Mathematical Models In Epidemiology Research Articles

Global dynamics of heterogeneous epidemic models with exponential and nonexponential latent period distributions

AbstractMany epidemic models assume an exponential distribution for the latent stage, but this may not accurately represent reality and could impact disease transmission predictions. Previous studies for short time scale models have shown that the choice of latency distribution affects estimates of the epidemic peak, time to peak, and infection eradication time, but has little effect on the final infection size. However, it is unclear if these conclusions hold for long time scale models. To address this, we investigate the impact of different latency distributions on disease dynamics in long‐term models, comparing them with short‐term models. We propose two susceptible‐exposed‐infected‐hospitalized‐recovered () models with multiple groups, using exponential and gamma distributions for latency. We derive the basic reproduction number () for both models and prove the global stability of the equilibrium points. We conduct numerical simulations and find that the gamma distribution may lead to larger epidemic peak sizes and longer peak times compared to the exponential distribution. However, the impact of latency distribution on estimating the peak and time to peak is smaller in long‐term models than in short‐term models. Additionally, the effect on the final total infected size is negligible regardless of the time scale. Therefore, when analyzing long‐term epidemic dynamics using heterogeneity models, the choice of latency distribution does not significantly affect the results. Assuming an exponential distribution for the latency is sufficient for simplifying the model and facilitating analysis. Our study provides valuable insights for selecting appropriate mathematical models in epidemiology.

Precision epidemiology at the nexus of mathematics and nanotechnology: Unraveling the dance of viral dynamics

The intersection of mathematical modeling, nanotechnology, and epidemiology marks a paradigm shift in our battle against infectious diseases, aligning with the focus of the journal on the regulation, expression, function, and evolution of genes in diverse biological contexts. This exploration navigates the intricate dance of viral transmission dynamics, highlighting mathematical models as dual tools of insight and precision instruments, a theme relevant to the diverse sections of Gene. In the context of virology, ethical considerations loom large, necessitating robust frameworks to protect individual rights, an aspect essential in infectious disease research. Global collaboration emerges as a critical pillar in our response to emerging infectious diseases, fortified by the predictive prowess of mathematical models enriched by nanotechnology. The synergy of interdisciplinary collaboration, training the next generation to bridge mathematical rigor, biology, and epidemiology, promises accelerated discoveries and robust models that account for real-world complexities, fostering innovation and exploration in the field. In this intricate review, mathematical modeling in viral transmission dynamics and epidemiology serves as a guiding beacon, illuminating the path toward precision interventions, global preparedness, and the collective endeavor to safeguard human health, resonating with the aim of advancing knowledge in gene regulation and expression.

Predictive models for health outcomes due to SARS-CoV-2, including the effect of vaccination: a systematic review

BackgroundThe interaction between modelers and policymakers is becoming more common due to the increase in computing speed seen in recent decades. The recent pandemic caused by the SARS-CoV-2 virus was no exception. Thus, this study aims to identify and assess epidemiological mathematical models of SARS-CoV-2 applied to real-world data, including immunization for coronavirus 2019 (COVID-19).MethodologyPubMed, JSTOR, medRxiv, LILACS, EconLit, and other databases were searched for studies employing epidemiological mathematical models of SARS-CoV-2 applied to real-world data. We summarized the information qualitatively, and each article included was assessed for bias risk using the Joanna Briggs Institute (JBI) and PROBAST checklist tool. The PROSPERO registration number is CRD42022344542.FindingsIn total, 5646 articles were retrieved, of which 411 were included. Most of the information was published in 2021. The countries with the highest number of studies were the United States, Canada, China, and the United Kingdom; no studies were found in low-income countries. The SEIR model (susceptible, exposed, infectious, and recovered) was the most frequently used approach, followed by agent-based modeling. Moreover, the most commonly used software were R, Matlab, and Python, with the most recurring health outcomes being death and recovery. According to the JBI assessment, 61.4% of articles were considered to have a low risk of bias.InterpretationThe utilization of mathematical models increased following the onset of the SARS-CoV-2 pandemic. Stakeholders have begun to incorporate these analytical tools more extensively into public policy, enabling the construction of various scenarios for public health. This contribution adds value to informed decision-making. Therefore, understanding their advancements, strengths, and limitations is essential.

Introduction to the special issue on mathematical economic epidemiology models

Introduction to the special issue on mathematical economic epidemiology models

Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

COVID-19 Pandemisinin Kompartman Modelleri: Sistematik Bir Literatür Taraması

As COVID-19 rapidly spread all around the world, different methods have been proposed to explore the dynamics of the pandemic, understand the transmission mechanism, and assess the preventive measures. Mathematical models are frequently used worldwide to predict various parameters and develop effective policies for disease control. Compartmental models are the most popular mathematical models in epidemiology. These models divide the population into distinct groups (compartments) based on their status and describe the movement of an individual from one compartment to another. Various compartmental models and their variations have been developed to model the pandemic dynamics and measure the efficiency and necessity of different initiatives such as lockdowns, face masks, and vaccination. This paper provides a systematic literature review on different compartmental models proposed to model the COVID-19 pandemic. These models are discussed in detail based on the compartmental structure in the model, aim of the model, variables, and methodological approaches.

Network-based uncertainty quantification for mathematical models in epidemiology

After the new Coronavirus disease (COVID-19) emerged in the end of January 2020 in Germany, a large number of individuals suffered from severe symptoms and eventually needed intensive care in hospitals. Due to the rapid spread of the disease, the number of deceased individuals increased as well, which is a motivation to prevent as many new infections as possible. Therefore, the knowledge about the current evolution of the virus spread is crucial to predict its future behavior and to react with suitable interventions. In this paper, the evolution of the COVID-19 pandemic in Germany is forecasted by a network-based inference method, in which the interactions of individuals are taken into account using a contact matrix. Then the results are compared to the predictions without considering a contact matrix as well as to the logistic regression, which shows the advantage of incorporating the contact matrix. Furthermore, the basic reproduction number of the pandemic in Germany using a neural network approach is estimated and used for further predictions of the evolution of COVID-19 in Germany. In order to mathematically model the different compartments of the population in the considered regions, the classical SIR model is employed. In this work, we deploy the LASSO (Least Absolute Shrinkage and Selection Operator) for the unknown parameter estimation. Furthermore, we calculate and illustrate the MAPE (Mean Absolute Percentage Error) of the estimations to show the accuracy of the predictions. The results include model parameter estimation and model validation, as well as the outbreak forecasting using network-informed algorithms. Our findings show that the network-inference based approach outperforms the logistic regression as well as the neural network approach and the SIR model calibration without a contact network. Furthermore according to the results, the network-inference based approach is particularly suitable for short- to mid-term predictions, even when there is not much information about the new disease. Moreover, the predictions based on the estimation of the reproduction number in Germany can yield more reliable results with increasing the availability of data, but could not outperform the network-inference based algorithm.

Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology

The review is devoted to an analysis of mathematical models used for describing epidemic processes. Our main focus is on the models that are based on partial differential equations (PDEs), especially those that were developed and used for the COVID-19 pandemic modeling. Most of our attention is given to the studies in which not only results of numerical simulations are presented but analytical results as well. In particular, traveling fronts (waves), exact solutions, and the estimation of key epidemic parameters of the epidemic models with governing PDEs (typically reaction–diffusion equations) are discussed. The review may serve as a valuable resource for researchers and practitioners in the field of mathematical modeling in epidemiology.

Optimal control analysis on the impact of non-pharmaceutical interventions and vaccination on the dynamics of COVID-19

COVID-19 is a serious issue, claiming millions of lives across the globe. Starting from the beginning of the pandemic, different controlling strategies, such as non-pharmaceutical interventions and vaccination, have been adapted to mitigate the virus. In this paper, we developed an epidemiological mathematical model to assess the transmission dynamics of COVID-19 with interventions. The impact of non-pharmaceutical interventions and vaccination of vulnerable individuals on its transmission has been discussed in the developed model. We have proved the mathematical well-posedness and epidemiological meaningfulness of the model. The disease free and endemic equilibrium points are calculated, and the stability analysis of the equilibria have been proved by obtaining the control as well as the basic reproduction numbers of the model. Furthermore, an optimal control analysis is implemented to assess the impacts of employing control interventions. The controls are employed as a time dependent self protective measures, vaccinating individuals, and treating the infected ones. Then we proved the existence and uniqueness of optimal controls, and analyzed the problem using Pontryagin’s Maximum Principle. We estimated the parameter values of the proposed model with collected epidemiological data. Different simulation cases were performed to compare the analytical results and identify the most appropriate control intervention strategies. After implementing the proposed control measures, and performing the efficiency analysis of the control strategies, we obtained a promising result for reducing the number of infected individuals by implementing the control measures properly.

Dynamics of Competitive Two-Strain Stochastic SIR Epidemics on Heterogeneous Networks

Mathematical modeling in epidemiology, biology, and life sciences requires the use of stochastic models. In this paper, we derive a competitive two-strain stochastic SIR epidemic model by considering the change in state of the epidemic process due to an event. Based on the density-dependent process theory, we construct a six-dimensional deterministic model that can be used to describe the diffusion limit of the stochastic epidemic on a heterogeneous network. Furthermore, we show the explicit expressions for the variances of infectious individuals with strain 1 and strain 2 when the level of infection is increasing exponentially. In particular, we find that the expressions of the variances are symmetric. Finally, simulations for epidemics spreading on networks are performed to confirm our analytical results. We find a close agreement between the simulations and theoretical predictions.

Mathematical Model of the Spread of Hantavirus Infection.

A mathematical epidemiological model incorporating the mobility of rodents and human groups among zones of less or major contact between them is presented. The hantavirus infection dynamics is expressed using a model type SEIR (Susceptible-Exposed-Infectious-Removed), which incorporates the displacement of the rodent and the human, between the urban and rural sector, the latter being subdivided in populated and non-populated. The results show the impact that rodent or human displacement may have on the propagation of hantavirus infection. Human mobility is more significant than rodents in increasing the number of hantavirus infection cases. The results found may be used as a reference by the health authorities to develop more specific campaigns on the territorial dynamics of the rodent, attend to the mobility of humans in these territories, mainly agricultural and forestry workers, and strengthen control-prevention actions in the community, to prevent future outbreaks that are fatal.

Epidemic modeling using differential equations with implementation in R

The COVID-19 pandemic, like past historical events such as the Vietnam War or 9/11, will shape a generation. Mathematics educators can seize this unprecedented opportunity to teach the principles of mathematical modeling in epidemiology. Compartmental epidemiological models, such as the SIR (susceptible-infected-recovered), are widely used by researchers and educators to study and teach infectious disease dynamics, particularly for COVID-19. The SIR model is considered a fundamental mathematical model for the spread of epidemic disease and employs Ordinary Differential Equations to determine the number of individuals in each compartment of a population at a given time. In this note, we demonstrate how to build a basic SIR model. We show how to simulate, plot, and interpret the results using numerical simulation with R, a popular programming language widely used in data analysis and scientific research.

An optimal control model for Covid-19 spread with impacts of vaccination and facemask

A non-linear system of differential equations was used to explain the spread of the COVID-19 virus and a SEIQR model was developed and tested to provide insights into the spread of the pandemic. This article, which is related to the aforementioned work as well as other work covering variations of SIR models, Hermite Wavelets Transform, and also the Generalized Compartmental COVID-19 model, we develop a mathematical control model and apply it to represent optimal vaccination strategy against COVID-19 using Pontryagin's Maximum Principle and also factoring in the effect of facemasks on the spread of the virus. As background work, we analyze the mathematical epidemiology model with the facemask effect on both reproduction number and stability, we also analyze the difference between confirmed COVID-19 cases of the Quarantine class and anonymous cases of the Infectious class that is expected to recover. We also apply control theory to mine insights for effective virus spread prevention strategies. Our models are validated using Matlab mathematical model validation tools. Statistical tests against data from Jordan are used to validate our work including the modeling of the relation between the facemask effect and COVID-19 spread. Furthermore, the relation between control measure ξ, cost, and Infected cases is also studied.

Environmental conditions modulate plant virus vertical transmission and survival of infected seeds.

Seed transmission is a major mode for plant virus persistence and dispersal, as it allows virus survival within the seed in unfavorable conditions and facilitate spread when they become more favorable. To access these benefits, viruses require infected seeds to remain viable and germinate in altered environmental conditions, which may be also advantageous for the plant. However, how environmental conditions and virus infection affect seed viability, and whether these effects modulate seed transmission rate and plant fitness, are unknown. To address these questions, we utilized turnip mosaic virus (TuMV), cucumber mosaic virus (CMV) and Arabidopsis thaliana as model systems. Using seeds from plants infected by these viruses, we analyzed seed germination rates, as a proxy of seed viability, and virus seed transmission rate under standard and altered temperature, CO2 and light intensity. With this data, we developed and parameterized a mathematical epidemiological model to explore the consequences of the observed alterations on virus prevalence and persistence. Altered conditions generally reduced overall seed viability and increased virus transmission rate as compared to standard conditions, which indicated that under environmental stress infected seeds are more viable. Hence, virus presence may be beneficial for the host. Subsequent simulations predicted that enhanced viability of infected seeds and higher virus transmission rate may increase virus prevalence and persistence in the host population under altered conditions. This work provides novel information on the influence of the environment in plant virus epidemics.

An algebraic formula, deep learning and a novel SEIR-type model for the COVID-19 pandemic.

The most extensively used mathematical models in epidemiology are the susceptible-exposed-infectious-recovered (SEIR) type models with constant coefficients. For the first wave of the COVID-19 epidemic, such models predict that at large times equilibrium is reached exponentially. However, epidemiological data from Europe suggest that this approach is algebraic. Indeed, accurate long-term predictions have been obtained via a forecasting model only if it uses an algebraic as opposed to the standard exponential formula. In this work, by allowing those parameters of the SEIR model that reflect behavioural aspects (e.g. spatial distancing) to vary nonlinearly with the extent of the epidemic, we construct a model which exhibits asymptoticly algebraic behaviour. Interestingly, the emerging power law is consistent with the typical dynamics observed in various social settings. In addition, using reliable epidemiological data, we solve in a numerically robust way the inverse problem of determining all model parameters characterizing our novel model. Finally, using deep learning, we demonstrate that the algebraic forecasting model used earlier is optimal.

The Identifiability of Mathematical Models in Epidemiology: Tuberculosis, HIV, COVID-19

The paper is devoted to the short review and application of sensitivity-based identifiability approaches for analyzing mathematical models of epidemiology and related processes described by systems of differential equations and agent-based models. It is shown that for structural identifiability of basic SIR models (describe the dynamic of Susceptible, Infected and Removed groups based on nonlinear ordinary differential equations) of epidemic spread and linear compartmental models it is possible to use a priori information about the process. It is demonstrated that a model can be structurally identifiable but be practically non-identifiable due to incomplete data. The paper uses methods for analyzing the sensitivity of parameters to data variation, as well as analyzing the sensitivity of model states to parameter variation, based on linear and differential algebra, Bayesian, and Monte Carlo approaches. It was shown that in the SEIR-HCD model of COVID-19 propagation, described by a system of seven ordinary differential equations and based on the mass balance law, the parameter of humoral immunity acquisition is the least sensitive to changes in the number of diagnosed, critical and mortality cases of COVID-19. The spatial SEIR-HCD model of COVID-19 propagation demonstrated an increase the sensitivity of the partial immunity duration parameter over time, as well as a decrease in the limits of change in the infectivity and infection parameters. In the case of the SEIR-HCD mean-field model of COVID-19 propagation, the sensitivity of the system to the self-isolation index and the lack of sensitivity of the stochastic parameters of the system are shown. In the case of the agent-based COVID-19 propagation model, the change in the infectivity parameter was reduced by more than a factor of 2 compared to the statistics. A differential model of co-infection HIV and tuberculosis spread with multiple drug resistance was developed and its local identifiability was shown.

Mathematical model of coffee tree’s rust control using snails as biological agents

Coffee rust is one of the main diseases that affect coffee plantations worldwide, causing large-scale ecological and economic damage. While multiple methods have been proposed to tackle this challenge, using snails as biological agents have shown to be the most consistent and promising approach. However, snails are an invasive species, and overusing them can cause devastating outcomes. In this paper, we develop and explore an ecological–epidemiological mathematical model for the coffee tree rust pandemic control using snails as biological agents. We analyze the equilibria of the proposed system with their stability properties. In addition, we perform numerical analysis to obtain the sensitivity of the system to different changes and manipulation of the snails pandemic control, under specific conditions. Finally, we propose an in silico mechanism to obtain an analytical connection between the system’s initial condition and the number of snails needed to optimally control the rust pandemic spread while preventing the snail population to grow unmanageably. Our model can be used to optimize the usage of snails as biological agents to control the rust pandemic in spatially-small areas, by predicting the number of snails one needs to introduce to the ecosystem in order to obtain a desired outcome.

Hybrid discrete-time-continuous-time models and a SARS CoV-2 mystery: Sub-Saharan Africa's low SARS CoV-2 disease burden.

Worldwide, the recent SARS-CoV-2 virus has infected more than 670 million people and killed nearly 67.0 million. In Africa, the number of confirmed COVID-19 cases was approximately 12.7 million as of January 11, 2023, that is about 2% of the infections around the world. Many theories and modeling techniques have been used to explain this lower-than-expected number of reported COVID-19 cases in Africa relative to the high disease burden in most developed countries. We noted that most epidemiological mathematical models are formulated in continuous-time interval, and taking Cameroon in Sub-Saharan Africa, and New York State in the USA as case studies, in this paper we developed parameterized hybrid discrete-time-continuous-time models of COVID-19 in Cameroon and New York State. We used these hybrid models to study the lower-than-expected COVID-19 infections in developing countries. We then used error analysis to show that a time scale for a data-driven mathematical model should match that of the actual data reporting.

Modified SEIRD model for describing the COVID-19 epidemic

This article is devoted to mathematical models in epidemiology, in particular SIR, SEIR, and SEIRD models. It explores the importance of these models in predicting the spread of infectious diseases and evaluating the effectiveness of control measures. These models allow for assessing important epidemic parameters such as the speed of infection transmission, the number of people infected, and the number of deaths. This data can help in making decisions regarding the imposition and lifting of quarantine restrictions, opening and closing of schools and other institutions, as well as in developing vaccination strategies and other control measures. In summary, mathematical models such as SIR, SEIR, and SEIRD are important tools in the fight against epidemics. They enable epidemiologists and medical professionals to predict and control the spread of diseases, thus preserving the health and lives of people.

A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host

In this paper, we construct a generalized epidemiological mathematical model to study toxoplasmosis dynamics, taking into consideration both cat and mouse populations. The model incorporates generalized proportions for the congenital transmission in the mouse and cat populations, along with the oocysts available in the environment. We focus on determining the conditions under which toxoplasmosis can be eradicated. We conduct a stability analysis in order to reveal the dynamics of toxoplasmosis in the cat and mouse populations; moreover, we compute the basic reproduction number R0, which is crucial for the long-term behavior of the toxoplasmosis disease in these populations as well as the steady states related to both populations. We find that vertical transmission in the cat population is essential, and affects the basic reproduction number R0. If full vertical transmission is considered in the mouse population and R0<1, we find that all solutions converge to the limit set comprised by the infinitely many toxoplasmosis-free-cat steady states, meaning that toxoplasmosis would vanish from the cat population regardless of the initial conditions. On the other hand, if R0>1, then there is only one toxoplasmosis-endemic steady state. When full vertical transmission is not considered in the mouse population, then a unique toxoplasmosis-free equilibrium exists and toxoplasmosis can be eradicated for both the cat and mouse populations. This has important public health implications. Numerical simulations are carried out to reinforce our theoretical stability analysis and observe the repercussion of some parameters on the dynamics.