A delayed dynamical model for COVID-19 therapy with defective interfering particles and artificial antibodies

Abstract

<p style='text-indent:20px;'>In this paper, we use delay differential equations to propose a mathematical model for COVID-19 therapy with both defective interfering particles and artificial antibodies. For this model, the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> is given and its threshold properties are discussed. When <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_0&lt;1 $\end{document}</tex-math></inline-formula>, the disease-free equilibrium <inline-formula><tex-math id="M3">\begin{document}$ E_0 $\end{document}</tex-math></inline-formula> is globally asymptotically stable. When <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ E_0 $\end{document}</tex-math></inline-formula> becomes unstable and the infectious equilibrium without defective interfering particles <inline-formula><tex-math id="M6">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> comes into existence. There exists a positive constant <inline-formula><tex-math id="M7">\begin{document}$ R_1 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M8">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> is globally asymptotically stable when <inline-formula><tex-math id="M9">\begin{document}$ R_1&lt;1&lt;\mathcal{R}_0 $\end{document}</tex-math></inline-formula>. Further, when <inline-formula><tex-math id="M10">\begin{document}$ R_1&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> loses its stability and infectious equilibrium with defective interfering particles <inline-formula><tex-math id="M12">\begin{document}$ E_2 $\end{document}</tex-math></inline-formula> occurs. There exists a constant <inline-formula><tex-math id="M13">\begin{document}$ R_2 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M14">\begin{document}$ E_2 $\end{document}</tex-math></inline-formula> is asymptotically stable without time delay if <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;R_1&lt;\mathcal{R}_0&lt;R_2 $\end{document}</tex-math></inline-formula> and it loses its stability via Hopf bifurcation as the time delay increases. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.</p>