A model for mosquito-borne epidemic outbreaks with information-dependent protective behaviour

The gist A model for mosquito-borne epidemic outbreaks with information-dependent protective behaviour investigates how human hosts adopt protective measures in response to information about disease prevalence, finding that these behavioral responses can either facilitate epidemic control or prolong disease persistence depending on various interaction parameters

Model Framework and Baseline Dynamics

The study develops a model for a mosquito-borne epidemic where human hosts may adopt protective behaviour in response to information about both past and current disease prevalence The baseline vector-borne model without information describes the dynamics using an SIR–SI framework, incorporating non-human hosts that do not contribute to transmission In the absence of protective behaviour, the dynamics are described by a system of ODEs (1) which includes terms for susceptible (S), infectious (I) human individuals and susceptible (S), infectious (I) mosquitoes The effective mosquito biting rate is interpreted as determined by a Holling-type II functional response, denoted as ˆb, which accounts for the effect on the average mosquito searching time of both non-competent hosts and the spatial distribution of humans

Impact of Static Protective Behaviour

When protective behaviour is assumed to be fixed, the Control Reproduction Number (CRN), Rc(p, q), is computed using the Next Generation Matrix (NGM) method Proposition 2 shows that condition (12) for Rc(p, q) F(p, q), where F(p, q) is defined by a specific mathematical expression involving p and q Corollary 4 states that when l = 0, Rc(p, q) ≥ R0 for all p ∈ [0, 1), which implies that static protective behaviour in the human population always increases the reproduction number R0 when it is only partially adopted or partially effective Proposition 5 demonstrates that if l > 0, then for every p ∈ [0, 1], Rc(p, q) attains a minimum value smaller than R0 at q = (1 − p)/(1 − p + l)

Information-Dependent Behavioural Changes

The model is modified to allow individuals to dynamically adjust their behaviour based on the information index J(t), which summarizes the publicly available information on infection at time t ≥ 0 The information index is defined as J(t): = Z t −∞ (IP (θ) + INP (θ))K(t − θ) dθ, where K is a non-negative measurable memory kernel satisfying R ∞ 0 K(τ) dτ = 1 The resulting system of ODEs incorporates terms for the adjustment rates a(J) and w(J), which are assumed to be positive and non-increasing functions of J

Asymptotic Dynamics and Stability

When the dynamics evolve on two distinct time scales, Geometric Singular Perturbation Theory (GSPT) is applied to reduce the original two-group model to a model for a homogeneous host population The study investigates the existence and stability of equilibria of system (40), which corresponds to an Established Equilibrium (EE) reached only due to behavioral changes during an outbreak Proposition 6 establishes that for system (5), limt→+∞ IP (t) = limt→+∞ INP (t) = limt→+∞ IM(t) = 0, indicating eventual epidemic extinction The stability of the EE depends on the specific form of the memory kernel K, with different kernels leading to different stability outcomes, such as local asymptotic stability or Hopf bifurcations resulting in self-sustained oscillations

Numerical Simulations and Conclusion

Numerical simulations compare model (20) with the fast system (35), showing that while protective behaviour can increase the reproduction number R0, the final epidemic size R∞ H is smaller than without protection The inclusion of information-induced protective behaviour may lead to prolonged epidemic waves, as solutions rapidly approach a slow manifold along which the infected population decays very slowly The analysis suggests that prolonged outbreaks are an intrinsic feature of information-dependent behavioral models Based on analytical and numerical results, the inclusion of information-induced protective behaviour may lead to prolonged epidemic waves The paper concludes by suggesting extensions such as including demography or seasonality to further model these dynamics

Appendix Details

The computation of Rˆc for model (20) in Section 3.1 provides the characteristic polynomial, from which the formula in (27) can be readily recovered The stability analysis of the EE involves deriving a characteristic equation (B5) which is then analyzed using Laplace transform techniques to find eigenvalues The results for Erlang-2 distributed memory show that there exist thresholds k+, k− ≥ 0 such that the EE is LAS for k > k+ and undergoes a Hopf bifurcation at k = k+ The paper also notes that the presence of other hosts, i.e. l > 0, always decreases the value of R0 regardless of the level of awareness or intervention, and even in their absence (i.e. when q = 1 or p = 0) The final size relations show that for l = 1 and ρ = 5.12, the model with l = 1 predicts a smaller final size than the model with l = 0.25

References

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1. Vector-borne epidemic outbreak modeling incorporating information-dependent protective behavior: The improved AI system can model how human hosts adopt protective behaviour in response to information on both past and current disease prevalence, allowing for scenarios where behavioural responses may either facilitate epidemic control or prolong disease persistence, potentially generating recurrent damped epidemic waves.

2. Geometric Singular Perturbation Theory (GSPT) application for transient dynamics: The system can effectively reduce the complex two-group model to a model for a homogeneous host population under the assumption that opinion dynamics evolves on a much faster time scale than disease transmission, enabling an investigation of the impact of information-induced behavioural changes on the transient dynamics of the epidemic.

3. Analysis of Control Reproduction Number (CRN) thresholds: The system can calculate the CRN, which is defined as the relevant threshold quantity when control strategies are implemented, and determine conditions ensuring that Rc(p, q) F(p, q) for control efficacy.

4. Characterization of information-induced behavioral changes: The model can incorporate information via an information index J(t), defined as a weighted sum of past and present infections with a memory kernel K, allowing the system to analyze how behavioural changes are driven by exponentially waning information on infection prevalence.

5. Investigation of time-scale separation effects: Using GSPT, the system can rigorously separate time scales where behavioural changes occur on a faster time scale than the epidemiological dynamics, leading to a reduced layer system that describes the fast dynamics and allows for an analysis of transient behaviour.