Dynamic analysis and optimization of syphilis spread: Simulations, integrating treatment and public health interventions

Abstract Throughout global history, the population has faced unprecedented challenges due to infectious spreads. Addressing the need to mitigate these infections requires well-directed and comprehensive efforts. Sexually transmitted infections, including syphilis, remain significant global health concerns affecting both developed and developing nations. Syphilis, resulting from the transmission of the Treponema pallidum bacterium through sexual contact, is estimated to affect around 12 million individuals annually worldwide. The objectives are achieved by launching the syphilis model, dividing the entire population into six compartments. Additionally, an ideal control plan is presented that integrates the most practicable medical measures to lower the quantity of afflicted persons encapsulate the dynamics of the prevailing degree of syphilis in a population. The devised model is validated by verifying essential features such as positivity, invariant region, and equilibrium points of the point for feasibility of solutions. This investigation focuses on examining the local stability of the syphilis model with a specific emphasis on considering limited observations a critical aspect of epidemic models. The reproductive number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {R}_{0} has been calculated to evaluate its impact across various sub-compartments playing a pivotal role in determining community-wide transmission rates. A sensitivity analysis of the models parameters has been performed. To gain numerical solutions, the advanced and well-established numerical technique nonstandard finite difference is employed to provide insights into the genuine behavior of the model. Additionally, to assist in achieving the fundamental aim of this research, an optimal control strategy is induced by considering control variables, namely, an educational awareness campaign and treatment protocols intended to reduce the prevalence of infected individuals. The purposes are attained by employing the Pontryagin maximum principle through mathematically and modeling.