A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes

Idris Ahmed; Goni Umar Modu; Abdullahi Yusuf; Poom Kumam; Ibrahim Yusuf

doi:10.1016/j.rinp.2020.103776

A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes

Results Phys. 2021 Feb:21:103776. doi: 10.1016/j.rinp.2020.103776. Epub 2021 Jan 6.

Authors

Idris Ahmed^{1

2}, Goni Umar Modu³, Abdullahi Yusuf^{4

5}, Poom Kumam^{2

6}, Ibrahim Yusuf⁷

Affiliations

¹ Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
² KMUTTFixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
³ Department of Statistics, Ramat Polytechnic Maiduguri, P. M. B 1070 Maiduguri, Borno State, Nigeria.
⁴ Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey.
⁵ Department of Mathematics, Federal University Dutse, Jigawa 7156, Nigeria.
⁶ Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan.
⁷ Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011 Kano, Nigeria.

Abstract

The research work in this paper attempts to describe the outbreak of Coronavirus Disease 2019 (COVID-19) with the help of a mathematical model using both the Ordinary Differential Equation (ODE) and Fractional Differential Equation. The spread of the disease has been on the increase across the globe for some time with no end in sight. The research used the data of COVID-19 cases in Nigeria for the numerical simulation which has been fitted to the model. We brought in the consideration of both asymptomatic and symptomatic infected individuals with the fact that an exposed individual is either sent to quarantine first or move to one of the infected classes with the possibility that susceptible individual can also move to quarantined class directly. It was found that the proposed model has two equilibrium points; the disease-free equilibrium point $(DFE)$ and the endemic equilibrium point $(E_{1})$ . Stability analysis of the equilibrium points shows $(E_{0})$ is locally asymptotically stable whenever the basic reproduction number, $R_{0} < 1$ and $(E_{1})$ is globally asymptotically stable whenever $R_{0} > 1$ . Sensitivity analysis of the parameters in the $R_{0}$ was conducted and the profile of each state variable was also depicted using the fitted values of the parameters showing the spread of the disease. The most sensitive parameters in the $R_{0}$ are the contact rate between susceptible individuals and the rate of transfer of individuals from exposed class to symptomatically infected class. Moreover, the basic reproduction number for the data is calculated as $R_{0} \approx 1.7031$ . Existence and uniqueness of solution established via the technique of fixed point theorem. Also, using the least square curve fitting method together with the fminsearch function in the MATLAB optimization toolbox, we obtain the best values for some of the unknown biological parameters involved in the proposed model. Furthermore, we solved the fractional model numerically using the Atangana-Toufik numerical scheme and presenting different forms of graphical results that can be useful in minimizing the infection.

Keywords: 34A12; 39A30; 47H10; ABC-fractional operator; Basic reproductive number; Corona virus; Existence and uniqueness; Mathematical model; Nonlinear differential equations; Sensitivity analysis.