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In this study, we developed an SIR-like mathematical model of disease transmission dynamics. Hantavirus is a neglected tropical disease, and this paper presents a mathematical model of hantavirus transmission among rodents and its effect on the number of hantavirus-infected humans. We review an existing SIR-SIR model of hantavirus transmission and analyze it in a standard mathematical epidemiology framework. The original SIR-SIR model is summarized, with emphasis on its structural assumptions, epidemiological interpretation, and analytical results, including the derivation of the basic reproduction number and the characterization of the stability of the disease-free and endemic equilibria. A critical evaluation of the original SIR-SIR model highlights several biological limitations of the baseline model, notably, the unrealistic assumption of homogeneous transmission and the absence of ecological seasonality. To address these gaps, an improved model incorporating periodic forcing in rodent recruitment and disease transmission is proposed. The use of sine and cosine functions introduces a biologically motivated phase shift between rodent recruitment and transmission, reflecting the fact that birth pulses and peak contact rates rarely occur simultaneously in natural rodent populations. The reproduction number for the extended system is constructed using a Floquet-style argument for DFE stability. A theorem connecting the stability of the DFE with the seasonal component is presented, resembling the well-known rule for non-seasonal hantavirus transmission but with more realistic assumptions. Numerical simulations demonstrate that seasonal variation can generate oscillatory outbreak patterns that more closely reflect empirical rodent population dynamics and human risk profiles. Overall, the results underscore the importance of ecological realism in zoonotic disease modeling and provide a foundation for more accurate prediction and control of the disease, especially in NTD elimination programs. Full article

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article