Nonlinear optimal control strategies for a mathematical model of COVID-19 and influenza co-infection

Infectious diseases have remained one of humanity's biggest problems for decades. Multiple disease infections, in particular, have been shown to increase the difficulty of diagnosing and treating infected people, resulting in worsening human health. For example, the presence of influenza in the population is exacerbating the ongoing COVID-19 pandemic. We formulate and analyze a deterministic mathematical model that incorporates the biological dynamics of COVID-19 and influenza to effectively investigate the co-dynamics of the two diseases in the public. The existence and stability of the disease-free equilibrium of COVID-19-only and influenza-only sub-models are established by using their respective threshold quantities. The result shows that the COVID-19 free equilibrium is locally asymptotically stable when $$ , whereas the influenza-only model, is locally asymptotically stable when $$ . Furthermore, the existence of the endemic equilibria of the sub-models is examined while the conditions for the phenomenon of backward bifurcation are presented. A generalized analytical result of the COVID-19-influenza co-infection model is presented. We run a numerical simulation on the model without optimal control to see how competitive outcomes between-hosts and within-hosts affect disease co-dynamics. The findings established that disease competitive dynamics in the population are determined by transmission probabilities and threshold quantities. To obtain the optimal control problem, we extend the formulated model by including three time-dependent control functions. The maximum principle of Pontryagin was used to prove the existence of the optimal control problem and to derive the necessary conditions for optimum disease control. A numerical simulation was performed to demonstrate the impact of different combinations of control strategies on the infected population. The findings show that, while single and twofold control interventions can be used to reduce disease, the threefold control intervention, which incorporates all three controls, will be the most effective in reducing COVID-19 and influenza in the population.