Mathematical Model of the Impact of Home-Based Care on Contagious Respiratory Illness Under Optimal Conditions

Abstract

Mathematical models are vital for understanding real-world phenomena without direct experimentation, particularly in epidemics, as they predict and analyze the effectiveness of various mitigation strategies. Given the rapid transmission of infectious respiratory diseases, public health measures aim to curb spread while managing impacts. This study assesses rapid contact tracing and testing, focusing on isolating confirmed cases through home-based care or traditional methods, on coronavirus transmission within a community. A deterministic mathematical model using ordinary differential equations segments the population into seven compartments: susceptible, exposed, asymptomatic, symptomatic, home-based care, hospitalized, and recovered. The basic reproduction number is determined via the next generation matrix. Local stability of the disease-free equilibrium is analyzed using the trace-determinant method, while global stability is confirmed with the Lyapunov-Krasovskii approach. A Python-based numerical simulation on NumPy and PyPlot uses parameters calibrated to previous studies and estimated for this research. Simulations indicate home-based care delays peak infection days and reduces peak population, providing time to bolster healthcare facilities. Optimal control methods, including media awareness, reduce susceptibility and encourage asymptomatic individuals to choose home-based care. Using Pontryagin's Maximum Principle, the study identifies optimal strategies, highlighting that media awareness effectively lowers susceptibility and optimal control directs asymptomatics to home-based care, reducing strain on healthcare facilities. In conclusion, home-based care is effective for managing mild symptomatic and asymptomatic cases, alleviating pressure on healthcare resources and prioritizing severe cases. Combining home-based care with other non-pharmaceutical strategies is recommended for maximum effectiveness.