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1. On the Effects of Type II Left Censoring in Stable and Chaotic Compartmental Models for Infectious Diseases: Do Small Sample Estimates Survive Censoring?

   https://doi.org/10.1609/aaaiss.v2i1.27715  

   Selvitella, Alessandro Maria; Foster, Kathleen Lois (January 2024, Proceedings of the AAAI Symposium Series)

   In this paper, we discuss a selection of tools from dynamical systems and order statistics, which are most often utilized separately, and combine them into an algorithm to estimate the parameters of mathematical models for infectious diseases in the case of small sample sizes and left censoring, which is relevant in the case of rapidly evolving infectious diseases and remote populations. The proposed method relies on the analogy between survival functions and the dynamics of the susceptible compartment in SIR-type models, which are both monotone decreasing in time and are both determined by a dual variable: the hazard function in survival prediction and the number of infected people in SIR-type models. We illustrate the methodology in the case of a continuous model in the presence of noisy measurements with different distributions (Normal, Poisson, Negative Binomial) and in a discrete model, reminiscent of the Ricker map, which admits chaotic dynamics. This estimation procedure shows stable results in experiments based on a popular benchmark dataset for SIR-type models and small samples. This manuscript illustrates how classical theoretical statistical methods and dynamical systems can be merged in interesting ways to study problems ranging from more fundamental small sample situations to more complex infectious disease and survival models, with the potential that this tools can be applied in the presence of a large number of covariates and different types of censored data. 

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2. Erlang-Distributed SEIR Epidemic Models with Cross-Diffusion

   https://doi.org/10.3390/math11092167  

   Chebotaeva, Victoria; Vasquez, Paula A. (May 2023, Mathematics)

   We explore the effects of cross-diffusion dynamics in epidemiological models. Using reaction–diffusion models of infectious disease, we explicitly consider situations where an individual in a category will move according to the concentration of individuals in other categories. Namely, we model susceptible individuals moving away from infected and infectious individuals. Here, we show that including these cross-diffusion dynamics results in a delay in the onset of an epidemic and an increase in the total number of infectious individuals. This representation provides more realistic spatiotemporal dynamics of the disease classes in an Erlang SEIR model and allows us to study how spatial mobility, due to social behavior, can affect the spread of an epidemic. We found that tailored control measures, such as targeted testing, contact tracing, and isolation of infected individuals, can be more effective in mitigating the spread of infectious diseases while minimizing the negative impact on society and the economy. 

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3. Investigating and forecasting infectious disease dynamics using epidemiological and molecular surveillance data

   https://doi.org/10.1016/j.plrev.2024.10.011  

   Chowell, Gerardo; Skums, Pavel (December 2024, Physics of Life Reviews)

   The integration of viral genomic data into public health surveillance has revolutionized our ability to track and forecast infectious disease dynamics. This review addresses two critical aspects of infectious disease forecasting and monitoring: the methodological workflow for epidemic forecasting and the transformative role of molecular surveillance. We first present a detailed approach for validating epidemic models, emphasizing an iterative workflow that utilizes ordinary differential equation (ODE)-based models to investigate and forecast disease dynamics. We recommend a more structured approach to model validation, systematically addressing key stages such as model calibration, assessment of structural and practical parameter identifiability, and effective uncertainty propagation in forecasts. Furthermore, we underscore the importance of incorporating multiple data streams by applying both simulated and real epidemiological data from the COVID-19 pandemic to produce more reliable forecasts with quantified uncertainty. Additionally, we emphasize the pivotal role of viral genomic data in tracking transmission dynamics and pathogen evolution. By leveraging advanced computational tools such as Bayesian phylogenetics and phylodynamics, researchers can more accurately estimate transmission clusters and reconstruct outbreak histories, thereby improving data-driven modeling and forecasting and informing targeted public health interventions. Finally, we discuss the transformative potential of integrating molecular epidemiology with mathematical modeling to complement and enhance epidemic forecasting and optimize public health strategies. 

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4. Adaptive human behavior and delays in information availability autonomously modulate epidemic waves

   https://doi.org/10.1093/pnasnexus/pgaf145  

   Mahmud, Md_Shahriar; Eshun, Solomon; Espinoza, Baltazar; Kadelka, Claus; Arenas, ed., Alex (May 2025, PNAS Nexus)

   Abstract The recurrence of epidemic waves has been a hallmark of infectious disease outbreaks. Repeated surges in infections pose significant challenges to public health systems, yet the mechanisms that drive these waves remain insufficiently understood. Most prior models attribute epidemic waves to exogenous factors, such as transmission seasonality, viral mutations, or implementation of public health interventions. We show that epidemic waves can emerge autonomously from the feedback loop between infection dynamics and human behavior. Our results are based on a behavioral framework in which individuals continuously adjust their level of risk mitigation subject to their perceived risk of infection, which depends on information availability and disease severity. We show that delayed behavioral responses alone can lead to the emergence of multiple epidemic waves. The magnitude and frequency of these waves depend on the interplay between behavioral factors (delay, severity, and sensitivity of responses) and disease factors (transmission and recovery rates). Notably, if the response is either too prompt or excessively delayed, multiple waves cannot emerge. Our results further align with previous observations that adaptive human behavior can produce nonmonotonic final epidemic sizes, shaped by the trade-offs between various biological and behavioral factors—namely, risk sensitivity, response stringency, and disease generation time. Interestingly, we found that the minimal final epidemic size occurs on regimes that exhibit a few damped oscillations. Altogether, our results emphasize the importance of integrating social and operational factors into infectious disease models, in order to capture the joint evolution of adaptive behavioral responses and epidemic dynamics. 

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5. Asymptotic and transient dynamics of SEIR epidemic models on weighted networks

   https://doi.org/10.1017/S0956792522000109  

   TIAN, CANRONG; LIU, ZUHAN; RUAN, SHIGUI (January 2022, European Journal of Applied Mathematics)

   We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission. 

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