Abstract
The novel coronavirus disease, coined as COVID-19, is a murderous and infectious disease initiated from Wuhan, China. This killer disease has taken a large number of lives around the world and its dynamics could not be controlled so far. In this article, the spatio-temporal compartmental epidemic model of the novel disease with advection and diffusion process is projected and analyzed. To counteract these types of diseases or restrict their spread, mankind depends upon mathematical modeling and medicine to reduce, alleviate, and anticipate the behavior of disease dynamics. The existence and uniqueness of the solution for the proposed system are investigated. Also, the solution to the considered system is made possible in a well-known functions space. For this purpose, a Banach space of function is chosen and the solutions are optimized in the closed and convex subset of the space. The essential explicit estimates for the solutions are investigated for the associated auxiliary data. The numerical solution and its analysis are the crux of this study. Moreover, the consistency, stability, and positivity are the indispensable and core properties of the compartmental models that a numerical design must possess. To this end, a nonstandard finite difference numerical scheme is developed to find the numerical solutions which preserve the structural properties of the continuous system. The M-matrix theory is applied to prove the positivity of the design. The results for the consistency and stability of the design are also presented in this study. The plausibility of the projected scheme is indicated by an appropriate example. Computer simulations are also exhibited to conclude the results.
Highlights
Transmission of different infectious diseases occur in various parts of the world
Numerical solutions and simulations help us to understand the behavior of the disease, especially when advection and diffusion terms are included in the model
We have analyzed a mathematical model concerned with the novel coronavirus disease and the existence and uniqueness of its solution in a closed and convex subset of a banach space is discussed
Summary
To understand the dynamics of these, mathematicians have developed and analysed the mathematical models by taking into account their different aspects [1,2,3,4,5,6,7]. After developing a suitable mathematical model of this infection, it is a challenge for the researchers to find the solutions to these models These models have a number of parameters (defining the rates with which various compartments are affected) so, in general, it becomes hard to find their analytical solutions. Numerical solutions and simulations help us to understand the behavior of the disease, especially when advection and diffusion terms are included in the model. In the best of our knowledge, the explicit estimates with the optimal balls for any epidemic model have never been discussed
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