Positive Numerical Approximation of Integro-Differential Epidemic Model

Abstract

In this paper, we study a dynamically consistent numerical method for the approximation of a nonlinear integro-differential equation modeling an epidemic with age of infection. The discrete scheme is based on direct quadrature methods with Gregory convolution weights and preserves, with no restrictive conditions on the step-length of integration h, some of the essential properties of the continuous system. In particular, the numerical solution is positive and bounded and, in cases of interest in applications, it is monotone. We prove an order of convergence theorem and show by numerical experiments that the discrete final size tends to its continuous equivalent as h tends to zero.

The effect of host population heterogeneity on epidemic outbreaks

“…The key point is that there is no user-friendly tool to solve (2.7) and that (2.8) is straightforward to implement. For the numerical analysis point of view, we refer to Messina et al [46][47][48][49][50].…”

“…The Renewal Equation (6.5) is a delay equation, i.e., a rule for extending a function of time towards the future on the basis of the, assumed to be, known past. Solving such equations numerically is not really difficult, but user-friendly software does not exist (but see Messina et al [46][47][48][49][50] for promising developments). A recently developed methodology for numerical bifurcation analysis via systematic approximation by ODE is described in [61].…”

New Trends on the Mathematical Models and Solitons Arising in Real-World Problems

“…Mathematical models in science and technology have recently attracted an increased amount of research attention with the aim to understand, describe, and predict the future behaviors of natural phenomena. Recent studies on fractional calculus have been particularly popular among researchers due to their favorable properties when analyzing real-world models associated with properties such as anomalous diffusion, non-Markovian processes, random walk, long range, and, most importantly, heterogeneous behaviors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The concept of local differential operators along with power law settings and non-local differential operators were suggested in order to accurately replicate the above-cited natural processes.…”