Mathematical Epidemiology Research Articles

Role of fractional derivatives in the mathematical modeling of the transmission of Chlamydia in the United States from 1989 to 2019.

Nowadays, the mathematical modeling of infectious diseases is a big trend worldwide. The mathematical models help us to forecast future outbreaks of diseases in the presence of present data. In this article, we represent a model of the transmission of Chlamydia in the United States by using data from 1989 to 2019. In the formulation of the model, we used integer and fractional derivatives. Several graphs are plotted for the various possible cases of the given parameters. The aim of this paper is to justify how the mathematical models in terms of fractional derivatives have more degree of freedom to explore disease dynamics for a particular data set and capture memory effects. The separate parameter estimation for each value of the fractional order increases the novelty of this work. The use of a real-data set of Chlamydia in the United States makes this study more visible and important to the literature.

Convincing the “Herd” of immunity: Lessons from smallpox vaccination in 19th century Germany

Although vaccination is a cost-effective way to control infectious diseases, it is often met with popular resistance. Studying smallpox in 19th century Germany, this paper explores how economic incentives contribute to this phenomenon. The paper adds to the literature by combining mathematical epidemiology and unpublished archival evidence from two German states - Baden and Wurttemberg. The two states are an intriguing case because their initial conditions and vaccination laws were similar. Despite this, Baden experienced lower smallpox prevalence and higher vaccination uptake than Wurttemberg. The epidemiological model predicts that incentives to vaccinate decline rapidly when immunization reduces prevalence. The archival evidence reveals that Baden offset this decline by creating a public vaccination system which reduced costs for vaccinees and vaccinators alike. This suggests that the high fixed costs of centralized immunization policies can be compensated by economies of scale and popular acceptance.

Applications of the Delay Stochastic Simulation Algorithm (DSSA) in Mathematical Epidemiology

The calculation of the probability of a minor outbreak is crucial in analyzing a stochastic epidemic model. For stochastic epidemic models with fixed delays, the linear chain trick is applied to transform the delayed models into a family of ODE models with increasing shape parameters. We then prove that the well-established results on the probability of a minor outbreak for continuous-time Markov chain (CTMC) epidemic models also hold for the stochastic epidemic models with fixed delays. All theoretical results are verified by numerical simulations implemented by the delay stochastic simulation algorithm (DSSA) in Python. It is shown that DSSA is able to generate exact realizations for underlying delayed models in the context of mathematical epidemiology, and therefore, provides insights into the effect of delays during the outbreak phases of epidemics.

Improving Estimates of Social Contact Patterns for Airborne Transmission of Respiratory Pathogens.

Data on social contact patterns are widely used to parameterize age-mixing matrices in mathematical models of infectious diseases. Most studies focus on close contacts only (i.e., persons spoken with face-to-face). This focus may be appropriate for studies of droplet and short-range aerosol transmission but neglects casual or shared air contacts, who may be at risk from airborne transmission. Using data from 2 provinces in South Africa, we estimated age mixing patterns relevant for droplet transmission, nonsaturating airborne transmission, and Mycobacterium tuberculosis transmission, an airborne infection where saturation of household contacts occurs. Estimated contact patterns by age did not vary greatly between the infection types, indicating that widespread use of close contact data may not be resulting in major inaccuracies. However, contact in persons >50 years of age was lower when we considered casual contacts, and therefore the contribution of older age groups to airborne transmission may be overestimated.

MODELLING OF PATHOGENS IMPACT ON THE HUMAN DISEASE TRANSMISSION WITH OPTIMAL CONTROL STRATEGIES

This study concentrates on a nonlinear deterministic mathematical model for the impact of pathogens on human disease transmission with optimal control strategies. Both pathogen-free and coexistence equilibria are computed. The basic reproduction number R0, which plays a vital role in mathematical epidemiology, was derived. The qualitative analysis of the model revealed the scenario for both pathogen-free and coexistence equilibria together with R0. The local stability of the equilibria is established via the Jacobian matrix and Routh-Hurwitz criteria, while the global stability of the equilibria is proven by using an appropriate Lyapunov function. Also, the normalized sensitivity analysis has been performed to observe the impact of different parameters on R0. The proposed model is extended into optimal control problem by incorporating three control variables, namely, preventive measure variable based on separation of susceptible from contacting the pathogens, integrated vector management based on chemical, biological control, ... etc. to kill pathogens and their carriers, and supporting infective medication variable based on the care of the infected individual in quarantine center. Optimal disease control analysis is examined using Pontryagin minimum principle. Numerical simulations are performed depending on analytical results and discussed quantitatively.

Global stability of a HCV dynamics model with cellular proliferation and delay

In this work, we propose and investigate a delay cell population model of hepatitis C virus (HCV) infection with cellular proliferation, absorption effect and a nonlinear incidence function. First of all, after having shown the existence of the local solutions of our model, we show the existence of the global solutions and positivity. Moreover, we determine the uninfected equilibrium point and the basic reproduction rate R0, which is a threshold number in mathematical epidemiology. After showing the existence and uniqueness of the infected equilibrium point, we proceed to the study of the local and global stability of this equilibrium. We show that if R0 < 1, the uninfected equilibrium point is globally asymptotically stable, which means that the disease will disappear and if R0 > 1, we have a unique infected equilibrium that is globally asymptotically stable under some conditions. Finally, we perform some numerical simulations to illustrate the obtained theoretical results.

The reproductive numbers of SARS-CoV-2 vary depending on the compartmental model used during epidemiological modelling

We illustrate the correct way to compare reproductive numbers obtained from various compartmental models used in mathematical epidemiology.

Application of mathematical models of infectious diseases in the evaluation of COVID-19 transmission risk at mass gatherings

The SARS-CoV-2 Omicron variant is rampant in Europe and the United States, and the Delta variant has caused several small-scale outbreaks in China. It is particularly important to simulate the transmission risk of novel coronavirus pneumonia (COVID-19) during large-scale events, so as to ensure a good preparation of personnel, materials, isolation sites and other support work in advance. Taking the Beijing 2022 Winter Olympic Games as an example, this study introduces the use of mathematical models to simulate the entry risks, closed-loop risks and prevention and control measures of athletes, officials and other stakeholders of the Olympic Games. In the simulation results on January 19, 2022, the estimated number of Olympic Games-related infections who were identified at borders was 357 (95%CI: 153-568) and the observed number was 323. The estimated number of "seed" cases that entered the closed-loop of Olympics Games was 195 (95%CI: 43-335), and the observed number of cases in the closed-loop was 212. This study demonstrates the important role of mathematical models of infectious diseases in the pragmatic application of preventive medicine and public health.

On the issue of the directions of development of morbidity forecasting in the modern conditions of the COVID-19 pandemic

The issues of forecasting the spread of diseases have been of particular importance for a long time, even more than 100 years ago, but over the past two decades there has been an increased interest in mathematical epidemiology in the world and in connection with the emergence of new and poorly studied infectious diseases. The purpose of this study is to analyze the trends in the development of morbidity forecasting in the modern conditions of the COVID-19 pandemic. Within the framework of the study, current literary and statistical data were provided, the analysis was based on various scientific publications on predicting the spread of diseases of the population in the Russian Federation and in the world; analysis of the situation and methods of combating the spread of a new coronavirus infection, the main factors affecting the spread of diseases were identified, the main characteristics were given and their significance was determined, such methods as: studying and generalizing experience, bibliographic, informational and analytical, statistical. Thus, methodological approaches to predicting the spread of diseases cannot be used separately from the entire existing healthcare system. As a result, competent planning of preventive and anti-epidemic measures is necessary, as well as effective allocation of health resources, especially during epidemics and pandemics; and the results of the study can serve as materials for further study of problems related to the threat of the spread of diseases, scientific search for solutions to the problems of the organization of the health system, as well as in the educational process at the stage of higher and additional medical education.

Impact of contact data resolution on the evaluation of interventions in mathematical models of infectious diseases.

Computational models offer a unique setting to test strategies to mitigate the spread of infectious diseases, providing useful insights to applied public health. To be actionable, models need to be informed by data, which can be available at different levels of detail. While high-resolution data describing contacts between individuals are increasingly available, data gathering remains challenging, especially during a health emergency. Many models thus use synthetic data or coarse information to evaluate intervention protocols. Here, we evaluate how the representation of contact data might affect the impact of various strategies in models, in the realm of COVID-19 transmission in educational and work contexts. Starting from high-resolution contact data, we use detailed to coarse data representations to inform a model of SARS-CoV-2 transmission and simulate different mitigation strategies. We find that coarse data representations estimate a lower risk of superspreading events. However, the rankings of protocols according to their efficiency or cost remain coherent across representations, ensuring the consistency of model findings to inform public health advice. Caution should be taken, however, on the quantitative estimations of those benefits and costs triggering the adoption of protocols, as these may depend on data representation.

ANALYSIS OF A NONLINEAR DYNAMICAL MODEL OF HEPATITIS B DISEASE

Mathematical epidemiology holds prime importance for comprehending the dynamics of infectious diseases. Consequently, mathematical model of hepatitis B with fractional-order derivative under Caputo sense is primarily focused in this research. The analysis of the required solution is qualitatively derived by applying the fixed-point theory approach. By perturbing the proposed model, the Ulam–Hyer’s stability techniques are further derived. To achieve the iterative series solution of the proposed system of hepatitis, the modified Euler method like Taylor’s series method is utilized. For validation and importance of the fractional operators, sufficient significant numerical results at various fractional orders are presented and compared them with the integer order. It is inferred from this research that, by using the fractional-order method, the transmission mechanism of hepatitis B disease can be acutely revealed. This study may provide positive theoretical support for the prevention and treatment of hepatitis B disease.

New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the “R0 alternative” of Van den Driessche and Watmough, which states that the local stability condition of the disease-free equilibrium may be expressed as R0<1, where R0 is the famous basic reproduction number, which also plays a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite common that 1. When R0>1, there exists a unique fixed endemic point, and 2. the endemic point is locally stable when R0>1. One would like to establish these properties for a large class of realistic epidemic models (and we do not include here epidemics without casualties). We have introduced recently a “simple” but broad class of “SIR-PH models” with varying populations, with the express purpose of establishing for these processes the two properties above. Since that seemed still hard, we have introduced a further class of “SIR-PH-FA” models, which may be interpreted as approximations for the SIR-PH models, and which include simpler models typically studied in the literature (with constant population, without loss of immunity, etc.). For this class, the first “endemic law” above is “almost established”, as explicit formulas for a unique endemic point are available, independently of the number of infectious compartments, and it only remains to check its belonging to the invariant domain. This may yet turn out to be always verified, but we have not been able to establish that. However, the second property, the sufficiency of R0>1 for the local stability of an endemic point, remains open even for SIR-PH-FA models, despite the numerous particular cases in which it was checked to hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH and SIR-PH-FA, and also for a second, more refined “intermediate approximation” SIR-PH-IA. We illustrate the current status-quo by presenting new results on a generalization of the SAIRS epidemic model.

Incorporating global dynamics to improve the accuracy of disease models: Example of a COVID-19 SIR model.

Mathematical models of infectious diseases exhibit robust dynamics, such as stable endemic, disease-free equilibriums or convergence of the solutions to periodic epidemic waves. The present work shows that the accuracy of such dynamics can be significantly improved by including global effects of host movements in disease models. To demonstrate improved accuracy, we extended a standard Susceptible-Infected-Recovered (SIR) model by incorporating the global dynamics of the COVID-19 pandemic. The extended SIR model assumes three possibilities for susceptible individuals traveling outside of their community:• They can return to the community without any exposure to the infection.• They can be exposed and develop symptoms after returning to the community.• They can be tested positively during the trip and remain quarantined until fully recovered.To examine the predictive accuracy of the extended SIR model, we studied the prevalence of the COVID-19 infection in six randomly selected cities and states in the United States: Kansas City, Saint Louis, San Francisco, Missouri, Illinois, and Arizona. The extended SIR model was parameterized using a two-step model-fitting algorithm. The extended SIR model significantly outperformed the standard SIR model and revealed oscillatory behaviors with an increasing trend of infected individuals. In conclusion, the analytics and predictive accuracy of disease models can be significantly improved by incorporating the global dynamics of the infection.

Fuzzy fractional mathematical model of COVID-19 epidemic

In this paper, we develop a mathematical model with a Caputo fractional derivative under fuzzy sense for the prediction of COVID-19. We present numerical results of the mathematical model for COVID-19 of most three infected countries such as the USA, India and Italy. Using the proposed model, we estimate predicting future outbreaks, the effectiveness of preventive measures and potential control strategies of the infection. We provide a comparative study of the proposed model with Ahmadian’s fuzzy fractional mathematical model. The results demonstrate that our proposed fuzzy fractional model gives a nearer forecast to the actual data. The present study can confirm the efficiency and applicability of the fractional derivative under uncertainty conditions to mathematical epidemiology.

ARBO: Arbovirus modeling and uncertainty quantification toolbox.

The ongoing pandemic of COVID-19 has highlighted the importance of mathematical tools to understand and predict outbreaks of severe infectious diseases, including arboviruses such as Zika. To this end, we introduce ARBO, a package for simulation and analysis of arbovirus nonlinear dynamics. The implementation follows a minimalist style, and is intuitive and extensible to many settings of vector-borne disease outbreaks. This paper outlines the main tools that compose ARBO, discusses how recent research works about the Brazilian Zika outbreak have explored the package’s capabilities, and describes its potential impact for future works on mathematical epidemiology.

A computationally tractable birth-death model that combines phylogenetic and epidemiological data.

Inferring the dynamics of pathogen transmission during an outbreak is an important problem in infectious disease epidemiology. In mathematical epidemiology, estimates are often informed by time series of confirmed cases, while in phylodynamics genetic sequences of the pathogen, sampled through time, are the primary data source. Each type of data provides different, and potentially complementary, insight. Recent studies have recognised that combining data sources can improve estimates of the transmission rate and the number of infected individuals. However, inference methods are typically highly specialised and field-specific and are either computationally prohibitive or require intensive simulation, limiting their real-time utility. We present a novel birth-death phylogenetic model and derive a tractable analytic approximation of its likelihood, the computational complexity of which is linear in the size of the dataset. This approach combines epidemiological and phylodynamic data to produce estimates of key parameters of transmission dynamics and the unobserved prevalence. Using simulated data, we show (a) that the approximation agrees well with existing methods, (b) validate the claim of linear complexity and (c) explore robustness to model misspecification. This approximation facilitates inference on large datasets, which is increasingly important as large genomic sequence datasets become commonplace.

A Scoping Review of Drug Epidemic Models.

The phenomenon of drug epidemics has been a global issue in the past decades, causing enormous damages to the physical and mental health of drug users and social well-being. Despite great efforts to curb drug epidemics at the governmental or social level, the total number of drug users has still been on the rise in recent years, along with illicit production and trafficking around the world. Inspired by dynamical epidemic models of infectious disease, a flourishment of promising results has been observed in the exploration of drug epidemic models. In this review, we aim to provide a scoping review of all existing drug epidemic modeling studies, and it has been shown that most studies focused on analyses of theoretical behaviors of the model systems, lacking emphasis on practical applications in real settings. We found that the drug epidemic models were characterized by a longer time scale, no incubation period, no significant prevention vaccines interfered, and population specificity. This review could assist policymakers and public health workers in gaining deeper insights into modeling tools, and help modelers improve their works, thus narrowing gaps between mathematical epidemiology and public health studies.

Global research activity on mathematical modeling of transmission and control of 23 selected infectious disease outbreak

BackgroundMathematical analysis and modeling allow policymakers to understand and predict the dynamics of an infectious disease under several different scenarios. The current study aimed to analyze global research activity on mathematical modeling of transmission and control of several infectious diseases with a known history of serious outbreaks.MethodsRelevant publications were retrieved using a comprehensive validated search query. The database used was SciVerse Scopus. Indicators related to evolution, growth of publications, infectious diseases encountered, key players, citations, and international research collaboration were presented.ResultsThe search strategy found 5606. The growth of publications started in 1967 and showed a sharp rise in 2020 and 2021. The retrieved articles received relatively high citations (h-index = 158). Despite being multidisciplinary, Plos One journal made the highest contribution to the field. The main findings of the study are summarized as follows: (a) COVID-19 had a strong impact on the number of publications in the field, specifically during the years 2020 and 2021; (b) research in the field was published in a wide range of journals, mainly those in the field of infectious diseases and mathematical sciences; (c) research in the field was mainly published by scholars in the United States and the United Kingdom; (d) international research collaboration between active countries and less developed countries was poor; (e) research activity relied on research groups with a large number of researchers per group indicative of good author-author collaboration; (f) HIV/AIDS, coronavirus disease, influenza, and malaria were the most frequently researched diseases; (g) recently published articles on COVID-19 received the highest number of citations; and (h) researchers in the Eastern Mediterranian and South-East Asian regions made the least contribution to the retrieved articles.ConclusionMathematical modeling is gaining popularity as a tool for understanding the dynamics of infectious diseases. The application of mathematical modeling on new emerging infectious disease outbreaks is a priority. Research collaboration with less developed countries in the field of mathematical epidemiology is needed and should be prioritized and funded.

Fractal–fractional model and numerical scheme based on Newton polynomial for Q fever disease under Atangana–Baleanu derivative

Scientists and researchers are increasingly interested in mathematical modelling of infectious diseases with non-integer order. It is self-evident that a fixed order can only characterize classical models in epidemiology, but models with fractional-order derivatives are not fixed order. When a derivative has a non-fixed order, it becomes more potent in representing real-world problems. Different fresh techniques concerning fractional operators, such as the exponential decay and the Mittag-Leffler kernel, have been proposed in recent years, which transcend the constraints of prior fractional-order derivatives. These novel operators are functional in modelling problems in science and engineering—a more recent operator in fractional calculus, known as the fractal–fractional operator. Fractal–fractional operators have not been frequently utilized to explore complex dynamics of infectious disease spread in an ecosystem. In this paper, a novel fractal–fractional operator is used to study the dynamics of Q fever in livestock and ticks. Fixed point theorem is used to prove the model’s existence and uniqueness under the Atangana–Baleanu (Mittag-Leffler kernel) fractal–fractional operator. A non-linear analysis is used to determine the model’s Hyers–Ulam stability. A numerical scheme for the Q fever disease using the newly constructed numerical scheme based on Newton polynomial is presented. The numerical simulation of the model shows some exciting dynamics of the disease in an ecosystem. The fractal–fractional dynamism shows that a change in the geometric pattern of nature also affects the spread of Q fever. For example, it is noticed that the fractal dimensions and fractional order produces different asymptotic stabilities for symptomatic livestock, Coxiella burnetii in the environment, susceptible and infected ticks. It is of a view that this study will also contribute to the rich dynamics of the fractal and the fractional perspective of disease spread in nature.

Demand Forecasting in Hospital and ICUs using a Modified Propagation Dynamic Model: A novel GSEIR approach

Compartmentalized mathematical modeling of infectious diseases is essential in understanding the epidemic's spread in the population. It helps to identify the best strategy for control and prevention. In this paper, we propose an epidemiological model to estimate disease burden statistics of infected, recovered, and deceased cases as a function of time, including two sensitive compartments, namely hospitalization and intensive care unit admissions. To formulate and solve the flow mutation from one compartment to another, we use a non-linear least squares algorithm. Our main goal is to forecast infection rates’ outcome trends to alert health authorities of potential risks of hospital overload. The approach developed has been validated using COVID-19 data. The results obtained demonstrate the effectiveness and robustness of the model. The model can also be applied as a benchmark for assessing disease burden in the hospital environment.