A Human-Vector Susceptible-Infected-Susceptible Model for Analyzing and Controlling the Spread of Vector-Borne Diseases

TL;DR

This work introduces the HV-SIS model, a parsimonious yet analytically tractable framework for vector-borne disease spread that couples a human SIS dynamic with vector birth–death processes and cross-species contagion. It leverages monotone dynamical systems theory to establish a threshold-driven phase transition between disease-free and endemic regimes, with global convergence results. The paper then incorporates two control actions—vector control ($u_1$) and protection incentives ($u_2$)—and derives an optimal-control formulation to minimize intervention costs while ensuring eradication when feasible. The results provide insights into which intervention strategy is more cost-effective under varying parameter regimes and costs, offering a principled basis for planning vector-borne disease control programs with analytical guarantees. The analysis highlights the potential impact of vector-focused strategies and sets the stage for more sophisticated, networked, and behavior-aware extensions.

Abstract

We propose an epidemic model for the spread of vector-borne diseases. The model, which is built extending the classical susceptible-infected-susceptible model, accounts for two populations -- humans and vectors -- and for cross-contagion between the two species, whereby humans become infected upon interaction with carrier vectors, and vectors become carriers after interaction with infected humans. We formulate the model as a system of ordinary differential equations and leverage monotone systems theory to rigorously characterize the epidemic dynamics. Specifically, we characterize the global asymptotic behavior of the disease, determining conditions for quick eradication of the disease (i.e., for which all trajectories converge to a disease-free equilibrium), or convergence to a (unique) endemic equilibrium. Then, we incorporate two control actions: namely, vector control and incentives to adopt protection measures. Using the derived mathematical tools, we assess the impact of these two control actions and determine the optimal control policy.

Figures (4)

• Figure 1: Schematic of the human-vector epidemic model. Solid arrows represent possible transitions of the state of humans (S and I for susceptible and infected, respectively) and vectors (N and C for non-carrier and carrier, respectively). Dashed arrows are associated with the vital dynamics of vectors. Dotted colored arrows indicate transitions that are triggered by interactions with humans or vectors with a specific state.
• Figure 2: Trajectories of the human-vector epidemic model. In (a) $\sigma_0<1$ and the trajectory converges to the DFE; in (b) $\sigma_0>1$ and the trajectory converges to the EE. The equilibrium predicted by Theorem \ref{['th:uncontrolled']} in the two cases is depicted with gray dashed horizontal lines. Common parameters are $\omega=\beta_h=\beta_v=0.2$, and $\gamma=0.4$.
• Figure 3: (a) Infected individuals and (b) carrier vectors at the EE for different values of the control inputs $u_1$ and $u_2$. Model parameters are $\omega=\beta_h=\beta_v=0.2$, $\gamma=0.4$, and $\mu=0.1$.
• Figure 4: In the blue area, vector control is preferable; in the yellow area, personal protection measures are preferable, according to Corollary \ref{['cor:linear']} Common model parameters are $\omega=0.2$, $\gamma=0.4$, and $\mu=0.1$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Proposition 1
• proof
• Remark 1
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof