Mathematical Framework for Epidemic Dynamics: Optimal Control and Global Sensitivity Analysis

TL;DR

This paper addresses robust epidemic control under parameter uncertainty by extending the SIHR model with time-dependent controls $u_1(t)$ (prevention) and $u_2(t)$ (treatment). It develops an integrated framework that solves the controlled SIHR system via an optimal control formulation (Forward-Backward Sweep) and quantifies outcome uncertainty using Polynomial Chaos Expansion (PCE) and Sobol indices, applied to Djibouti's COVID-19 dynamics. The main contributions are (i) formulation of a two-control SIHR model with explicit control shapes, (ii) a PCE-Sobol uncertainty quantification pipeline for controlled epidemics, and (iii) numerical demonstrations showing that combined $u_1$ and $u_2$ strategies substantially reduce infections and hospitalizations and that $\beta$ dominates early dynamics while $\gamma,\alpha,\lambda$ become important later. The findings highlight the value of integrating optimization with uncertainty analysis for designing robust, resource-aware intervention policies in low-resource settings, and provide a general framework extendable to co-infections and other diseases.

Abstract

This study develops and analyzes an extended Susceptible, Infected, Hospitalized and Recovered (SIHR) model incorporating time dependent control functions to capture preventive measures (e.g., distancing, mask use) and resource limited therapeutic interventions. This formulation provides a realistic mathematical framework for modeling public health responses beyond classical uncontrolled epidemic models. The control design is integrated into the model via an optimal control framework, solved numerically using the Forward Backward Sweep method, enabling the exploration of intervention strategies on epidemic dynamics, including infection prevalence, hospitalization burden, and the effective reproduction number. To assess the robustness of these strategies under uncertainty, we employ Polynomial Chaos Expansion combined with Sobol sensitivity indices, quantifying the influence of key epidemiological parameters (transmission, recovery, hospitalization rates) on model outcomes. Numerical simulations, calibrated to Djiboutian COVID 19 data, show that combined preventive and therapeutic interventions substantially mitigate epidemic burden, though their effectiveness depends critically on transmission related uncertainties. The originality of this work lies in combining optimal control theory with global sensitivity analysis, thus bridging numerical methods, optimization, and epidemic modeling. This integrated approach offers a general mathematical framework for designing and evaluating control strategies in infectious disease outbreaks, with applications to low resource settings and beyond.

Figures (12)

• Figure 1: Optimal control functions $u_1$ (preventive interventions such as social distancing, mask usage, and vaccination campaigns) and $u_2$ (treatment/hospitalization) over time. $u_1$ shows a delayed but sustained increase, stabilizing at a high level, whereas $u_2$ peaks early to curb the initial epidemic surge and then declines.
• Figure 2: Temporal evolution of the effective reproduction number $\mathcal{R}_e(t)$ under optimal control. The combination of preventive measures ($u_1$) and treatment/hospitalization ($u_2$) reduces $\mathcal{R}_e$ below the epidemic threshold of 1, indicating effective suppression of COVID-19 transmission. The curve highlights periods of maximum intervention impact.
• Figure 3: 3D surface of the effective reproduction number $\mathcal{R}_e$ as a function of preventive ($u_1$) and treatment/hospitalization ($u_2$) control intensities. The red point indicates the control values at the midpoint of the intervention period. This visualization demonstrates the nonlinear interaction between interventions and their joint effect on reducing $\mathcal{R}_e$.
• Figure 4: Temporal dynamics of the SIHR compartments under uncontrolled and optimally controlled scenarios.
• Figure 5: First–order Sobol indices $S_i(t)$ (PCE) for all compartments with control $(S_c,I_c,H_c,R_c)$ under full intervention ($u_1\neq 0$, $u_2\neq 0$).

Theorems & Definitions (8)

• Theorem 1: Effectiveness and Optimization of Realistic Controls
• proof
• Remark 1
• Theorem 2: Well-posedness, positivity, and boundedness under controls
• proof
• Theorem 3: Effective reproduction number and stability of the DFE
• proof
• Remark 2