Unveiling Activated NK Cell Dynamics in Salmonella Typhi Infection Under Optimum Double-Drug (Azithromycin and IFN−γ) Therapy: An Immuno-Mathematical Model Analysis

This article presents a four-dimensional nonlinear immuno-mathematical model to elucidate the crucial role of activated natural killer (NK) cells during the interaction between Salmonella Typhi (S. Typhi) bacteria and macrophages. Analytically, we have demonstrated the direct involvement of NK cell-mediated immune responses through Interferon - $\gamma$ ( $IFN-\gamma$) signaling in the necessary conditions for disease persistence. The stability criteria for disease-free and endemic equilibria are theoretically validated using the basic reproduction number ( ${ℛ}_0$) and the Routh-Hurwitz criterion. Our system exhibits a transcritical bifurcation at ${ℛ}_0=1$, indicating changes in the stability of the disease-free and endemic equilibria. We have also investigated the effects of a dual-drug regimen (Azithromycin and $IFN-\gamma$) by introducing two surrogate control parameters into our model, utilizing optimal control theory. The derived optimal drug dosage pair offers a cost-effective and efficient treatment strategy. For model calibration via numerical simulations, we have collected some parameter values from certain published mouse model studies and also performed sensitivity analysis for ${ℛ}_0$ to assess the relative importance of these parameters for our mathematical system. To evaluate the robustness of model predictions against parameter variations, we have computed sensitivity indices and model sustainability across different parameter regions. Numerical results substantiate the outcomes of the control-induced system, and we have compared the results under various maximum control efforts. Our study suggests that combination therapy using Azithromycin and $IFN-\gamma$ with optimum drug efficacy can eradicate 100% S. Typhi infection from infected macrophages within 10 days.