically stable at E1 under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

Mathematical Modeling of Malaria Transmission Dynamics: Case of Burundi

Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, R0, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if R0 > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (σ), the vector population of mosquitoes (Nv), the probability of being infected for a human bitten by an infectious mosquito per unit of time (b) and the probability of being infected for a mosquito per unit of time (c) must be reduced by applying optimal control measures.

Analysis of a hand-foot-mouth disease model

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease （HFMD） infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number （R0） of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls （two treatment controls with quarantine control）. We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows tile reliability of our model from the practical point of view.

Effects of Nonmonotonic Functional Responses on a Disease Transmission Model:Modeling and Simulation

In this article,a novel susceptible-infected-recovered epidemic model with nonmonotonic incidence and treatment rates is proposed and analyzed mathematically.The Monod-Haldane functional response is considered for nonmonotonic behavior of both incidence rate and treatment rate.The model analysis shows that the model has two equilibria which are named as disease-free equilibrium(DFE)and endemic equilibrium(EE).The stability analysis has been performed for the local and global behavior of the DFE and EE.With the help of the basic reproduction number(R_(0)),we investigate that DFE is locally asymptotically stable when R_(0)<1 and unstable when R_(0)>1.The local stability of DFE at R_(0)=1 has been analyzed,and it is obtained that DFE exhibits a forward transcritical bifurcation.Further,we identify conditions for the existence of EE and show the local stability of EE under certain conditions.Moreover,the global stability behavior of DFE and EE has been investigated.Lastly,numerical simulations have been done in the support of our theoretical findings.

Effects on prey-predator with different functional responses

In this paper, we investigate the effects on prey of two predators which are also related in terms of prey-predator relationship. Different types of functional responses are con- sidered to formulate the mathematical model for predator and generalist predator of our proposed model. Harvesting effort for the generalist predator is considered and the density-dependent mortality rate for predator and generalist predator is incorporated in our proposed model. Local stability as well as global stability for the system is dis- cussed. We analyze the different bifurcation parameters to evaluate Hopf bifurcation in the neighborhood of interior equilibrium point. Finally, some numerical simulations and graphical figures are provided to verify our analytical results with the help of different sets of parameters.