Modeling SARS-CoV-2 and HBV co-dynamics with optimal control

Abstract

Clinical reports have shown that chronic hepatitis B virus (HBV) patients co-infected with SARS-CoV-2 have a higher risk of complications with liver disease than patients without SARS-CoV-2. In this work, a co-dynamical model is designed for SARS-CoV-2 and HBV which incorporates incident infection with the dual diseases. Existence of boundary and co-existence endemic equilibria are proved. The occurrence of backward bifurcation, in the absence and presence of incident co-infection, is investigated through the proposed model. It is noted that in the absence of incident co-infection, backward bifurcation is not observed in the model. However, incident co-infection triggers this phenomenon. For a special case of the study, the disease free and endemic equilibria are shown to be globally asymptotically stable. To contain the spread of both infections in case of an endemic situation, the time dependent controls are incorporated in the model. Also, global sensitivity analysis is carried out by using appropriate ranges of the parameter values which helps to assess their level of sensitivity with reference to the reproduction numbers and the infected components of the model. Finally, numerical assessment of the control system using various intervention strategies is performed, and reached at the conclusion that enhanced preventive efforts against incident co-infection could remarkably control the co-circulation of both SARS-CoV-2 and HBV.

Keywords: Backward bifurcation; HBV; Incident co-infection; Lyapunov functions; Optimal control; SARS-CoV-2.

Fig. 1

The Model’s schematic diagram.

Fig. 2

Graphical results of the sensitivity analysis with the SARS-CoV-2, HBV, co-infection related reproduction numbers as response functions. Reasonable ranges of the parameter values given in Table 1 are used.

Fig. 3

Graphical results of the sensitivity analysis with different epidemiological states and total infected population as response functions. Reasonable ranges of the parameter values given in Table 1 are used.

Fig. 4

Impact of SARS-CoV-2 prevention control. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.

Fig. 5

Impact of HBV prevention control. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.

Fig. 6

Impact of control against co-infection with both diseases. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.