Stability and bifurcations of a minimal model for the effect of PrEP-related risk compensation in epidemics of sexually transmitted infections

HIV pre-exposure prophylaxis (PrEP) drastically reduces the risk of HIV infection if taken as prescribed, providing almost perfect protection even during unprotected sexual intercourse. Although this has been transformative in reducing new HIV infections among high-risk populations, it has also been linked to an increase in risk practices – a phenomenon known as risk compensation – thereby favoring the spread of other sexually transmitted infections (STIs) deemed less severe. In this paper, we study a minimal compartmental model describing the effect of risk awareness and risk compensation due to PrEP on the spread of other STIs among a high-infection-risk group of men who have sex with men (MSM). The model integrates three key elements of risk-mediated behavior and PrEP programs: (i) HIV risk awareness drives self-protective behaviors (such as condom use and voluntary STI screening); (ii) individuals on PrEP are subject to risk compensation, but (iii) are required to screen for asymptomatic STIs frequently. We derived the basic reproduction number of the system, $R_0$, and found a transcritical bifurcation at $R_0=1$, where the disease-free equilibrium becomes unstable and an endemic equilibrium emerges. This endemic equilibrium is asymptotically stable wherever it exists. We identified critical thresholds in behavioral and policy parameters that separate these regimes and analyzed typical values for plausible parameter choices. Beyond the specific epidemiological context, the model serves as a general framework for studying nonlinear interactions between behavioral adaptation, preventive interventions, and disease dynamics, providing insights into how feedback mechanisms can lead to non-trivial responses in epidemic systems. Finally, our model can be easily extended to study the effect of interventions and risk compensation in other STIs.

💡 Research Summary

This paper presents a mathematical modeling study investigating the unintended consequences of HIV pre-exposure prophylaxis (PrEP) on the epidemiology of other sexually transmitted infections (STIs). While PrEP is highly effective in preventing HIV acquisition, its use has been associated with “risk compensation”—a decrease in protective behaviors like condom use due to a lowered perception of HIV risk. This behavioral shift can potentially facilitate the spread of other, often less severe but highly prevalent bacterial STIs like chlamydia. The study aims to untangle the complex interplay between PrEP use, risk perception, behavioral adaptation, and public health interventions (specifically STI screening mandates within PrEP programs) using a dynamic compartmental model.

The authors construct a minimal “susceptible-infectious-treated-susceptible” (SITS) model stratified by symptom status (asymptomatic I_a and symptomatic I_s infections). The model’s core innovation is the integration of three key feedback mechanisms driven by two primary exogenous parameters: the level of PrEP uptake (P) and the perceived HIV risk awareness (H). First, risk awareness (H) drives voluntary protective behavior (modeled by a mitigation function m(H)), which reduces the effective transmission rate. Second, individuals on PrEP are subject to risk compensation, modeled by a “risk assimilation” parameter (ξ). When ξ=1, PrEP users behave identically to non-users; when ξ=0, they exhibit full risk compensation (no protective behavior). Third, the model differentiates between voluntary asymptomatic testing driven by risk perception (at rate λ_H(H)) and mandatory regular screening required for PrEP users (at rate λ_P). Treated individuals move to a temporarily immune compartment (T).

The analysis begins by establishing the model’s well-posedness, proving the positivity and boundedness of solutions under biologically reasonable assumptions. The system is reduced to three dimensions by assuming a constant total population. The central analytical task is the derivation of the basic reproduction number (R0) using the next-generation matrix method. R0 is expressed as a function of all behavioral and policy parameters (P, H, ξ, λ_P, etc.).

The main theoretical result is the identification of a transcritical bifurcation at R0=1. For R0 < 1, the disease-free equilibrium (DFE) is locally asymptotically stable, leading to disease elimination. As R0 crosses the threshold of 1, the DFE loses stability, and a unique endemic equilibrium (EE) emerges and is asymptotically stable wherever it exists. The analysis confirms that this EE is stable across its entire range of existence, and no more complex bifurcations (like backward bifurcations or bistability) are present in this minimal model. This indicates a straightforward transition between disease extinction and sustained endemicity based on the magnitude of R0.

The paper further complements the analytical findings with numerical explorations. It visualizes the phase space defined by PrEP uptake (P) and risk awareness (H), plotting the R0=1 contour that separates the “disease-free” and “endemic” regimes. This boundary is nonlinear, revealing parameter regions where increasing PrEP coverage can paradoxically trigger an STI epidemic if population risk awareness is below a critical level. Sensitivity analyses illustrate the opposing effects of risk compensation (increasing R0) and mandatory screening (decreasing R0), providing quantitative insights into the screening frequency required to offset the negative behavioral effects of PrEP.

In conclusion, this study provides a foundational mathematical framework for understanding the nonlinear interactions between behavioral adaptation, preventive biomedical interventions, and disease dynamics. The model successfully captures the essence of the PrEP risk-compensation dilemma and offers a tool for exploring critical thresholds in behavioral and policy parameters. While simplified, the model is readily extensible to study other STIs or more sophisticated behavioral mechanisms. The work underscores the importance of integrating behavioral surveillance and robust screening protocols into PrEP programs to mitigate potential adverse effects on STI epidemics, offering valuable insights for both public health policy and theoretical epidemiology.