Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Sebastian Bonhoeffer et al. [2], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar type of approximate model incorporating time delay in the process of infection on the healthy T cells which, in turn, implies inclusion of a similar delay in the process of viral replication. The model is studied both analytically and numerically. We also include a similar delay in the killing rate of infected CD4 + T cells by Cytotoxic TLymphocyte (CTL) and in the stimulation of CTL and analyze two resulting models numerically.The models with no time delay present have two equilibria: one where there is no infection and a non-trivial equilibrium where the infection can persist. If there is no time delay then the non-trivial equilibrium is locally asymptotically stable. Both our analytical results (for the first model) and our numerical results (for all three models) indicate that introduction of a time delay can destabilize the non-trivial equilibrium. The numerical results indicate that such destabilization occurs at realistic time delays and that there is a threshold time delay beneath which the equilibrium with infection present is locally asymptotically stable and above which this equilibrium is unstable and exhibits oscillatory solutions of increasing amplitude.
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.
In recent, non-pharmaceutical intervention (lockdown, quarantine, expended testing) and the pharmaceutical intervention (use of commonly used drugs) are the only available strategies to control the COVID-19 disease. Though the scientist all over the world are engaging themselves to find the way out the vaccine of COVID-19, still it is persisted unanswered how to oust the pandemic epidemic from the world. Generally, social distancing, using the mask, etc. are the only available policy to control the pandemic. In this situation uses of common drugs (Azithromycin, HCQ, Antiprotozoal with Doxycycline, Levocetirizine with Montelukast) are common but effective treatment for the reported and hospitalized patient. These drugs activate the immune system of our body to fight against the disease progression. We have formulated a seven compartmental SEIQR type model to explore the COVID-19 disease progression. We have studied the effect of pharmaceutical and non pharmaceutical intervention as a control input and it effect to reduce the number of the infected population while reducing the cost related with the awareness and drug in a specific time frame. Analytical finding tells that the system behavior depends on basic reproduction
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