Oscillatory Regimes in a Game-Theoretic Model for Mosquito Population Dynamics under Breeding Site Control

TL;DR

This work addresses how spontaneous household decision-making, driven by perceived risk, shapes mosquito population dynamics under breeding-site control. It develops a game-theoretic framework where imitation dynamics determine the fraction of households performing control, coupled to a behavior-dependent carrying capacity $K_v(w)$ that links human actions to vector growth, and a prevalence-dependent payoff in the full model. The paper derives four locally stable equilibria in the constant-payoff case and introduces a fifth interior equilibrium $E_{05}$ when payoffs depend on prevalence, showing that a Hopf bifurcation can yield sustained oscillations in both mosquito abundance and household engagement; numerical analyses detail how oscillation amplitude and period vary with $N$ and $r_c/r_d$. These findings highlight how behavioral feedback can generate nontrivial vector dynamics, informing public-health strategies and communications aimed at sustaining household participation in breeding-site removal.

Abstract

Mosquito-borne diseases remain a major public-health threat, and the effective control of mosquito populations requires sustained household participation in removing breeding sites. While environmental drivers of mosquito oscillations have been extensively studied, the influence of spontaneous household decision-making on the dynamics of mosquito populations remains poorly understood. We introduce a game-theoretic model in which the fraction of households performing breeding site control evolves through imitation dynamics driven by perceived risks. Household behavior regulates the carrying capacity of the aquatic mosquito stage, creating a feedback between control actions and mosquito population growth. For a simplified model with constant payoffs, we characterize four locally stable equilibria, corresponding to full or no household control and the presence or absence of mosquito populations. When the perceived risk of not controlling breeding sites depends on mosquito prevalence, the system admits an additional equilibrium with partial household engagement. We derive conditions under which this equilibrium undergoes a Hopf bifurcation, yielding sustained oscillations arising solely from the interaction between mosquito abundance and household behavior. Numerical simulations and parameter explorations further describe the amplitude and phase properties of these oscillatory regimes.

Figures (5)

• Figure 1: We model the dynamic impact of household mosquito control measures on breeding site availability. Here, $w$ and $\bar{w}$ denote the proportions of households performing and not performing breeding site control, respectively. Dashed arrows indicate how the entomological and behavioral components are integrated: (i) the growth rate of aquatic mosquitoes, $\Lambda(w)$, as a function of the proportion of households performing control; and (ii) the perceived payoff for not performing control, $f_d(A_v)$, as a function of the adult mosquito population $A_v$.
• Figure 2: Parameter regions of stability and sample trajectories for the simplified System \ref{['eq:simplified_model_1']} with constant payoffs. The central colormap shows the four regions in which the steady states $E_{01} - E_{04}$ are LAS. For each region, we exhibit trajectories converging to the corresponding steady state. Parameter values taken from dumont_mosquito_control: $r = 0.5$, $\nu_L = 0.067\,\text{days}^{-1}$, $\mu_L = 0.62\,\text{days}^{-1}$, and $\mu_A = 0.04\,\text{days}^{-1}$. Other parameters set to plausible values: $K_{v_{\max}} = 2\times 10^{6}$, $K_{v_{\min}} = 1\times 10^{6}$, $t_{\text{span}} = [0,100]$ days, $k = 0.8\,\text{days}^{-1}$, and $b$ ranging from $1$ to $15$ to generate different $N$ values. Initial conditions: $L_0 = 20000$, $A_0 = 20000$, and $w_0 = 0.5$.
• Figure 3: Parameter regions of stability and sample trajectories of the full game-theoretic model \ref{['eq:behavioral_entomological_system']}. The colormap highlights four distinct regions where steady states are LAS and one where $E_{05}$ is unstable and oscillations emerge. For each region, corresponding system trajectories converge to a steady state or oscillate. Parameter values taken from dumont_mosquito_control: $r = 0.5$, $\nu_L = 0.067\,\text{days}^{-1}$, $\mu_L = 0.62\,\text{days}^{-1}$, and $\mu_A = 0.04\,\text{days}^{-1}$. Other parameters set to plausible values: $K_{v_{\max}} = 2\times 10^{6}$, $K_{v_{\min}} = 1\times 10^{5}$, $t_{\text{span}} = [0,300]$ days, $k \in[0.5,0.8]$ days $^{-1}$, $m = 0.3\,\text{mosquito}^{-1}$, and $b$ ranging from $1$ to $15$ to generate different $N$ values. Initial conditions: $L_0 = 20000$, $A_0 = 20000$, and $w_0 = 0.3$.
• Figure 4: Amplitude and period of sustained oscillations in mosquito populations and household behavior. Oscillations in aquatic mosquitoes ($L_v$), adult mosquitoes ($A_v$), and household mechanical control ($w$) vary with the basic offspring number $N$ and the perceived risk ratio $r_c/r_d$. Larger values of $N$ produce oscillations with higher amplitudes and shorter periods, while smaller values of $N$ result in lower amplitudes and longer periods. Parameter values: $r = 0.5$, $b$ ranging from $1$ to $15$ to generate different $N$ values, $\nu_L = 0.04\,\text{days}^{-1}$, $\mu_L = 0.03\,\text{days}^{-1}$, and $\mu_A = 0.2\,\text{days}^{-1}$. Other parameters set to plausible values: $K_{v_{\max}} = 2\times 10^{6}$, $K_{v_{\min}} = 1\times 10^{5}$, $t_{\text{span}} = [0,1000]$ days, $k = 0.8\,\text{days}^{-1}$, $m = 0.3\,\text{mosquito}^{-1}$. Initial conditions: $L_0 = 20000$, $A_0 = 20000$, and $w_0 = 0.3$.
• Figure 5: Parameter regions of stability and sample trajectories for the simplified model with constant public health intervention ($\gamma > 0$). The central colormap shows the four regions in which the steady states are locally asymptotically stable (LAS). For each region, representative system trajectories are displayed, illustrating convergence to the corresponding steady state (dashed line). Here we chose $\gamma = 0.4$. Parameter values taken from dumont_mosquito_control: $r = 0.5$, $\nu_L = 0.067\,\text{days}^{-1}$, $\mu_L = 0.62\,\text{days}^{-1}$, and $\mu_A = 0.04\,\text{days}^{-1}$. Other parameters set to plausible values: $K_{v_{\max}} = 2\times 10^{6}$, $K_{v_{\min}} = 1\times 10^{6}$, $t_{\text{span}} = [0,100]$ days, $k = 0.8\,\text{days}^{-1}$, and $b$ ranging from $1$ to $15$ to generate different $N$ values. Initial conditions: $L_0 = 20000$, $A_0 = 20000$, and $w_0 = 0.5$.

Theorems & Definitions (18)

• Proposition 1
• Theorem 1
• proof
• Remark 1
• Proposition 2
• Theorem 2
• Theorem 3
• Theorem 4: Hopf Bifurcation
• proof
• Remark 2