A positivity- and threshold-preserving nonstandard finite difference scheme for a reservoir-driven visceral leishmaniasis model

Abstract

A dynamically consistent numerical method for an epidemic model should reproduce the main qualitative features of the underlying continuous system. This requirement is especially important in vector-borne infections with reservoir transmission, where standard explicit methods may create negative populations, spurious oscillations, or step-size-dependent threshold behavior. This paper develops and analyzes a nonstandard finite difference (NSFD) scheme for a structured model of visceral leishmaniasis involving humans, sandfly vectors, and animal reservoir hosts. The continuous model includes human latency, treatment, temporary immunity, vector incubation, asymptomatic and symptomatic reservoir infection, host-attractiveness-based bite allocation, and density-regulated vector demography. Positivity, boundedness, a positively invariant feasible region, the disease-free equilibrium, the basic reproduction number, and local stability of the disease-free equilibrium are established for the continuous model. The basic reproduction number decomposes into human-vector and reservoir-vector transmission contributions, giving a direct interpretation of the two infection loops that can sustain invasion. A structure-preserving NSFD discretization is then constructed using a denominator function and nonlocal approximations of incidence, transfer, and density-dependent vector-regulation terms. The discrete scheme is shown to preserve nonnegativity and the biological bounds for all admissible time steps, to share the same disease-free equilibrium as the continuous model, and to preserve the threshold condition determined by the next-generation number. In particular, the disease-free equilibrium of the discrete system is locally asymptotically stable when \(R_0<1\) and unstable when \(R_0>1\). Numerical experiments compare the NSFD method with forward Euler discretization under subcritical and supercritical transmission regimes. The results illustrate that the proposed scheme preserves biologically meaningful trajectories and threshold behavior for larger time steps, whereas Euler discretization can lose positivity or become unstable. The framework provides a reliable computational basis for reservoir-driven leishmaniasis models and related vector-borne disease systems.