Mathematical Models in Epidemiology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (30 September 2020) | Viewed by 55135

Special Issue Editor


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Guest Editor
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Interests: mathematical epidemiology; mathematical biology; differential equations; dynamical systems; numerical analysis

Special Issue Information

Dear Colleagues,

In recent years, mathematical models of infectious disease transmission have attracted a great deal of attention and been studied by many researchers from a broad range of mathematical viewpoints. These models are usually formulated as nonlinear systems of ordinary, delay or partial differential equations, and various mathematical theories have been developed for them. One of the most important concepts in this field is the basic reproduction number Ro, which is the expected number of secondary cases produced by a typical infected individual in a fully susceptible population. Ro is important from both of the mathematical and epidemiological viewpoints as it can be a threshold not only for the occurrence of the initial outbreak of disease but also for the long-term persistence of disease in the sense of the global stability of positive endemic equilibrium.

The purpose of this Special Issue is to establish a collection of papers that provide novel insights on mathematical theories of epidemic models. Papers of all mathematical backgrounds are welcome including ordinary, delay and partial differential equations, dynamical systems, stability and bifurcation theory, control theory and network theory, but not limited to them.

Dr. Toshikazu Kuniya
Guest Editor

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Keywords

  • Epidemic models
  • Basic reproduction number
  • Ordinary differential equations
  • Delay differential equations
  • Partial differential equations
  • Dynamical systems
  • Stability analysis
  • Bifurcation
  • Optimal control
  • Network

Published Papers (10 papers)

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Research

22 pages, 3511 KiB  
Article
A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function
by Domingo Benítez, Gustavo Montero, Eduardo Rodríguez, David Greiner, Albert Oliver, Luis González and Rafael Montenegro
Mathematics 2020, 8(11), 2000; https://doi.org/10.3390/math8112000 - 09 Nov 2020
Cited by 4 | Viewed by 1968
Abstract
A novel phenomenological epidemic model is proposed to characterize the state of infectious diseases and predict their behaviors. This model is given by a new stochastic partial differential equation that is derived from foundations of statistical physics. The analytical solution of this equation [...] Read more.
A novel phenomenological epidemic model is proposed to characterize the state of infectious diseases and predict their behaviors. This model is given by a new stochastic partial differential equation that is derived from foundations of statistical physics. The analytical solution of this equation describes the spatio-temporal evolution of a Gaussian probability density function. Our proposal can be applied to several epidemic variables such as infected, deaths, or admitted-to-the-Intensive Care Unit (ICU). To measure model performance, we quantify the error of the model fit to real time-series datasets and generate forecasts for all the phases of the COVID-19, Ebola, and Zika epidemics. All parameters and model uncertainties are numerically quantified. The new model is compared with other phenomenological models such as Logistic Grow, Original, and Generalized Richards Growth models. When the models are used to describe epidemic trajectories that register infected individuals, this comparison shows that the median RMSE error and standard deviation of the residuals of the new model fit to the data are lower than the best of these growing models by, on average, 19.6% and 35.7%, respectively. Using three forecasting experiments for the COVID-19 outbreak, the median RMSE error and standard deviation of residuals are improved by the performance of our model, on average by 31.0% and 27.9%, respectively, concerning the best performance of the growth models. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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15 pages, 1854 KiB  
Article
Dynamics of Epidemic Spreading in the Group-Based Multilayer Networks
by Dong Wang, Yi Zhao and Hui Leng
Mathematics 2020, 8(11), 1895; https://doi.org/10.3390/math8111895 - 31 Oct 2020
Cited by 1 | Viewed by 1842
Abstract
The co-evolution between information and epidemic in multilayer networks has attracted wide attention. However, previous studies usually assume that two networks with the same individuals are coupled into a multiplex network, ignoring the context that the individuals of each layer in the multilayer [...] Read more.
The co-evolution between information and epidemic in multilayer networks has attracted wide attention. However, previous studies usually assume that two networks with the same individuals are coupled into a multiplex network, ignoring the context that the individuals of each layer in the multilayer network are often different, especially in group structures with rich collective phenomena. In this paper, based on the scenario of group-based multilayer networks, we investigate the coupled UAU-SIS (Unaware-Aware-Unaware-Susceptible-Infected-Susceptible) model via microscopic Markov chain approach (MMCA). Importantly, the evolution of such transmission process with respective to various impact factors, especially for the group features, is captured by simulations. We further obtain the theoretical threshold for the onset of epidemic outbreaks and analyze its characteristics through numerical simulations. It is concluded that the growth of the group size of information (physical) layer effectively suppresses (enhances) epidemic spreading. Moreover, taking the context of epidemic immunization into account, we find that the propagation capacity and robustness of this type of network are greater than the conventional multiplex network. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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18 pages, 2366 KiB  
Article
Modeling and Controlling Epidemic Outbreaks: The Role of Population Size, Model Heterogeneity and Fast Response in the Case of Measles
by Kezban Yagci Sokat and Benjamin Armbruster
Mathematics 2020, 8(11), 1892; https://doi.org/10.3390/math8111892 - 31 Oct 2020
Cited by 1 | Viewed by 1920
Abstract
Modelers typically use detailed simulation models and vary the fraction vaccinated to study outbreak control. However, there is currently no guidance for modelers on how much detail (i.e., heterogeneity) is necessary and how large a population to simulate. We provide theoretical and numerical [...] Read more.
Modelers typically use detailed simulation models and vary the fraction vaccinated to study outbreak control. However, there is currently no guidance for modelers on how much detail (i.e., heterogeneity) is necessary and how large a population to simulate. We provide theoretical and numerical guidance for those decisions and also analyze the benefit of a faster public health response through a stochastic simulation model in the case of measles in the United States. Theoretically, we prove that the outbreak size converges as the simulation population increases and that the outbreaks are slightly larger with a heterogeneous community structure. We find that the simulated outbreak size is not sensitive to the size of the simulated population beyond a certain size. We also observe that in case of an outbreak, a faster public health response provides benefits similar to increased vaccination. Insights from this study can inform the control and elimination measures of the ongoing coronavirus disease (COVID-19) as measles has shown to have a similar structure to COVID-19. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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21 pages, 782 KiB  
Article
Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis
by Dmitry Gromov and Ethan O. Romero-Severson
Mathematics 2020, 8(9), 1500; https://doi.org/10.3390/math8091500 - 04 Sep 2020
Viewed by 1643
Abstract
Chronic viral infections can persist for decades spanning thousands of viral generations, leading to a highly diverse population of viruses with its own complex evolutionary history. We propose an expandable mathematical framework for understanding how the emergence of genetic and phenotypic diversity affects [...] Read more.
Chronic viral infections can persist for decades spanning thousands of viral generations, leading to a highly diverse population of viruses with its own complex evolutionary history. We propose an expandable mathematical framework for understanding how the emergence of genetic and phenotypic diversity affects the population-level control of those infections by both non-curative treatment and chemo-prophylactic measures. Our frameworks allows both neutral and phenotypic evolution, and we consider the specific evolution of contagiousness, resistance to therapy, and efficacy of prophylaxis. We compute both the controlled and uncontrolled, population-level basic reproduction number accounting for the within-host evolutionary process where new phenotypes emerge and are lost in infected persons, which we also extend to include both treatment and prophylactic control efforts. We used these results to discuss the conditions under which the relative efficacy of prophylactic versus therapeutic methods of control are superior. Finally, we give expressions for the endemic equilibrium of these models for certain constrained versions of the within-host evolutionary model providing a potential method for estimating within-host evolutionary parameters from population-level genetic sequence data. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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16 pages, 443 KiB  
Article
A Mathematical Model of Epidemics—A Tutorial for Students
by Yutaka Okabe and Akira Shudo
Mathematics 2020, 8(7), 1174; https://doi.org/10.3390/math8071174 - 17 Jul 2020
Cited by 16 | Viewed by 27041
Abstract
This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. [...] Read more.
This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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14 pages, 2007 KiB  
Article
Comparing Direct and Indirect Transmission in a Simple Model of Veterinary Disease
by Kezban Yagci Sokat, Stefan Edlund, Kenneth Clarkson and James Kaufman
Mathematics 2019, 7(11), 1039; https://doi.org/10.3390/math7111039 - 03 Nov 2019
Cited by 1 | Viewed by 3742
Abstract
Foodborne diseases are a longstanding worldwide public health concern. Modeling the transmission pathways of foodborne pathogens accurately and effectively can aid in understanding the spread of pathogens and facilitate decision making for intervention. A new compartmental model is reported that integrates the effects [...] Read more.
Foodborne diseases are a longstanding worldwide public health concern. Modeling the transmission pathways of foodborne pathogens accurately and effectively can aid in understanding the spread of pathogens and facilitate decision making for intervention. A new compartmental model is reported that integrates the effects of both direct and indirect transmission. Depending on the choice of epidemiological parameters, the model can be tuned to be purely direct, purely indirect, or used to explore the dynamics in an intermediate regime. Steady state analysis of the model and limiting cases are studied. A numerical simulation is employed to study the impact of different epidemiological parameters and dose response. Direct transmission can surpass the effect of indirect transmission for the same range of parameter values and result in an earlier epidemic. The rate at which the pathogens are removed from the environment can lead to a faster epidemic. The environmental contamination can decrease the time to reach the steady state depending on the dose response. These results can inform policy makers for control strategies to reduce foodborne pathogen transmission. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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21 pages, 451 KiB  
Article
Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity
by Imane Abouelkheir, Fadwa El Kihal, Mostafa Rachik and Ilias Elmouki
Mathematics 2019, 7(5), 420; https://doi.org/10.3390/math7050420 - 10 May 2019
Cited by 15 | Viewed by 3388
Abstract
Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when [...] Read more.
Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin’s maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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24 pages, 744 KiB  
Article
Role of Media and Effects of Infodemics and Escapes in the Spatial Spread of Epidemics: A Stochastic Multi-Region Model with Optimal Control Approach
by Fadwa El Kihal, Imane Abouelkheir, Mostafa Rachik and Ilias Elmouki
Mathematics 2019, 7(3), 304; https://doi.org/10.3390/math7030304 - 25 Mar 2019
Cited by 8 | Viewed by 3125
Abstract
Mass vaccination campaigns play major roles in the war against epidemics. Such prevention strategies cannot always reach their goals significantly without the help of media and awareness campaigns used to prevent contacts between susceptible and infected people. Feelings of fear, infodemics, and misconception [...] Read more.
Mass vaccination campaigns play major roles in the war against epidemics. Such prevention strategies cannot always reach their goals significantly without the help of media and awareness campaigns used to prevent contacts between susceptible and infected people. Feelings of fear, infodemics, and misconception could lead to some fluctuations of such policies. In addition to the vaccination strategy, the movement restriction approach is essential because of the factor of mobility or travel. However, anti-epidemic border measures may also be disturbed if some infected travelers manage to escape and infiltrate into a safer region. In this paper, we aim to study infection dynamics related to the spatial spread of an epidemic in interconnected regions in the presence of random perturbations caused by the three above-mentioned reasons. Therefore, we devise a stochastic multi-region epidemic model in which contacts between susceptible and infected populations, vaccination-based and movement restriction optimal control approaches are all assumed to be unpredictable, and then, we discuss the effectiveness of such policies. In order to reach our goal, we employ a stochastic maximum principle version for noised systems, state and prove the sufficient and necessary conditions of optimality, and finally provide the numerical results obtained using a stochastic progressive-regressive schemes method. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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25 pages, 482 KiB  
Article
Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks
by Carlos Rodríguez Lucatero and Luis Angel Alarcón Ramos
Mathematics 2019, 7(1), 30; https://doi.org/10.3390/math7010030 - 29 Dec 2018
Cited by 1 | Viewed by 2789
Abstract
In this paper, we justify by the use of Enumerative Combinatorics, the applicability of an asymptotic stability result on Discrete-Time Epidemics in Complex Networks, where the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation [...] Read more.
In this paper, we justify by the use of Enumerative Combinatorics, the applicability of an asymptotic stability result on Discrete-Time Epidemics in Complex Networks, where the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation process are analyzed, and, because of that, are good candidates to be controlled in the network in order to stabilize the network to reach the extinction state. The epidemic model analyzed was proposed and published in 2011 by of Gómez et al. The asymptotic stability result obtained in the present article imply that it is not necessary to control all nodes, but only a minimal set of nodes if the topology of the network is not regular. This result could be important in the spirit of considering policies of isolation or quarantine of those nodes to be controlled. Simulation results using a refined version of the asymptotic stability result were presented in another paper of the second author for large free-scale and regular networks that corroborate the theoretical findings. In the present article, we justify the applicability of the controllability result obtained in the mentioned paper in almost all the cases by means of the use of Combinatorics. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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12 pages, 6526 KiB  
Article
Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies
by Yanli Ma, Jia-Bao Liu and Haixia Li
Mathematics 2018, 6(12), 328; https://doi.org/10.3390/math6120328 - 14 Dec 2018
Cited by 25 | Viewed by 4203
Abstract
In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which [...] Read more.
In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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