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We prove that, for Poisson transmission and recovery processes, the classic Susceptible $\to$ Infected $\to$ Recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time $t>0$, a strict lower bound on the expected…

Populations and Evolution · Quantitative Biology 2017-02-13 Robert R. Wilkinson , Frank G. Ball , Kieran J. Sharkey

The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection…

Populations and Evolution · Quantitative Biology 2020-10-07 E. Sadurní , G. Luna-Acosta

The SIR pandemic model suffers from an unrealistic assumption: The rate of removal from the infectious class of individuals is assumed to be proportional to the number of infectious individuals. This means that a change in the rate of…

Populations and Evolution · Quantitative Biology 2021-02-23 Paul A. Reiser

Epidemiological modeling is vital in understanding disease dynamics and guiding public health interventions. This study presents a time-fractional SEIR model to describe the transmission dynamics of Mpox, incorporating memory effects via…

Numerical Analysis · Mathematics 2026-01-29 Gaurav Saini , Bappa Ghosh , Sunita Chand , Jugal Mohapatra

We discuss the criticality in the stochastic SIR model for infectious diseases. We adopt the path-integral formalism for the propagation of infections among susceptible, infectious, and removed individuals, and perform the perturbative and…

Populations and Evolution · Quantitative Biology 2022-02-14 Shigehiro Yasui , Yutaka Hatakeyama , Yoshiyasu Okuhara

This paper deals with a new epidemiological model of SIRS with stochastic perturbations. The primary objective is to establish the existence of a unique non-negative nonlocal solution. Using the basic reproduction number $\mathscr{R}_0$…

Dynamical Systems · Mathematics 2025-03-10 Achraf Zinihi , Moulay Rchid Sidi Ammi , Matthias Ehrhardt

We define and study an open stochastic SIR (Susceptible -- Infected -- Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same…

Probability · Mathematics 2019-03-28 Pierre Montagnon

In the present work, we have introduced and studied two epidemic models that are constructed with Caputo fractional derivative. We considered standard incidence rate and varying population dynamics for SIS and SIRS mathematical models.…

Dynamical Systems · Mathematics 2019-12-03 E. Okyere , J. Ackora-Prah , F. T. Oduro

We study the problem of optimal control of the stochastic SIR model. Models of this type are used in mathematical epidemiology to capture the time evolution of highly infectious diseases such as COVID-19. Our approach relies on…

Populations and Evolution · Quantitative Biology 2020-05-04 Andrew Lesniewski

This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity of the infectious individuals has been reduced, so…

Probability · Mathematics 2023-04-05 Anicet Mougabe-Peurkor , Ibrahima Drame , Modeste N'zi , Etienne Pardoux

In this paper, an epidemic model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size…

Numerical Analysis · Mathematics 2022-04-05 Bálint Takács , Yiannis Hadjimichael

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible -> exposed (=latent) -> infective -> removed) epidemic with a contact tracing scheme, in which removed individuals may name some of their infectious…

Probability · Mathematics 2015-12-08 Frank G Ball , Edward S Knock , Philip D O'Neill

Motivated by the classical Susceptible-Infected-Recovered (SIR) epidemic models proposed by Kermack and Mckendrick, we consider a class of stochastic compartmental dynamical systems with a notion of partial ordering among the compartments.…

Populations and Evolution · Quantitative Biology 2019-01-03 Wasiur R. KhudaBukhsh , Boseung Choi , Eben Kenah , Grzegorz A. Rempala

This paper utilizes a stochastic Susceptible-Infected-recovered (SIR) model with a non-linear incidence rate to perform a detailed mathematical study of optimal lock-down intensity and vaccination rate under the COVID-19 pandemic…

Optimization and Control · Mathematics 2022-09-29 Paramahansa Pramanik

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for…

Physics and Society · Physics 2010-07-23 Brian Karrer , M. E. J. Newman

The Susceptible-Infectious-Recovered (SIR) model is the canonical model of epidemics of infections that make people immune upon recovery. Many of the open questions in computational epidemiology concern the underlying contact structure's…

Physics and Society · Physics 2021-06-09 Petter Holme

The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not…

Dynamical Systems · Mathematics 2021-05-04 U. A. Rozikov , S. K. Shoyimardonov

Susceptible-Infected-Recovered (SIR) models have been used for decades to understand epidemic outbreak dynamics. We develop an SIR model specifically designed to study the effects of population behavior with respect to health and…

Populations and Evolution · Quantitative Biology 2023-08-03 Jasmin Nunuvero , Angelique Santiago , Moshe Cohen , Anca Radulescu

We introduce a kinetic framework for modeling the time evolution of the statistical distributions of the population densities in the three compartments of susceptible, infectious, and recovered individuals, under epidemic spreading driven…

Analysis of PDEs · Mathematics 2025-12-16 Giorgio Martalò , Giuseppe Toscani , Mattia Zanella

We introduce an epidemic model with varying infectivity and general exposed and infectious periods, where the infectivity of each individual is a random function of the elapsed time since infection, those function being i.i.d. for the…

Probability · Mathematics 2021-06-01 Raphael Forien , Guodong Pang , Etienne Pardoux