Once bitten, twice shy: A modeling framework for incorporating heterogeneous mosquito biting into transmission models

TL;DR

This work highlights that standard mosquito-borne disease models, which assume a single bite per gonotrophic cycle, overlook substantial heterogeneity in biting behavior and its linkage to the gonotrophic cycle. By adopting phase-type distributions and the Generalized Linear Chain Trick, the authors develop a flexible framework that yields tractable formulas for the basic offspring number $ N_0$ and the basic reproduction number $ R_0$ under diverse biting-process specifications. Through a detailed case study spanning empirical, phenomenological, and mechanistic models, they show that the relationship between $ ext{GCD}$ and $ R_0$ can be linear, saturating, or non-monotonic, and that reduction in $ ext{GCD}$ can sometimes decrease $ R_0$ when multiple biting per cycle varies with disruption. Sensitivity analyses identify probing and ingestion-related rates and probabilities as key levers for transmission, underscoring how individual-level biting interventions may scale to population-level risk and informing temperature- and host-defense–related questions in vector-borne disease dynamics.

Abstract

The risk of mosquito-borne disease outbreaks is tightly linked to the frequency at which mosquitoes feed on blood, also known as the biting rate. However, standard models of mosquito-borne disease transmission inherently assume that mosquitoes bite only once per reproductive cycle -- an assumption commonly violated in nature. Drivers of multiple biting also affect the mosquito gonotrophic cycle duration (GCD), the quantity customarily used to estimate biting rates. Here, we present a novel framework for incorporating more complex mosquito biting behaviors into transmission models, accounting for heterogeneity and linkages between mosquito biting rates and multiple biting. We provide general formulas for the basic offspring number, $\mathcal{N}_0$, and basic reproduction number, $\mathcal{R}_0$, threshold measures for mosquito population and pathogen transmission persistence, respectively. To exhibit its flexibility, we expand on specific models derived from the framework that arise from empirical, phenomenological, or mechanistic modeling perspectives. Using the gonotrophic cycle duration as a standard quantity to make comparisons among the models, we show that assumptions about the biting process strongly affect the relationship between GCD and $\mathcal{R}_0$. While under the standard assumption of one bite per reproductive cycle, $\mathcal{R}_0$ is an increasing linear function of the inverse of the GCD, alternative models of the biting process can exhibit saturating or concave relationships. Critically, from a mechanistic perspective, decreases in the GCD can lead to substantial decreases in $\mathcal{R}_0$. This work highlights the importance of incorporating the behavioral dynamics of mosquitoes into transmission models and provides a method for evaluating how individual-level interventions against mosquito biting scale up to determine population-level mosquito-borne disease risk.

Figures (8)

• Figure 1: Compartmental diagram for the mosquito demographic model, system \ref{['eq:orig_model']}. The matrix $\boldsymbol{A}^T -\boldsymbol{D}$ indicates the transition rates within and among the biting stages. $\boldsymbol{D}=\boldsymbol{D}({\operatorname{diag}\boldsymbol{A}-(\boldsymbol{A\vec{1})^T}})$ is the diagonal matrix whose entries are given by the vector $\operatorname{diag}\boldsymbol{A}-(\boldsymbol{A\vec{1})^T}$, where $\operatorname{diag}\boldsymbol{A}$ is itself the column vector whose entries are the diagonal entries of $\boldsymbol{A}$.
• Figure 2: Compartmental diagram of the transmission model represented by system \ref{['eq:epi_model']}. The oviposition and resting stages of the mosquito population, which do not contribute directly to transmission, are omitted for space considerations. Arrows indicate direct transitions between compartments. The purple double lines indicate the influence of compartments on rates of transition to infected stages, i.e., the forces of infection.
• Figure 3: Compartmental diagram for the mechanistic model described in Section \ref{['subsubsec:mech_model']}. Newly emerged mosquitoes initially enter the host-seeking state. If they are successful throughout the biting process, mosquitoes proceed from host-seeking ($Q$) to landing on the host ($L$), then probing for a blood vessel ($P$), and finally ingesting blood ($G$), before leaving the host to find a location to lay eggs ($V$). Mosquitoes may be unsuccessful in their biting attempts, leading them to reattempt landing on the same host (blue lines) or seek out a new host (red lines). For visual clarity, the juvenile compartment, mortality rates, and self-loops are not drawn.
• Figure 4: Probability density functions (PDFs) for the gonotrophic cycle duration for the five model types. The distributions are compared by setting them to have equal means ($\theta$, see equation \ref{['eq:theta_dfn']}, represented by a dashed gray line): six hours (A), twelve hours (B), and twenty-four hours (C). See section \ref{['subsubsec:mech_model']} for details on how the mechanistic model probability density function is approximated for the given mean. The PDFs associated with the standard and exponential models are equivalent. The distributions for the mechanistic model obtained from changing $\lambda_Q$, $p_P$, and $p_G$ were visually identical, so only the curve for $\lambda_Q$ is shown here. For the mechanistic model, baseline values for 'Persistent' type mosquitoes were used (see Table \ref{['tab:mech_parameters']}). All other parameters are given in Table \ref{['tab:parameters']}.
• Figure 5: Relationships between $\mathcal{R}_0$ and the standard biting rate, the inverse of the gonotrophic cycle duration, for each model type. Tick marks indicate the critical minimum standard biting rate, the standard biting rate at which $\mathcal{R}_0$ first exceeds one (see Table \ref{['tab:R0_summary']}). The gray horizontal line indicates where $\mathcal{R}_0$ equals one. For the mechanistic models, the standard biting rate increases as the parameters listed in parentheses are increased (indicated by arrowheads along the curves). The mechanistic model uses parameters from the 'Persistent' mosquito parameter set in Table \ref{['tab:mech_parameters']}. The $\beta_H$ and $\beta_B$ values for the standard and the mechanistic ($p_G$) models were reduced to 8.75% to highlight the qualitative differences among the model types better visually. All other parameters are given in Table \ref{['tab:parameters']}.

Theorems & Definitions (9)

• Proposition 1
• proof
• Theorem 1
• proof
• Corollary 1
• proof
• Corollary 2
• Theorem 2
• proof