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Peer-reviewedCC BY 4.0PLOS One

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

  • Martin Anokye1
  • Luca Guerrini2
  • John Amoah Mensah3
  • Michael Fosu Ofori4
  • Stefania Ragni5
5 affiliations
  1. 1Department of Mathematics, University of Cape Coast, Cape Coast, Ghana
  2. 2Department of Management, Polytechnic University of Marche, Ancona, Italy
  3. 3Department of Computer Science, Sunyani Technical University, Sunyani, Ghana
  4. 4Department of Statistical Sciences, Kumasi Technical University, Kumasi, Ghana
  5. 5Department of Economics and Management, University of Ferrara, Ferrara, Italy
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Abstract

We extend the delayed reaction–diffusion SI model with nonlinear incidence proposed by Song et al. by replacing the fixed transmission lag with weak and strong gamma-distributed memory kernels. Because the model has no exposed compartment, the memory term is interpreted phenomenologically as a distributed timing of transmission-related effects of past infectious states rather than as an explicit latent-stage mechanism. Using the same parameter set adopted by Song et al., we show that the distributed-memory models have a stability structure that is qualitatively different from that of the discrete-delay model. For the spatially homogeneous mode, both gamma kernels exhibit a small stabilising spectral Hopf crossing at a mean memory time near 0.08. The discrete-delay model has a comparable small stabilising crossing but, in addition, a much larger destabilising crossing near τ=1.335. Consequently, at the nominal values 1.2 and 1.6 used in the reference simulations, the gamma models converge to the selected endemic equilibrium, whereas the discrete-delay model crosses into an oscillatory regime. The first spatial mode is different: for the gamma formulations its modal constant term is negative for every positive memory rate, so this mode is unstable for every positive mean-memory value. We therefore interpret this behavior as a memory-independent spatial-mode instability, distinct from both a memory-induced Hopf crossing and a classical Turing instability.

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Cite

BibTeXRIS
BibTeX
@article{anokye2026gammadistributed,
  title = {{Gamma–distributed transmission memory in a reaction–diffusion epidemic model with nonlinear incidence}},
  author = {Martin Anokye and Luca Guerrini and John Amoah Mensah and Michael Fosu Ofori and Stefania Ragni},
  journal = {PLOS One},
  year = {2026},
  month = oct,
  doi = {10.1371/journal.pone.0360508},
  url = {https://doi.org/10.1371/journal.pone.0360508},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Gamma–distributed transmission memory in a reaction–diffusion epidemic model with nonlinear incidence
AU  - Martin Anokye
AU  - Luca Guerrini
AU  - John Amoah Mensah
AU  - Michael Fosu Ofori
AU  - Stefania Ragni
PY  - 2026
DA  - 2026/10/14
JO  - PLOS One
DO  - 10.1371/journal.pone.0360508
UR  - https://doi.org/10.1371/journal.pone.0360508
AB  - We extend the delayed reaction–diffusion SI model with nonlinear incidence proposed by Song et al. by replacing the fixed transmission lag with weak and strong gamma-distributed memory kernels. Because the model has no exposed compartment, the memory term is interpreted phenomenologically as a distributed timing of transmission-related effects of past infectious states rather than as an explicit latent-stage mechanism. Using the same parameter set adopted by Song et al., we show that the distributed-memory models have a stability structure that is qualitatively different from that of the discrete-delay model. For the spatially homogeneous mode, both gamma kernels exhibit a small stabilising spectral Hopf crossing at a mean memory time near 0.08. The discrete-delay model has a comparable small stabilising crossing but, in addition, a much larger destabilising crossing near τ=1.335. Consequently, at the nominal values 1.2 and 1.6 used in the reference simulations, the gamma models converge to the selected endemic equilibrium, whereas the discrete-delay model crosses into an oscillatory regime. The first spatial mode is different: for the gamma formulations its modal constant term is negative for every positive memory rate, so this mode is unstable for every positive mean-memory value. We therefore interpret this behavior as a memory-independent spatial-mode instability, distinct from both a memory-induced Hopf crossing and a classical Turing instability.
SN  - 1932-6203
ER  -