In this research article, we establish a fractional-order mathematical model to explore the infections of the coronavirus disease (COVID-19) caused by the novel SARS-CoV-2 virus. We introduce a set of fractional differential equations taking uninfected epithelial cells, infected epithelial cells, SARS-CoV-2 virus, and CTL response cell accounting for the lytic and non-lytic effects of immune responses. We also include the effect of a commonly used antiviral drug in COVID-19 treatment in an optimal control-theoretic approach. The stability of the equilibria of the fractional ordered system using qualitative theory. Numerical simulations are presented using an iterative scheme in Matlab in support of the analytical results.

The current emergence of coronavirus (SARS-CoV-2) puts the world in threat. The structural research on the receptor recognition by SARS-CoV-2 has identified the key interactions between SARS-CoV-2 spike protein and its host (epithelial cell) receptor, also known as angiotensin-converting enzyme 2 (ACE2). It controls both the cross-species and human-to-human transmissions of SARS-CoV-2. In view of this, we propose and analyze a mathematical model for investigating the effect of CTL responses over the viral mutation to control the viral infection when a postinfection immunostimulant drug (pidotimod) is administered at regular intervals. Dynamics of the system with and without impulses have been analyzed using the basic reproduction number. This study shows that the proper dosing interval and drug dose both are important to eradicate the viral infection.

Recent development in antiretroviral treatment against HIV can help AIDS patients to fight against HIV. But the question that whether the disease is to be partially or totally eradicated from HIV infected individuals still remains unsolved. Usually, the most effective treatment for the disease is HAART which can only control the disease progression. But as the immune system becomes weak, the patients can not fight against other diseases. Immune cells are activated and proliferated by IL-2 after the identification of antigen. IL-2 production is impaired in HIV positive patients and intermitted administration of immune activator IL-2 together with HAART which is a more effective treatment to fight against the disease. Thus, its expediency is essential and is yet to be explored. In this article we anticipated a mathematical model of the effect of IL-2 together with RTIs therapy in HIV positive patients. Our analytical as well as numerical study shows that the optimal schedule of treatment for best result is to be obtained by systematic drug therapy. But at the last stage of treatment, the infection level raises again due to minimisation of drug dosage. Thus we study the perfect adherence of the drugs and found out if RTIs are taken with sufficient interval then for fixed interval of IL-2 therapy, certain amount of drug dosages may be able to sustain the immune system at pre-infection stage and the infected CD4⁺T cells are going towards extinction.