A vector-host epidemic model with spatial structure and seasonality

Recently, Li and Zhao [5] (Bull. Math. Biol., 83(5), 43, 25 pp (2021)) proposed and studied a periodic reaction-diffusion model of Zika virus with seasonality and spatial heterogeneous structure in host and vector populations. They found the basic reproduction ratio R0, which is a threshold parameter. In this short paper we shall use the upper and lower solutions method to study the model of [5] with Neumann boundary conditions replaced by general boundary conditions.

💡 Research Summary

In this paper the authors extend the periodic reaction‑diffusion model for Zika virus originally proposed by Li and Zhao (Bull. Math. Biol., 2021) by replacing the homogeneous Neumann boundary conditions with a much broader class of boundary conditions. Specifically, they consider either pure Neumann (aₖ=0, bₖ=1) or Robin‑type conditions (aₖ=1, bₖ(x,t)≥0) for both the infected host density Hᵢ and the vector densities Vᵤ, Vᵢ. This generalization allows the model to capture more realistic ecological situations where the domain boundary may permit fluxes or partial absorption.

The analytical framework is built on the theory of periodic parabolic operators. The authors introduce a family of linear operators Lₖ = ∂ₜ – aᵢⱼᵏ(x,t)Dᵢⱼ + bᵢᵏ(x,t)Dᵢ together with the boundary operators Bₖ = aₖ∂_ν + bₖ(x,t). Under the assumptions that the coefficients are T‑periodic, Hölder continuous, and that the combined operator (L, H, B) is regular and satisfies a strong maximum principle, they apply the Krein‑Rutman theorem to guarantee the existence of a unique principal eigenvalue λ(L, H, B; T) with a strictly positive eigenfunction.

The model is then reduced to a subsystem for the total vector population V = Vᵤ + Vᵢ, which satisfies a scalar periodic parabolic equation (3.2). The associated linear eigenvalue problem (3.3) yields a principal eigenvalue ζ(μ₁, β). If ζ≥0 the only periodic solution is V≡0, implying disease extinction. If ζ<0, a unique positive periodic solution V>0 exists and is globally asymptotically stable.

Assuming ζ<0, the authors focus on the coupled subsystem for the infected host Hᵢ and infected vector Vᵢ. Linearizing around the disease‑free state leads to the eigenvalue problem (3.5) with principal eigenvalue λ(V). Theorem 3.1 establishes that λ(V)<0 is necessary and sufficient for the existence of a positive periodic solution (Hᵢ, Vᵢ); moreover, such a solution is unique and satisfies Vᵢ<V. Conversely, λ(V)≥0 precludes any positive periodic solution.

Crucially, λ(V) is directly related to the basic reproduction number R₀ introduced by Li and Zhao: R₀≤1 ⇔ λ(V)≥0 and R₀>1 ⇔ λ(V)<0. Hence R₀ retains its role as a sharp threshold for disease persistence even under the generalized boundary conditions.

The core methodological contribution is the systematic use of the upper‑and‑lower solutions technique. By perturbing the vector equilibrium V with a small multiple ε φ of the positive eigenfunction φ of (3.3), the authors construct auxiliary problems (3.6) and (3.7) that incorporate the perturbation. They define a perturbed principal eigenvalue λ(V; ε) and show, via continuity, that for sufficiently small ε the sign of λ(V; ε) matches that of λ(V). For the linearized problem (3.10) they build explicit upper and lower solutions using the principal eigenfunction η of the host diffusion operator, which guarantees a positive solution for Hᵢ. The pair (Hᵢ, V+εφ) then serves as a strict upper solution for the nonlinear system, while a scaled version of the eigenfunction (ϕ₁^ε, ϕ₂^ε) provides a lower solution. Application of the monotone iteration scheme yields at least one positive periodic solution of the perturbed system. Uniqueness is proved by a scaling argument: assuming two distinct positive solutions, one can find a maximal scaling factor s<1 such that s solution₂ ≤ solution₁, leading to a contradiction via the strong maximum principle.

Finally, the authors verify that all the above arguments hold for both Neumann and Robin boundary conditions, because the regularity and strong maximum principle are preserved under the assumptions on aₖ and bₖ. Consequently, the paper demonstrates that the threshold behavior governed by R₀ is robust to a wide class of boundary conditions, and it provides a considerably simpler proof of the global dynamics compared with the earlier works of Magal et al. and Li & Zhao. The results broaden the applicability of spatially heterogeneous, seasonally forced vector‑host epidemic models to more realistic ecological settings.