Mathematical Methods and Models in Epidemiology

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

Deadline for manuscript submissions: 31 October 2024 | Viewed by 16435

Special Issue Editors

Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA
Interests: mathematical biology; control theory
Special Issues, Collections and Topics in MDPI journals
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA
Interests: systems and their applications; computational modeling; mathematics education
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue of Mathematical Biology will focus on recent developments in methods and models in epidemiology. The Special Issue is thus seeking articles that compare and contrast between existing models and between newer and older models in addition to those providing an overview of the development of new models and numerical studies. It is expected that several of these articles will deal with general theory, while others will cover more specific epidemic and endemic diseases. To be considered for submission, articles should not be purely based on conjectural possibilities but should be related to a current situation or connect a current situation to previous ones.

We hope to hear from authors throughout the world, especially those who are carefully studying the dynamics in the most affected areas.

Prof. Dr. James P. Braselton
Prof. Dr. Martha L. Abell
Guest Editors

Manuscript Submission Information

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Keywords

  • mathematical biology
  • epidemiology
  • epidemic models
  • endemic models
  • COVID-19

Published Papers (10 papers)

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Research

19 pages, 3347 KiB  
Article
Mathematical Analysis of an Anthroponotic Cutaneous Leishmaniasis Model with Asymptomatic Infection
by Muntaser Safan and Alhanouf Altheyabi
Mathematics 2023, 11(10), 2388; https://doi.org/10.3390/math11102388 - 21 May 2023
Cited by 1 | Viewed by 908
Abstract
This work aims mainly to study the impact of experiencing asymptomatic anthroponotic cutaneous leishmaniasis (ACL) infection on the overall dynamics and outcomes of the disease. Therefore, a deterministic model for the transmission dynamics of ACL of type SEAIS in the human host and [...] Read more.
This work aims mainly to study the impact of experiencing asymptomatic anthroponotic cutaneous leishmaniasis (ACL) infection on the overall dynamics and outcomes of the disease. Therefore, a deterministic model for the transmission dynamics of ACL of type SEAIS in the human host and SI in sandfly populations is proposed and mathematically analyzed. The model is shown to be well-posed. Its equilibrium and stability analyses are shown. The equilibrium analysis shows that the model has an ACL-free equilibrium that is proven to be locally and globally asymptotically stable if and only if R0<1. In addition, the model has a unique ACL-endemic equilibrium that is shown to exist and be locally asymptotically stable if and only if R0>1. Numerical simulations are performed to show the asymptotic stability of these equilibriums. In addition, the effect of ignoring asymptomatic infections is studied and the analysis shows that ignoring the development of asymptomatic infections overestimates the effort required to eliminate the infection. Moreover, it implies inaccurate measures of controlling ACL infection, especially those based on either using insecticide sprays or bednets. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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18 pages, 1256 KiB  
Article
Using Data of a Lassa Fever Epidemic in Nigeria: A Mathematical Model Is Shown to Capture the Dynamics and Point to Possible Control Methods
by Obiora Cornelius Collins and Kevin Jan Duffy
Mathematics 2023, 11(5), 1181; https://doi.org/10.3390/math11051181 - 28 Feb 2023
Cited by 2 | Viewed by 1718
Abstract
Lassa fever is a deadly viral illness that is endemic in some parts of West Africa, including Nigeria. A deterministic model in the form of a non-linear system of differential equations is developed to analyse the dynamics and possible control of the disease. [...] Read more.
Lassa fever is a deadly viral illness that is endemic in some parts of West Africa, including Nigeria. A deterministic model in the form of a non-linear system of differential equations is developed to analyse the dynamics and possible control of the disease. The model is tested by fitting it to data from Nigeria’s Lassa fever outbreak using a least-squares fitting routine and the model is shown to provide a reasonable fit to the data. Parameters representing various control measures in the model are estimated using the model fitting. Important epidemiological features of the model such as the basic reproduction number (R0), the disease-free equilibrium, and the endemic equilibrium are determined and analysed. The disease-free equilibrium is shown to be asymptotically stable when R0<1. A bifurcation about R0=1 was determined using the Center Manifold Theorem. Using numerical simulations of the model future dynamics of Lassa fever disease in Nigeria are predicted and the impact of control measures on the disease determined. The use of approved rodenticides is shown to be the most effective control followed by reducing person-to-person and rodent-to-person contacts, respectively. Isolation and treatment of infected individuals are shown to be less effective when compared with the other control measures. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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27 pages, 4974 KiB  
Article
A New Polymorphic Comprehensive Model for COVID-19 Transition Cycle Dynamics with Extended Feed Streams to Symptomatic and Asymptomatic Infections
by Yas Al-Hadeethi, Intesar F. El Ramley, Hiba Mohammed and Abeer Z. Barasheed
Mathematics 2023, 11(5), 1119; https://doi.org/10.3390/math11051119 - 23 Feb 2023
Cited by 1 | Viewed by 961
Abstract
This work presents a new polymorphic, reusable, and comprehensive mathematical model for COVID-19 epidemic transition cycle dynamics. This model has the following characteristics: (1) The core SEIR model includes asymptomatic and symptomatic infections; (2) the symptomatic infection is a multi-variant; (3) the recovery [...] Read more.
This work presents a new polymorphic, reusable, and comprehensive mathematical model for COVID-19 epidemic transition cycle dynamics. This model has the following characteristics: (1) The core SEIR model includes asymptomatic and symptomatic infections; (2) the symptomatic infection is a multi-variant; (3) the recovery stage provides a partial feed to the symptomatic infection; and (4) the symptomatic and asymptomatic stages have additional feed streams from the protected stage. The proposed formalisation template is a canonical way to achieve different models for the underlying health control environment. This template approach endows the model with polymorphic and reusable capability across different scenarios. To verify the model’s reliability and validity, this work utilised two sets of initial conditions: date range and COVID-19 data for Canada and Saudi Arabia. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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18 pages, 1226 KiB  
Article
High Resolution Spatio-Temporal Model for Room-Level Airborne Pandemic Spread
by Teddy Lazebnik and Ariel Alexi
Mathematics 2023, 11(2), 426; https://doi.org/10.3390/math11020426 - 13 Jan 2023
Cited by 5 | Viewed by 1599
Abstract
Airborne pandemics have caused millions of deaths worldwide, large-scale economic losses, and catastrophic sociological shifts in human history. Researchers have developed multiple mathematical models and computational frameworks to investigate and predict pandemic spread on various levels and scales such as countries, cities, large [...] Read more.
Airborne pandemics have caused millions of deaths worldwide, large-scale economic losses, and catastrophic sociological shifts in human history. Researchers have developed multiple mathematical models and computational frameworks to investigate and predict pandemic spread on various levels and scales such as countries, cities, large social events, and even buildings. However, attempts of modeling airborne pandemic dynamics on the smallest scale, a single room, have been mostly neglected. As time indoors increases due to global urbanization processes, more infections occur in shared rooms. In this study, a high-resolution spatio-temporal epidemiological model with airflow dynamics to evaluate airborne pandemic spread is proposed. The model is implemented, using Python, with high-resolution 3D data obtained from a light detection and ranging (LiDAR) device and computing model based on the Computational Fluid Dynamics (CFD) model for the airflow and the Susceptible–Exposed–Infected (SEI) model for the epidemiological dynamics. The pandemic spread is evaluated in four types of rooms, showing significant differences even for a short exposure duration. We show that the room’s topology and individual distribution in the room define the ability of air ventilation to reduce pandemic spread throughout breathing zone infection. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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19 pages, 1529 KiB  
Article
Network Thermodynamics-Based Scalable Compartmental Model for Multi-Strain Epidemics
by Joseph Pateras, Ashwin Vaidya and Preetam Ghosh
Mathematics 2022, 10(19), 3513; https://doi.org/10.3390/math10193513 - 26 Sep 2022
Viewed by 1522
Abstract
SARS-CoV-2 continues to upend human life by posing novel threats related to disease spread and mutations. Current models for the disease burden of SARS-CoV-2 consider the aggregate nature of the virus without differentiating between the potency of its multiple strains. Hence, there is [...] Read more.
SARS-CoV-2 continues to upend human life by posing novel threats related to disease spread and mutations. Current models for the disease burden of SARS-CoV-2 consider the aggregate nature of the virus without differentiating between the potency of its multiple strains. Hence, there is a need to create a fundamental modeling framework for multi-strain viruses that considers the competing viral pathogenic pathways. Alongside the consideration that other viral pathogens may coexist, there is also a need for a generalizable modeling framework to account for multiple epidemics (i.e., multi-demics) scenarios, such as influenza and COVID-19 occurring simultaneously. We present a fundamental network thermodynamics approach for assessing, determining, and predicting viral outbreak severity, which extends well-known standard epidemiological models. In particular, we use historical data from New York City’s 2011–2019 influenza seasons and SARS-CoV-2 spread to identify the model parameters. In our model-based analysis, we employ a standard susceptible–infected–recovered (SIR) model with pertinent generalizations to account for multi-strain and multi-demics scenarios. We show that the reaction affinities underpinning the formation processes of our model can be used to categorize the severity of infectious or deceased populations. The spontaneity of occurrence captured by the change in Gibbs free energy of reaction (G) in the system suggests the stability of forward occurring population transfers. The magnitude of G is used to examine past influenza outbreaks and infer epidemiological factors, such as mortality and case burden. This method can be extrapolated for wide-ranging utility in computational epidemiology. The risk of overlapping multi-demics seasons between influenza and SARS-CoV-2 will persist as a significant threat in forthcoming years. Further, the possibility of mutating strains requires novel ways of analyzing the network of competing infection pathways. The approach outlined in this study allows for the identification of new stable strains and the potential increase in disease burden from a complex systems perspective, thereby allowing for a potential response to the significant question: are the effects of a multi-demic greater than the sum of its individual viral epidemics? Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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15 pages, 1361 KiB  
Article
Efficient Numerical Solutions to a SIR Epidemic Model
by Mohammad Mehdizadeh Khalsaraei, Ali Shokri, Higinio Ramos, Shao-Wen Yao and Maryam Molayi
Mathematics 2022, 10(18), 3299; https://doi.org/10.3390/math10183299 - 11 Sep 2022
Cited by 3 | Viewed by 1966
Abstract
Two non-standard predictor-corrector type finite difference methods for a SIR epidemic model are proposed. The methods have useful and significant features, such as positivity, basic stability, boundedness and preservation of the conservation laws. The proposed schemes are compared with classical fourth order Runge–Kutta [...] Read more.
Two non-standard predictor-corrector type finite difference methods for a SIR epidemic model are proposed. The methods have useful and significant features, such as positivity, basic stability, boundedness and preservation of the conservation laws. The proposed schemes are compared with classical fourth order Runge–Kutta and non-standard difference methods (NSFD). The stability analysis is studied and numerical simulations are provided. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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28 pages, 703 KiB  
Article
A Multi-Scale Model for Cholera Outbreaks
by Beryl Musundi, Johannes Müller and Zhilan Feng
Mathematics 2022, 10(17), 3114; https://doi.org/10.3390/math10173114 - 30 Aug 2022
Cited by 2 | Viewed by 1540
Abstract
Cholera, caused by the pathogenic Vibrio cholerae bacteria, remains a severe public health threat. Although a lot of emphasis has been placed on the population-level spread of the disease, the infection itself starts within the body. As such, we formulated a multi-scale model [...] Read more.
Cholera, caused by the pathogenic Vibrio cholerae bacteria, remains a severe public health threat. Although a lot of emphasis has been placed on the population-level spread of the disease, the infection itself starts within the body. As such, we formulated a multi-scale model that explicitly connects the within-host and between-host dynamics of the disease. To model the within-host dynamics, we assigned each susceptible individual with a pathogen load that increases through the uptake of contaminated food and water (booster event). We introduced minimal and maximal times when the booster events happen and defined a time since the last booster event. We then scaled the within-host dynamics to the population where we structured the susceptible population using the two variables (pathogen load and time since the last booster event). We analyzed the pathogen load’s invariant distribution and utilized the results and time scale assumptions to reduce the dimension of the multi-scale model. The resulting model is an SIR model whose incidence function has terms derived from the multi-scale model. We finally conducted numerical simulations to investigate the long-term behavior of the SIR model. The simulations revealed parameter regions where either no cholera cases happen, where cholera is present at a low prevalence, and where a full-blown cholera epidemic takes off. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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17 pages, 5175 KiB  
Article
Exploring Radial Kernel on the Novel Forced SEYNHRV-S Model to Capture the Second Wave of COVID-19 Spread and the Variable Transmission Rate
by Fehaid Salem Alshammari and Ezgi Akyildiz Tezcan
Mathematics 2022, 10(9), 1501; https://doi.org/10.3390/math10091501 - 01 May 2022
Cited by 2 | Viewed by 1320
Abstract
The transmission rate of COVID-19 varies over time. There are many reasons underlying this mechanism, such as seasonal changes, lockdowns, social distancing, and wearing face masks. Hence, it is very difficult to directly measure the transmission rate. The main task of the present [...] Read more.
The transmission rate of COVID-19 varies over time. There are many reasons underlying this mechanism, such as seasonal changes, lockdowns, social distancing, and wearing face masks. Hence, it is very difficult to directly measure the transmission rate. The main task of the present paper was to identify the variable transmission rate (β1) for a SIR-like model. For this, we first propose a new compartmental forced SEYNHRV-S differential model. We then drive the nonlinear differential equation and present the finite difference technique to obtain the time-dependent transmission rate directly from COVID-19 data. Following this, we show that the transmission rate can be represented as a linear combination of radial kernels, where several forms of radial kernels are explored. The proposed model is flexible and general, so it can be adapted to monitor various epidemic scenarios in various countries. Hence, the model may be of interest for policymakers as a tool to evaluate different possible future scenarios. Numerical simulations are presented to validate the prediction of our SEYNHRV and forced SEYNHRV-S models, where the data from confirmed COVID-19 cases reported by the Ministry of Health in Saudi Arabia were used. These confirmed cases show the second wave of the infected population in Saudi Arabia. By using the COVID-19 data, we show that our model (forced SEYNHRV-S) is able to predict the second wave of infection in the population in Saudi Arabia. It is well known that COVID-19 epidemic data cannot be accurately represented by any compartmental approach with constant parameters, and this is also true for our SEYNHRV model. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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18 pages, 1815 KiB  
Article
Fractional Growth Model with Delay for Recurrent Outbreaks Applied to COVID-19 Data
by Fernando Alcántara-López, Carlos Fuentes, Carlos Chávez, Jesús López-Estrada and Fernando Brambila-Paz
Mathematics 2022, 10(5), 825; https://doi.org/10.3390/math10050825 - 04 Mar 2022
Viewed by 1862
Abstract
There are a great many epidemiological models that have been implemented to describe COVID-19 data; however, few attempted to reproduce the entire phenomenon due to the complexity of modeling recurrent outbreaks. In this work a fractional growth model with delay is developed that [...] Read more.
There are a great many epidemiological models that have been implemented to describe COVID-19 data; however, few attempted to reproduce the entire phenomenon due to the complexity of modeling recurrent outbreaks. In this work a fractional growth model with delay is developed that implements the Caputo fractional derivative with 0<β1. Furthermore, in order to preserve the nature of the phenomenon and ensure continuity in the derivatives of the function, a method is proposed to construct an initial condition function to implement in the model with delay. This model is analyzed and generalized to model recurrent outbreaks. The model is applied to fit data of cumulative confirmed cases from Mexico, the United States, and Russia, obtaining excellent fitting corroborated by the coefficient of determination, where R2>0.9995 in all cases. Lastly, as a result of the implementation of the delay effect, the global phenomenon was decomposed into its local parts, allowing for directly comparing each outbreak and its different characteristics. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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27 pages, 2285 KiB  
Article
A Fractional Ordered COVID-19 Model Incorporating Comorbidity and Vaccination
by Meghadri Das, Guruprasad Samanta and Manuel De la Sen
Mathematics 2021, 9(21), 2806; https://doi.org/10.3390/math9212806 - 04 Nov 2021
Cited by 8 | Viewed by 1525
Abstract
The primary goal of this research is to investigate COVID-19 transmission patterns in West Bengal, India in 2021; the first Coronavirus illness (COVID-19) in West Bengal was revealed on 17 March 2020. We employed the modified Susceptible-Asymptomatic-Vaccinated-Comorbidity-Infectious-Recovered (SAVICR) compartmental model as part of [...] Read more.
The primary goal of this research is to investigate COVID-19 transmission patterns in West Bengal, India in 2021; the first Coronavirus illness (COVID-19) in West Bengal was revealed on 17 March 2020. We employed the modified Susceptible-Asymptomatic-Vaccinated-Comorbidity-Infectious-Recovered (SAVICR) compartmental model as part of fractional orders because of the uncertainty created by the limited Coronavirus (COVID-19) information. In this article, two sub-compartments (Normal Infected and Infected with Co-morbidity) has been considered with vaccinated class, which is relevant in the present situation. We have studied the dynamical analysis of the system and also studied sensitivity of the parameters for West Bengal framework. We have also considered an optimal control problem taking social distancing (non-pharmaceutical treatments) as a control parameter along with vaccination. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)
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