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Mathematical modeling is critical to our understanding of how infectious diseases spread at the individual and population levels. This book gives readers the necessary skills to correctly formulate and analyze mathematical models in infectious disease epidemiology, and is the first treatment of the subject to integrate deterministic and stochastic models and methods. Mathematical Tools for Understanding Infectious Disease Dynamicsfully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided. Covers the latest research in mathematical modeling of infectious disease epidemiologyIntegrates deterministic and stochastic approachesTeaches skills in model construction, analysis, inference, and interpretationFeatures numerous exercises and their detailed elaborationsMotivated by real-world applications throughout

As the fear of infection is a crucial factor in the progress of the disease in the population. We aim, in this study, to investigate a susceptible-protected-infected-recovered (SPIR) epidemic model with mixed diffusion modeled by local and nonlocal diffusions. These types of diffusion are used to model the fear effect of being infected by the population. The model is shown to be well-posed; the solution exists, is positive, and is unique. The variational expression is obtained to determine threshold role of , also known as the basic reproduction number. Indeed, for , we show that the epidemic will extinct, corresponding to the global asymptotic stability of the infection-free equilibrium state. However, when , the existence of the infection equilibrium state and the uniform persistence of the solution have been proved. The Lyapunov function have been used to show the global asymptotic stability of the infection equilibrium state. Moreover, we compared the obtained results with the classical SIR epidemic model for determining the required protection function for stopping the disease, which can be obtained by reducing below one.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease’s persistence or eradication depends on the basic reproduction number R0s. If R0s<1, the disease is eradicated with a probability of 1, while R0s>1 results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if R0s>1, the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

This paper focuses on a combined SIR-SI epidemic model to evaluate the transmission dynamics of dengue fever, integrating the susceptible-infected-recovered (SIR) framework for the human population with the susceptible-infected (SI) framework for mosquitoes. The model is formulated as a system of nonlinear differential equations and is further extended by incorporating fractional-order derivatives in the Caputo sense to capture memory effects in disease transmission. A thorough investigation of the disease-free and endemic equilibrium points is conducted, encompassing both local and global stability at the disease-free state. The basic reproduction number, , is derived, and a sensitivity analysis is performed to identify the key parameters influencing the transmission dynamics. To ensure mathematical rigor, the existence and uniqueness of the model’s solutions are also examined. For numerical approximation, the two-step Lagrange polynomial method is applied, enabling simulation of the model under various fractional orders and parameter settings. The results demonstrate that the fractional-order approach offers deeper insights into the dynamics of dengue transmission, highlighting the importance of memory effects. These findings provide valuable guidance for medical professionals, policymakers, and public health authorities in designing more effective control strategies.

Fitting Susceptible-Infected-Recovered (SIR) models to incidence data is problematic when not all infected individuals are reported. Assuming an underlying SIR model with general but known distribution for the time to recovery, this paper derives the implied differential-integral equations for observed incidence data when a fixed fraction of newly infected individuals are not observed. The parameters of the resulting system of differential equations are identifiable. Using these differential equations, we develop a stochastic model for the conditional distribution of current disease incidence given the entire past history of reported cases. We estimate the model parameters using Bayesian Markov Chain Monte-Carlo sampling of the posterior distribution. We use our model to estimate the transmission rate and fraction of asymptomatic individuals for the current Coronavirus 2019 outbreak in eight American Countries: the United States of America, Brazil, Mexico, Argentina, Chile, Colombia, Peru, and Panama, from January 2020 to May 2021. Our analysis reveals that the fraction of reported cases varies across all countries. For example, the reported incidence fraction for the United States of America varies from 0.3 to 0.6, while for Brazil it varies from 0.2 to 0.4.

Background The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. Results A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number R 0 . Then the stability of the equilibria is given in terms of R 0 , and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if R 0 < 1 and R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when R 0 ≤ 1 . The system has a transcritical bifurcation when R 0 = 1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

We study a fast–slow version of an SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. We use GSPT to study the model, taking into account that the infection period is much shorter than the average duration of immunity. We show that the dynamics occurs through a sequence of fast and slow flows, that can be described through 2-dimensional maps that, under some assumptions, can be approximated as 1-dimensional maps. Using this method, together with numerical bifurcation tools, we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

The perception of susceptible individuals naturally lowers the transmission probability of an infectious disease but has been often ignored. In this paper, we formulate and analyze a diffusive SIS epidemic model with memory-based perceptive movement, where the perceptive movement describes a strategy for susceptible individuals to escape from infections. We prove the global existence and boundedness of a classical solution in an n -dimensional bounded smooth domain. We show the threshold-type dynamics in terms of the basic reproduction number R 0 : when R 0 < 1 , the unique disease-free equilibrium is globally asymptotically stable; when R 0 > 1 , there is a unique constant endemic equilibrium, and the model is uniformly persistent. Numerical analysis exhibits that when R 0 > 1 , solutions converge to the endemic equilibrium for slow memory-based movement and they converge to a stable periodic solution when memory-based movement is fast. Our results imply that the memory-based movement cannot determine the extinction or persistence of infectious disease, but it can change the persistence manner.

Communities are commonly not isolated but interact asymmetrically with each other, allowing the propagation of infectious diseases within the same community and between different communities. To reveal the impact of asymmetrical interactions and contact heterogeneity on disease transmission, we formulate a two-community SIR epidemic model, in which each community has its contact structure while communication between communities occurs through temporary commuters. We derive an explicit formula for the basic reproduction number R0, give an implicit equation for the final epidemic size z, and analyze the relationship between them. Unlike the typical positive correlation between R0 and z in the classic SIR model, we find a negatively correlated relationship between counterparts of our model deviating from homogeneous populations. Moreover, we investigate the impact of asymmetric coupling mechanisms on R0. The results suggest that, in scenarios with restricted movement of susceptible individuals within a community, R0 does not follow a simple monotonous relationship, indicating that an unbending decrease in the movement of susceptible individuals may increase R0. We further demonstrate that network contacts within communities have a greater effect on R0 than casual contacts between communities. Finally, we develop an epidemic model without restriction on the movement of susceptible individuals, and the numerical simulations suggest that the increase in human flow between communities leads to a larger R0.

In this paper, we are concerned with two SIS epidemic reaction–diffusion models with mass action infection mechanism of the form SI , and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings.