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Symmetry
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Mathematical Modeling of the Infectious Diseases and Their Controls
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Mathematical Modeling of the Infectious Diseases and Their Controls

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 19786

Special Issue Editor

Dr. Muhammad Altaf Khan

E-Mail Website
Guest Editor
Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa
Interests: mathematics

Special Issue Information

Dear Colleagues,

The mathematical modeling of infectious diseases and the mitigating effects of controls implemented by humans is of great importance for public health. There are many existing and newly emerging diseases that continuously affect the population through infections and deaths. Mathematical and statistical modeling is a great tool to study such infectious diseases and to determine their complex behavior. Further, potential controls can be measured in line with biological or clinical suggestions, making it possible to determine the optimal cost-effective controls that should be used to curtail these diseases. Infectious diseases are disorders usually caused by an organism, such as fungi, bacteria, viruses, or parasites, and are the leading causes of death in humans. Researchers and health authorities are continuously working to reduce the spread of the disease and to prevent their transmission amongst the population, yet there are many diseases that need further study to reduce their spread. From a mathematical point of view, models in the form of mathematical formulae or statistical models are widely used to study infectious diseases. In recent eras, researchers have developed novel methods to create infectious diseases models via differential or difference equations. Usually, researchers study and analyze the disease models in the form of ordinary differential equations, partial differential equations (age-structured models, etc.,) stochastic differential equations, and/or delay differential equations. Besides this, the infectious diseases models have also been studied using fractional derivatives, fractal–fractional operators, etc. The main focus of this Special Issue is to model and analyze various infectious disease models of a complex nature and provide useful recommendations about ways to control and, hopefully, eliminate them. Further, it will contain articles that aim to develop new algorithms or techniques to solve models based on the differential equations. We would like to invite authors to contribute to this Special Issue by submitting original and novel research papers regarding the modeling and simulation of infectious diseases. 

Potential topics to be included in this Special Issue consist of, but are not limited to, the following: 

  • Modeling with ordinary differential equations, or difference equations;
  • Modeling with partial differential equations;
  • Modeling with stochastic differential equations;
  • Modeling with delay differential equations;
  • Modeling the disease with fuzzy differential equations (ODE and Pde types); 
  • Modeling with fractional differential equations; 
  • Modeling with fractal–fractional differential equations; 
  • Modeling with fractal derivatives; 
  • Modeling with neural networks; 
  • Comparison of the numerical technique for the solution of the disease models for integer-order and non-integer orders;
  • Modeling with optimal controls (integer and noninteger orders);

Please note that all submission should full in the scope of Symmetry Journal.

Dr. Muhammad Altaf Khan
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical modeling
  • infectious diseases
  • optimal controls
  • fractional operators
  • numerical methods

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Published Papers (12 papers)

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Research

21 pages, 7291 KiB  
Article
Implications of Asymptomatic Carriers for Tuberculosis Transmission and Control in Thailand: A Modelling Approach
by Md Abdul Kuddus, Sazia Khatun Tithi and Thitiya Theparod
Symmetry 2024, 16(11), 1538; https://doi.org/10.3390/sym16111538 - 17 Nov 2024
Viewed by 783
Abstract
Tuberculosis (TB) is a regional and global bacterial illness that has been expanding and affecting individuals in every generation. An unknown percentage of asymptomatic hosts have TB, and the infection can spread while exhibiting no symptoms. These asymptomatic TB carriers, who contribute to [...] Read more.
Tuberculosis (TB) is a regional and global bacterial illness that has been expanding and affecting individuals in every generation. An unknown percentage of asymptomatic hosts have TB, and the infection can spread while exhibiting no symptoms. These asymptomatic TB carriers, who contribute to the spread of the illness yet go mostly undetected, may make it more difficult to prevent transmission. In this study, we utilized the concept of symmetry to construct a manageable disease modelling framework for TB transmission and control. We developed a TB model to investigate the potential influence of asymptomatic carriers, symptomatic infections, and the entirety of TB prevalence on different approaches to treatment and prevention in Thailand. Annual TB incidence data from Thailand from 2000 to 2022 were used to calibrate the model parameters. We assessed the potential for reaching conflicting results about the management and spread of tuberculosis in Thailand. Our results showed that some TB strategies that were thought to be effective in reducing transmission may have the opposite impact, or that an intervention’s effectiveness might be overestimated, making it seem unfeasible in certain scenarios. For example, the objective of TB treatment, which attempts to decrease the occurrence of symptomatic TB infections, is to decrease the TB infection and propagation rates if the relative carrier (η) is less than one. Nonetheless, our results indicate that this strategy may increase the frequency of asymptomatic TB patients, symptomatic TB viral infections, and overall TB prevalence if η has been sufficiently understated. We also found that reducing only the progression rate of symptomatic TB infections cannot stop asymptomatic TB carriers and total TB prevalence, even when the relative infection of carriers (η) is less than unity. Our research provides a better understanding of the role of asymptomatic patients in spreading TB and highlights the need to accurately include bearers in models that guide Thailand’s TB control strategy. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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17 pages, 16100 KiB  
Article
A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine
by Roman Cherniha, Vasyl’ Dutka and Vasyl’ Davydovych
Symmetry 2024, 16(11), 1411; https://doi.org/10.3390/sym16111411 - 23 Oct 2024
Viewed by 484
Abstract
A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 [...] Read more.
A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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19 pages, 1122 KiB  
Article
Comparative Analysis of Influenza Modeling Using Novel Fractional Operators with Real Data
by Mohamed A. Abdoon and Abdulrahman B. M. Alzahrani
Symmetry 2024, 16(9), 1126; https://doi.org/10.3390/sym16091126 - 30 Aug 2024
Viewed by 817
Abstract
In this work, the efficacy of fractional models under Atangana–Baleanu–Caputo, Caputo–Fabrizio, and Caputo is compared to the performance of integer-order models in the forecasting of weekly influenza cases using data from the Kingdom of Saudi Arabia. The suggested fractional influenza model was effectively [...] Read more.
In this work, the efficacy of fractional models under Atangana–Baleanu–Caputo, Caputo–Fabrizio, and Caputo is compared to the performance of integer-order models in the forecasting of weekly influenza cases using data from the Kingdom of Saudi Arabia. The suggested fractional influenza model was effectively verified using fractional calculus. Our investigation uncovered the topic’s essential properties and deepened our understanding of disease progression. Furthermore, we analyzed the numerical scheme’s positivity, limitations, and symmetry. The fractional-order models demonstrated superior accuracy, producing smaller root mean square error (RMSE) and mean absolute error (MAE) than the classical model. The novelty of this work lies in introducing the Atangana–Baleanu–Caputo fractional model to influenza forecasting to incorporate memory of an epidemic, which leads to higher accuracy than traditional models. These models effectively captured the peak and drop of influenza cases. Based on these findings, it can be concluded that fractional-order models perform better than typical integer-order models when predicting influenza dynamics. These insights should illuminate the importance of fractional calculus in addressing epidemic threats. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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28 pages, 6255 KiB  
Article
Spatial Predictive Modeling of Liver Fluke Opisthorchis viverrine (OV) Infection under the Mathematical Models in Hexagonal Symmetrical Shapes Using Machine Learning-Based Forest Classification Regression
by Benjamabhorn Pumhirunroj, Patiwat Littidej, Thidarut Boonmars, Atchara Artchayasawat, Narueset Prasertsri, Phusit Khamphilung, Satith Sangpradid, Nutchanat Buasri, Theeraya Uttha and Donald Slack
Symmetry 2024, 16(8), 1067; https://doi.org/10.3390/sym16081067 - 19 Aug 2024
Cited by 2 | Viewed by 1376
Abstract
Infection with liver flukes (Opisthorchis viverrini) is partly due to their ability to thrive in habitats in sub-basin areas, causing the intermediate host to remain in the watershed system throughout the year. Spatial modeling is used to predict water source infections, [...] Read more.
Infection with liver flukes (Opisthorchis viverrini) is partly due to their ability to thrive in habitats in sub-basin areas, causing the intermediate host to remain in the watershed system throughout the year. Spatial modeling is used to predict water source infections, which involves designing appropriate area units with hexagonal grids. This allows for the creation of a set of independent variables, which are then covered using machine learning techniques such as forest-based classification regression methods. The independent variable set was obtained from the local public health agency and used to establish a relationship with a mathematical model. The ordinary least (OLS) model approach was used to screen the variables, and the most consistent set was selected to create a new set of variables using the principal of component analysis (PCA) method. The results showed that the forest classification and regression (FCR) model was able to accurately predict the infection rates, with the PCA factor yielding a reliability value of 0.915. This was followed by values of 0.794, 0.741, and 0.632, respectively. This article provides detailed information on the factors related to water body infection, including the length and density of water flow lines in hexagonal form, and traces the depth of each process. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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19 pages, 2365 KiB  
Article
Multidimensional Epidemiological Survey Data Aggregation Scheme Based on Personalized Local Differential Privacy
by Xueyan Liu, Qiong Liu, Jia Wang and Hao Sun
Symmetry 2024, 16(3), 294; https://doi.org/10.3390/sym16030294 - 2 Mar 2024
Viewed by 1061
Abstract
In recent years, with the rapid development of intelligent technology, information security and privacy issues have become increasingly prominent. Epidemiological survey data (ESD) research plays a vital role in understanding the laws and trends of disease transmission. However, epidemiological investigations (EI) involve a [...] Read more.
In recent years, with the rapid development of intelligent technology, information security and privacy issues have become increasingly prominent. Epidemiological survey data (ESD) research plays a vital role in understanding the laws and trends of disease transmission. However, epidemiological investigations (EI) involve a large amount of privacy-sensitive data which, once leaked, will cause serious harm to individuals and society. Collecting EI data is also a huge task. To solve these problems and meet personalized privacy protection requirements in EIs, we improve the uOUE protocol based on utility-optimized local differential privacy to improve the efficiency and accuracy of data coding. At the same time, aiming at the collection and processing of ESD, a multidimensional epidemiological survey data aggregation scheme based on uOUE is designed. By using Paillier homomorphic encryption and an identity-based signature scheme to further prevent differential attacks and achieve multidimensional data aggregation, the safe, efficient, and accurate aggregation processing of ESD is executed. Through security proof and performance comparison, it is verified that our algorithm meets the requirements of local differential privacy and unbiased estimation. The experimental evaluation results on two data sets show that the algorithm has good practicability and accuracy in ESD collection and provides reliable and effective privacy protection. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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27 pages, 810 KiB  
Article
Role of Differential Susceptibility and Infectiousness on the Dynamics of an SIRS Model for Malaria Transmission
by Muntaser Safan, Derdei Bichara, Kamuela E. Yong, Amira Alharthi and Carlos Castillo-Chavez
Symmetry 2023, 15(10), 1950; https://doi.org/10.3390/sym15101950 - 21 Oct 2023
Cited by 1 | Viewed by 1390
Abstract
A deterministic model for the transmission dynamics of SIRS-type malaria in hosts and SI in mosquito populations is proposed. The host population is differentiated between naive, primary, and secondary susceptible individuals. Primary and secondary infected individuals (and also recovered) are differentiated from each [...] Read more.
A deterministic model for the transmission dynamics of SIRS-type malaria in hosts and SI in mosquito populations is proposed. The host population is differentiated between naive, primary, and secondary susceptible individuals. Primary and secondary infected individuals (and also recovered) are differentiated from each other according to their degree of infectiousness. The impact of changing the relative susceptibilities of primary and secondary (with respect to naive) susceptible individuals on the dynamics is investigated. Also, the impact of changing the relative infectiousness of secondary infected, primary, and secondary recovered individuals (with respect to primary infected) on the transmission dynamics of malaria is studied. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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15 pages, 618 KiB  
Article
Analysis of the Competition System Using Parameterized Fractional Differential Equations: Application to Real Data
by Mahmoud H. DarAssi, Muhammad Altaf Khan, Fatmawati and Marei Saeed Alqarni
Symmetry 2023, 15(2), 542; https://doi.org/10.3390/sym15020542 - 17 Feb 2023
Cited by 2 | Viewed by 1705
Abstract
Natural symmetries exist in several processes of chemistry, physics, and biology. Symmetries possess interesting dynamical characteristics that cannot be seen in non-symmetric systems. The present paper investigates the competition between two banking systems, rural and commercial, in Indonesia, in parameterized fractional order Caputo [...] Read more.
Natural symmetries exist in several processes of chemistry, physics, and biology. Symmetries possess interesting dynamical characteristics that cannot be seen in non-symmetric systems. The present paper investigates the competition between two banking systems, rural and commercial, in Indonesia, in parameterized fractional order Caputo derivative. A novel numerical method is used to discretize the competition system using the real data of rural and commercial banks in Indonesia for the period 2004–2014. The new scheme is more suitable and reliable for data fitting results and has good accuracy. The integer model is formulated in Caputo derivative and their stability results are presented. With the available parameters, the data for the model is analyzed using various scenarios. We shall compare the result with the previous method used in the literature and show that the present method is better than the previous method in the literature. It is shown that fractional order α and the parameter ρ involved in the numerical scheme provide excellent fitting. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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17 pages, 482 KiB  
Article
A Numerical Confirmation of a Fractional-Order COVID-19 Model’s Efficiency
by Iqbal M. Batiha, Ahmad Obeidat, Shameseddin Alshorm, Ahmed Alotaibi, Hajid Alsubaie, Shaher Momani, Meaad Albdareen, Ferjeni Zouidi, Sayed M. Eldin and Hadi Jahanshahi
Symmetry 2022, 14(12), 2583; https://doi.org/10.3390/sym14122583 - 7 Dec 2022
Cited by 19 | Viewed by 2652
Abstract
In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this [...] Read more.
In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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27 pages, 677 KiB  
Article
A Stochastic Mathematical Model for Understanding the COVID-19 Infection Using Real Data
by Fehaid Salem Alshammari, Fahir Talay Akyildiz, Muhammad Altaf Khan, Anwarud Din and Pongsakorn Sunthrayuth
Symmetry 2022, 14(12), 2521; https://doi.org/10.3390/sym14122521 - 29 Nov 2022
Cited by 9 | Viewed by 2107
Abstract
Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real [...] Read more.
Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real infection data of COVID-19 in Saudi Arabia and present its detailed mathematical results. We first present the existence and uniqueness of the deterministic model and later study the dynamical properties of the deterministic model and determine the global asymptotic stability of the system for R0≤1. We then study the dynamic properties of the stochastic model and present its global unique solution for the model. We further study the extinction of the stochastic model. Further, we use the nonlinear least-square fitting technique to fit the data to the model for the deterministic and stochastic case and the estimated basic reproduction number is R0≈1.1367. We show that the stochastic model provides a good fitting to the real data. We use the numerical approach to solve the stochastic system by presenting the results graphically. The sensitive parameters that significantly impact the model dynamics and reduce the number of infected cases in the future are shown graphically. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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23 pages, 1762 KiB  
Article
Mathematical Analysis of an SIVRWS Model for Pertussis with Waning and Naturally Boosted Immunity
by Muntaser Safan, Kamal Barley, Mohamed M. Elhaddad, Mohamed A. Darwish and Samir H. Saker
Symmetry 2022, 14(11), 2288; https://doi.org/10.3390/sym14112288 - 1 Nov 2022
Viewed by 2257
Abstract
This work aims mainly to study the controllability of pertussis infection in the presence of waning and natural booster of pertussis immunity and to study their impact on the overall dynamics and disease outcomes. Therefore, an SIVRWS (Susceptible-Infected-Vaccinated-Recovered-Waned-Susceptible) model for pertussis infection spread [...] Read more.
This work aims mainly to study the controllability of pertussis infection in the presence of waning and natural booster of pertussis immunity and to study their impact on the overall dynamics and disease outcomes. Therefore, an SIVRWS (Susceptible-Infected-Vaccinated-Recovered-Waned-Susceptible) model for pertussis infection spread in a demographically stationary, homogeneous, and fully symmetric mixing population is introduced. The model has been mathematically analyzed, where both equilibrium and stability analyses have been established, and uniform persistence of the model has been shown. The conditions on model parameters that ensure effective control of the infection have been derived. The effects of the interplay between waning and boosting pertussis immunity by re-exposure to Bordetella pertussis and vaccination on the dynamics have been investigated. The analytical results have been numerically confirmed and explained. The analysis reveals that ignoring the natural booster of immunity overestimates the endemic prevalence of the infection. Moreover, ignoring the differential susceptibility between secondary and primary susceptible individuals overestimates the critical vaccination coverage required to eliminate the infection. Moreover, the shorter the period of immunity acquired by either vaccination or experiencing natural infection, the higher the reproduction number and the endemic prevalence of infection, and therefore, the higher the effort needed to eliminate the infection. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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15 pages, 3270 KiB  
Article
Stability and Sensitivity Analysis of the COVID-19 Spread with Comorbid Diseases
by Jonner Nainggolan and Moch. Fandi Ansori
Symmetry 2022, 14(11), 2269; https://doi.org/10.3390/sym14112269 - 29 Oct 2022
Cited by 10 | Viewed by 1939
Abstract
This research investigates a model of the spread of COVID-19 in Indonesia by paying attention to comorbid disease, self-quarantine, government-provided quarantine, and vaccination factors. The symmetrical aspects of the model are studied. The evaluation of the model reveals non-endemic and endemic equilibrium points [...] Read more.
This research investigates a model of the spread of COVID-19 in Indonesia by paying attention to comorbid disease, self-quarantine, government-provided quarantine, and vaccination factors. The symmetrical aspects of the model are studied. The evaluation of the model reveals non-endemic and endemic equilibrium points and the basic reproduction number (BRN). We provide the local and global stability analysis of the equilibriums. According to the sensitivity analysis of the BRN, the key parameters impacting the spread of COVID-19 are the susceptible recruitment rate, contact rate, infection death rate, and probability of infected individuals having no comorbidities. In addition, we provide a sensitivity analysis to examine the effect of parameter changes in each subpopulation. We discovered that the natural death rate is the most sensitive parameter based on the sensitivity index after reaching equilibrium. Symmetry aspects appear in some of the visualizations of the model’s solution and the sensitivity of the BRN and parameters. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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12 pages, 308 KiB  
Article
A Comparative Study of Three Mathematical Models for the Interaction between the Human Immune System and a Virus
by Florian Munteanu
Symmetry 2022, 14(8), 1594; https://doi.org/10.3390/sym14081594 - 3 Aug 2022
Cited by 2 | Viewed by 1591
Abstract
In this paper, we will consider three deterministic models for the study of the interaction between the human immune system and a virus: the logistic model, the Gompertz model, and the generalized logistic model (or Richards model). A qualitative analysis of these three [...] Read more.
In this paper, we will consider three deterministic models for the study of the interaction between the human immune system and a virus: the logistic model, the Gompertz model, and the generalized logistic model (or Richards model). A qualitative analysis of these three models based on dynamical systems theory will be performed by studying the local behavior of the equilibrium points and obtaining the local dynamics properties from the linear stability point of view. Additionally, we will compare these models in order to understand which is more appropriate to model the interaction between the human immune system and a virus. Some natural medical interpretations will be obtained, which are available for all three models and can be useful to the medical community. Full article
(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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