Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse

The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number $\mathcal{R}_{0}$ R 0 is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

Dynamics of a stochastic delayed avian influenza model with mutation and temporary immunity

In this paper, the dynamic behaviors are studied for a stochastic delayed avian influenza model with mutation and temporary immunity. First, we prove the existence and uniqueness of the global positive solution for the stochastic model. Second, we give two different thresholds [Formula: see text] and [Formula: see text], and further establish the sufficient conditions of extinction and persistence in the mean for the avian-only subsystem and avian-human system, respectively. Compared with the corresponding deterministic model, the thresholds affected by the white noises are smaller than the ones of the deterministic system. Finally, numerical simulations are carried out to support our theoretical results. It is concluded that the vaccination immunity period can suppress the spread of avian influenza during poultry and human populations, while prompt the spread of mutant avian influenza in human population.

Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

<abstract><p>In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.</p></abstract>