Reaction-diffusion model of HIV infection of two target cells under optimal control strategy

In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number $R_0$ was established, and it is proved that $R_0$ serves as a threshold parameter: When $R_0<1$, the disease-free steady state is globally asymptotically stable, and the disease is extinct; when $R_0>1$, the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as $R_0$ increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs.