A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus

Abstract

Fractional differential equations are beginning to gain widespread usage in modeling physical and biological processes. It is worth mentioning that the standard mathematical models of integer-order derivatives, including nonlinear models, do not constitute suitable framework in many cases. In this work, a mathematical model for COVID-19 and Hepatitis B Virus (HBV) co-interaction is developed and studied using the Atangana-Baleanu fractional derivative. The necessary conditions of the existence and uniqueness of the solution of the proposed model are studied. The local stability analysis is carried out when the reproduction number is less than one. Using well constructed Lyapunov functions, the disease free and endemic equilibria are proven to be globally asymptotically stable under certain conditions. Employing fixed point theory, the stability of the iterative scheme to approximate the solution of the model is discussed. The model is fitted to real data from the city of Wuhan, China, and important parameters relating to each disease and their co-infection, are estimated from the fitting. The approximate solutions of the model are compared using the integer and fractional order derivatives. The impact of the fractional derivative on the proposed model is also highlighted. The results proven in this paper illustrate that HBV and COVID-19 transmission rates can greatly impact the dynamics of the co-infection of both diseases. It is concluded that to control the co-circulation of both diseases in a population, efforts must be geared towards preventing incident infection with either or both diseases.

Keywords: COVID-19; Co-infection; Fractional derivative; HBV; Lyapunov functions; Stability.

Fig. 1

Model fitting.

Fig. 2

Solution profiles for $\mathcal{S}\left(t\right)$, ${\mathcal{G}}_C\left(t\right)$, ${\mathcal{K}}_C\left(t\right)$, ${\mathcal{G}}_H\left(t\right)$, ${\mathcal{K}}_H\left(t\right)$, ${\mathcal{K}}_{CH}\left(t\right)$ and $\mathcal{R}\left(t\right)$ via integer and fractional order derivatives. Here, $\beta {}_{\text{C}}=6.29\times 10^{-8};\beta {}_{\text{H}}=5.0\times 10^{-8};\beta {}_{\text{CH}}=7.2\times 10^{-8}$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=3.2648>1$. Other parameters are the same as given in Table 1.

Fig. 3

Solution profiles for $\mathcal{S}\left(t\right)$, ${\mathcal{G}}_C\left(t\right)$, ${\mathcal{K}}_C\left(t\right)$, ${\mathcal{G}}_H\left(t\right)$, ${\mathcal{K}}_H\left(t\right)$, ${\mathcal{K}}_{CH}\left(t\right)$ and $\mathcal{R}\left(t\right)$ at different fractional orders. Here, $\beta {}_{\text{C}}=6.29\times 10^{-8};\beta {}_{\text{H}}=5.0\times 10^{-8};\beta {}_{\text{CH}}=7.2\times 10^{-8}$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=3.2648>1$. Other parameters are the same as given in Table 1.

Fig. 4

Solution profiles, assessing the impact of COVID-19 and HBV transmission rates. Other parameters are the same as given in Table 1.