Biophysical Reviews and Letters

In this research work, we have developed and analyzed a deterministic epidemiological model with a system of nonlinear differential equations for controlling the spread of Ebola virus disease (EVD) in a population with vital dynamics (where birth and death rates are not equal). The model examines the disease transmission dynamics with isolation from exposed and infected human class and effect of vaccination in susceptible human population through stability analysis and bifurcation analysis. The model exhibits two steady state equilibria, namely, disease-free and endemic equilibrium. Next generation matrix method is used to find the expression for $R_0$ (the basic reproduction number). Local and global stability of diseases-free equilibrium are shown using nonsingular M-matrix technique and Lyapunov’s theorem, respectively. The existence and local stability of endemic equilibrium are explored under certain conditions. All numerical data entries are supported by various authentic sources. The simulation study is done using MATLAB code 45 which uses Runge–Kutta method of fourth order and we plot the time series and bifurcation diagrams which support our analytical findings. Stability analysis of the model shows that the disease-free equilibrium is locally as well as globally asymptotically stable if $R_0<1$ and endemic equilibrium is locally asymptotically stable in absence of vaccination if $R_0>1$. Using central manifold theorem, the presence of transcritical bifurcation for a threshold value of the transmission rate parameter $\beta$ when $R_0$ passes through unity and backward bifurcation (i.e. transcritical bifurcation in opposite direction) for some higher value of $R_0$ are established. Our simulation study shows that isolation of exposed and infected individuals can be used as a more effective tool to control the spreading of EVD than only vaccination.

This work describes a delayed pest–plant ecological model with infection in the pest population. The interactions between plant and susceptible pest and also between susceptible and infected pest are taken as Holling type II responses. Well-posedness of the system has been discussed. Stability analysis of all equilibria has been performed. The effect of time-delay has been studied, where the delay may be regarded as the incubation period of the infected pest. Existence of a Hopf-bifurcation around interior equilibrium has been established by considering the amount of delay as bifurcation parameter. The length of delay is estimated for which the stability continues to hold. Numerical simulation with a hypothetical set of data has been presented to validate analytical findings.