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In this work, we propose a time‐delayed reaction–diffusion model to describe the propagation of infectious viral diseases like COVID‐19 The model is a two‐dimensional system of partial differential equations that describes the interactions between disjoint groups of a human population More precisely, we assume that the population is conformed by individuals who are susceptible to the virus, subjects who have been exposed to the virus, members who are infected and show symptoms, asymptomatic infected individuals, and recovered subjects Various realistic assumptions are imposed upon the model, including the consideration of a time‐delay parameter which takes into account the effects of social distancing and lockdown We obtain the equilibrium points of the model and analyze them for stability Moreover, we examine the bifurcation of the system in terms of one of the parameters of the model To simulate numerically this mathematical model, we propose a time‐splitting nonlocal finite‐difference scheme The properties of the model are thoroughly established, including its capability to preserve the positivity of solutions, its consistency, and its stability Some numerical experiments are provided for illustration purposes

In this manuscript, we follow a nonlinear approach in the problem of numerically estimating solutions of a generic, diffusive partial differential equation with nonpolynomial advection/convection, and reaction laws. The nonlocal, finite-difference methodology presented in this manuscript has the advantage of conditionally preserving the properties of non-negativity, boundedness, and temporal and spatial monotonicity of approximations, which are characteristics that are inherent to some kink-like solutions of particular forms of the mathematical model under investigation. We establish theorems on the stability of the technique, on the existence and uniqueness of non-negative and bounded solutions of our computational model, and on the property of preservation of monotonicity. As corollaries, we show that the method is capable of preserving the spatial and temporal monotonicity of numerical approximations. We provide some illustrative, numerical simulations that support our analytical results. Our simulations show that the method provides good approximations to the exact solutions of the particular problems considered and that the properties of non-negativity, boundedness, and monotonicity (both temporal and spatial) are conditionally preserved at each iteration. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 652–669, 2015

In this manuscript, we propose a simple, two-step, finite-difference scheme to approximate the solutions of an advective Fisher's equation. The method proposed is nonlinear, explicit and, in the linear regime, it approximates the solutions of the equation of interest with a consistency of first order in time and second order in space. We prove that the technique is capable of preserving the positive, the bounded, and the temporally and spatially monotone characters of initial approximations; moreover, we establish that the method is conditionally stable under suitable constraints on the model and numerical parameters. Some simulations are provided to evince the validity of our analytical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

We develop an explicit finite-difference method to approximate solutions of modified, Fermi–Pasta–Ulam media, which consider the presence of parameters, such as external damping, relativistic mass, a coefficient for the nonlinear term, and a coefficient of coupling in the case of discrete systems. We propose discrete expressions to approximate consistently the total energy of the system and the average energy flux, and prove that the discrete rate of change of energy is a consistent estimate of its continuous counterpart. The method is thoroughly tested in the energy domain, and our results show that the method gives an approximately constant energy in the case of conservative systems, which fluctuates within a narrow margin of error that may be attributed to truncation errors. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010