Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate

The current study deals with the stochastic reaction-diffusion epidemic model numerically with two proposed schemes. Such models have many applications in the disease dynamics of wildlife, human life, and others. During the last decade, it is observed that the epidemic models cannot predict the accurate behavior of infectious diseases. The empirical data gained about the spread of the disease shows non-deterministic behavior. It is a strong challenge for researchers to consider stochastic epidemic models. The effect of the stochastic process is analyzed. So, the SIR epidemic model is considered under the influence of the stochastic process. The time noise term is taken as the stochastic source. The coefficient of the stochastic term is a Borel function, and it is used to control the random behavior in the solutions. The proposed stochastic backward Euler scheme and the proposed stochastic implicit finite difference scheme (IFDS) are used for the numerical solution of the underlying model. Both schemes are consistent in the mean square sense. The stability of the schemes is proven with Von-Neumann criteria and schemes are unconditionally stable. The proposed stochastic backward Euler scheme converges toward a disease-free equilibrium and does not converge toward an endemic equilibrium but also possesses negative behavior. The proposed stochastic IFD scheme converges toward disease-free equilibrium and endemic equilibrium. This scheme also preserves positivity. The graphical behavior of the stochastic SIR model is much similar to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The efficacy of the proposed scheme is shown in the graphical behavior of the test problem for the various values of the parameters.

Analytical and Numerical Boundedness of a Model with Memory Effects for the Spreading of Infectious Diseases

In this study, an integer-order rabies model is converted into the fractional-order epidemic model. To this end, the Caputo fractional-order derivatives are plugged in place of the classical derivatives. The positivity and boundedness of the fractional-order mathematical model is investigated by applying Laplace transformation and its inversion. To study the qualitative behavior of the non-integer rabies model, two steady states and the basic reproductive number of the underlying model are worked out. The local and global stability is investigated at both the steady states of the fractional-order epidemic model. After analytic treatment, a structure-preserving numerical template is constructed to numerically solve the fractional-order epidemic model. Moreover, the positivity, boundedness and symmetry of the numerical scheme are examined. Lastly, numerical experiment and simulations are accomplished to substantiate the significant traits of the projected numerical design. Consequences of the study are highlighted in the closing section.

Fractional Calculus—Theory and Applications

In recent years, fractional calculus has witnessed tremendous progress in variousareas of sciences and mathematics [...]