Special Issue "Mathematical Models in Epidemiology"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 30 September 2019

Special Issue Editor

Guest Editor
Dr. Toshikazu Kuniya

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Website | E-Mail
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 (4 papers)

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Research

Open AccessArticle
Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity
Mathematics 2019, 7(5), 420; https://doi.org/10.3390/math7050420
Received: 18 March 2019 / Revised: 25 April 2019 / Accepted: 26 April 2019 / Published: 10 May 2019
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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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Open AccessArticle
Role of Media and Effects of Infodemics and Escapes in the Spatial Spread of Epidemics: A Stochastic Multi-Region Model with Optimal Control Approach
Mathematics 2019, 7(3), 304; https://doi.org/10.3390/math7030304
Received: 23 January 2019 / Revised: 19 February 2019 / Accepted: 6 March 2019 / Published: 25 March 2019
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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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Open AccessArticle
Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks
Mathematics 2019, 7(1), 30; https://doi.org/10.3390/math7010030
Received: 24 November 2018 / Revised: 23 December 2018 / Accepted: 24 December 2018 / Published: 29 December 2018
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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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Open AccessArticle
Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies
Mathematics 2018, 6(12), 328; https://doi.org/10.3390/math6120328
Received: 18 October 2018 / Revised: 4 December 2018 / Accepted: 6 December 2018 / Published: 14 December 2018
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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 R0, 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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