Optimal Control of Nonlinear Competitive Epidemic Spreading Processes With Memory in Heterogeneous Networks

Modeling and control of epidemic-like spreading processes, such as contagions, rumors or products in complex networks, holds profound significance for understanding transmission mechanisms, formulating effective prevention and optimizing resource allocation. This paper investigates the optimal control problems for competing processes modeled by fractional-order bi-virus systems with memory in complex networks for the first time. The proposed model performs exceptionally well in capturing the dynamics of virus transmission and immune. First, the key indicators: bi-basic reproduction numbers (BBRNs) are introduced for the metapopulation model with nonlinear infectious functions in heterogenous networks. Then they are utilized to classify the equilibria, including extinction, absolute dominance and coexistence ones. Second, sufficient conditions for the stability of the equilibria are derived. Leveraging this result, an optimal therapeutic protocol is presented to eradicate viruses and minimize the cost function simultaneously. The corresponding optimal control solutions are generated via gradient descent algorithm, which can converge relatively quickly by iteratively adjusting control variables. Particularly, the role of fractional-order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> in system modeling and optimization is deeply probed into. Finally, numerical examples are also included at the end of this paper to further shed light on the effectiveness of our results.