Dynamics of Secondary Infectious Diseases with Spatiotemporal Heterogeneity
- 1School of Mathematics and Data Science, Jiangnan University, Wuxi 214122, China
- 2School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
Abstract
Secondary infections and reactivation of latent pathogens pose prominent public-health challenges in the post-pandemic era. To account for spatiotemporal heterogeneity, host relapse transitions, and the coupling of multi-stage infection processes, this paper develops a six-compartment reaction-diffusion dynamical model consisting of susceptible, primary-infected, primary-recovered, secondary-infected, secondary-recovered, and fully recovered compartments. The model captures two distinct transmission pathways: exogenous secondary infection and endogenous reactivation of latent pathogens within hosts. Within the framework of infinite-dimensional dynamical systems, we rigorously prove the existence of a global exponential attractor for this non-autonomous spatiotemporal model, which constrains the long-term behavior of all solution trajectories. By means of upper–lower control systems and the Krein–Rutman theorem, we derive threshold criteria based on principal eigenvalues to judge disease extinction and endemic persistence. Numerical simulations are carried out using epidemiological data of herpes zoster following COVID-19 infection in China. We compare epidemic dynamics under homogeneous and spatiotemporally heterogeneous conditions, investigate the regulatory effects of the transmission coefficient, relapse rate, and state-transition parameters, and implement sensitivity analysis for key model parameters. Numerical experiments demonstrate that spatiotemporal heterogeneity triggers periodic spatiotemporal oscillations of infection burden; relapse elevates the magnitude of secondary infections; secondary cases exhibit remarkable time-lag responses even when the primary epidemic is well controlled. This work provides theoretical foundations for understanding spatiotemporal transmission of multi-stage infectious diseases, establishing early-warning frameworks and designing targeted intervention strategies for secondary infectious diseases.
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BibTeX
@article{zhu2026dynamics,
title = {{Dynamics of Secondary Infectious Diseases with Spatiotemporal Heterogeneity}},
author = {Cheng-Cheng Zhu and Jiang Zhu},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15110810},
url = {https://www.mdpi.com/2075-1680/15/11/810},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Dynamics of Secondary Infectious Diseases with Spatiotemporal Heterogeneity
AU - Cheng-Cheng Zhu
AU - Jiang Zhu
PY - 2026
DA - 2026/10/31
JO - Axioms
DO - 10.3390/axioms15110810
UR - https://www.mdpi.com/2075-1680/15/11/810
AB - Secondary infections and reactivation of latent pathogens pose prominent public-health challenges in the post-pandemic era. To account for spatiotemporal heterogeneity, host relapse transitions, and the coupling of multi-stage infection processes, this paper develops a six-compartment reaction-diffusion dynamical model consisting of susceptible, primary-infected, primary-recovered, secondary-infected, secondary-recovered, and fully recovered compartments. The model captures two distinct transmission pathways: exogenous secondary infection and endogenous reactivation of latent pathogens within hosts. Within the framework of infinite-dimensional dynamical systems, we rigorously prove the existence of a global exponential attractor for this non-autonomous spatiotemporal model, which constrains the long-term behavior of all solution trajectories. By means of upper–lower control systems and the Krein–Rutman theorem, we derive threshold criteria based on principal eigenvalues to judge disease extinction and endemic persistence. Numerical simulations are carried out using epidemiological data of herpes zoster following COVID-19 infection in China. We compare epidemic dynamics under homogeneous and spatiotemporally heterogeneous conditions, investigate the regulatory effects of the transmission coefficient, relapse rate, and state-transition parameters, and implement sensitivity analysis for key model parameters. Numerical experiments demonstrate that spatiotemporal heterogeneity triggers periodic spatiotemporal oscillations of infection burden; relapse elevates the magnitude of secondary infections; secondary cases exhibit remarkable time-lag responses even when the primary epidemic is well controlled. This work provides theoretical foundations for understanding spatiotemporal transmission of multi-stage infectious diseases, establishing early-warning frameworks and designing targeted intervention strategies for secondary infectious diseases.
SN - 2075-1680
ER -