Gamma-distributed memory changes stability in a nonlinear reaction–diffusion epidemic model

A mathematical analysis of a reaction–diffusion susceptible–infected epidemic model with nonlinear incidence found that replacing a fixed transmission delay with gamma-distributed memory produces a qualitatively different stability pattern. In simulations using the parameter set of an earlier discrete-delay model, both gamma-memory formulations converged to the selected endemic equilibrium at nominal memory values of 1.2 and 1.6, while the fixed-delay model entered an oscillatory regime at the higher value. The study examined model dynamics and linear stability; it did not evaluate epidemics in human populations.

The investigators began with the delayed susceptible–infected model of Song et al. and introduced two memory kernels: a weak, one-stage gamma kernel with an exponential shape and a strong, two-stage gamma kernel. Both average the infected density over past times rather than evaluating it at one fixed lag. A linear-chain construction converted the weak and strong memory terms into local systems, producing cubic and quartic characteristic polynomials for each spatial mode. The selected endemic equilibrium was unchanged because constant histories are unchanged by averaging.

The memory variables were treated as phenomenological filters rather than explicit exposed or latent-stage compartments. This distinction matters because the model contains no separate compartment for infection before infectiousness. The weak kernel places its greatest weight at the present and decays monotonically into the past, whereas the strong kernel suppresses zero-lag weight and concentrates influence around a positive time. Although their forms differ, the researchers varied the memory rate so that the mean memory time could be compared directly across the two models.

For spatially uniform perturbations, both gamma models displayed a small, stabilising spectral crossing at a mean memory time near 0.08. Below that threshold, the homogeneous endemic equilibrium was unstable; above it, the mode was stable. The weak- and strong-kernel thresholds were close, indicating that the mean memory time was the principal factor controlling this first transition for the selected parameter set. The crossings were transverse, and the paper described them as spectral Hopf-type crossings without claiming that a full nonlinear Hopf bifurcation had been demonstrated.

The fixed-delay model behaved differently. It also had a small stabilising crossing, but an additional, much larger destabilising crossing whose numerical value was not available in the supplied text. This larger threshold explains the contrast in the reference simulations. At the nominal value of 1.2, the fixed-delay and gamma models all converged to the endemic level, although they did so for different spectral reasons: the fixed-delay model remained below its large destabilising threshold, while the gamma models were already well beyond their small stabilising crossing. At 1.6, the fixed-delay trajectory developed growing oscillations, whereas the weak- and strong-gamma trajectories converged to the endemic equilibrium.

Numerically integrated homogeneous reduced systems supported these stability calculations. The researchers used a classical fourth-order Runge–Kutta method for the gamma systems and the method of steps with linear interpolation for the discrete-delay equation. Initial conditions and memory states were set to be equivalent across the formulations, so the differing transients were not attributed to inconsistent initialization. Spectral calculations showed that the largest real part of the homogeneous characteristic roots became negative after the crossing near 0.08 for both gamma kernels.

The first spatial mode followed a separate mechanism. Its modal constant term was negative for every positive memory rate in both gamma formulations, implying instability throughout the positive-memory range. The same mode was already unstable at zero memory, so its growth was not generated by a positive-memory spectral crossing. The authors therefore classified it as a memory-independent spatial-mode instability rather than a memory-induced Hopf instability. They also did not classify it as a classical Turing instability because the homogeneous zero-memory mode was itself unstable for the reference parameter set. All higher spatial modes satisfied the reported Routh–Hurwitz conditions and had no spectral crossing.

Overall, the simulations and linear analyses indicate that averaging transmission history over a gamma distribution can suppress the large-delay oscillatory mechanism associated with a single fixed lag. The weak and strong kernels produced similar first homogeneous thresholds, although their transfer functions and transient behaviour were not identical. The study is limited to local stability analysis and numerical behaviour around one endemic branch for one parameter set; the reported spatial visualization was a linear modal reconstruction rather than a full nonlinear reaction–diffusion simulation. The authors also stated that different epidemiological regimes or more nonlinear behaviour could make kernel shape more consequential, but those questions were not resolved by the present analysis.

Subject of Research: Gamma-distributed transmission memory and stability in a nonlinear reaction–diffusion SI epidemic model

Article Title: Gamma–distributed transmission memory in a reaction–diffusion epidemic model with nonlinear incidence

Article References: Anokye, M., Guerrini, L., Mensah, J. A., Ofori, M. F., & Ragni, S. (2026). Gamma–distributed transmission memory in a reaction–diffusion epidemic model with nonlinear incidence. PLOS One, 21(10), e0360508. https://doi.org/10.1371/journal.pone.0360508

Image Credits: AI Generated

DOI: 10.1371/journal.pone.0360508

Keywords: epidemic modeling, reaction–diffusion systems, transmission memory, gamma kernels, distributed delay, stability analysis, spectral crossing, spatial instability, nonlinear incidence, SI model, Gamma, distributed