Mathematical Epidemiology Research Articles

A fractional order mathematical model for the omicron: a new variant of COVID-19

For the purpose of analyzing the Omicron pandemic, we build a novel SEIaIsIoHR mathematical model. The fundamental properties of the model are studied, including boundedness, uniqueness, and existence. The boundedness of the model’s solution is examined by solving the fractional Gronwall’s inequality using the Laplace transform method. Using the Picard-Lindelöf theorem, we verify the solution’s existence and uniqueness. The next-generation matrix is used to compute Ro (the basic reproduction number), which is significant in the mathematical modeling of infectious diseases. It is demonstrated that both the endemic and disease-free equilibrium solutions are globally and locally asymptotically stable. We also conducted a sensitivity analysis of the parameters based on Ro . Moreover, the model is extended by employing optimal control theory to examine the effects of certain control strategies. Finally, numerical simulations are performed to validate the model.

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

Mathematical modelling for pandemic preparedness in Canada: Learning from COVID-19.

The COVID-19 pandemic underlined the need for pandemic planning but also brought into focus the use of mathematical modelling to support public health decisions. The types of models needed (compartment, agent-based, importation) are described. Best practices regarding biological realism (including the need for multidisciplinary expert advisors to modellers), model complexity, consideration of uncertainty and communications to decision-makers and the public are outlined. A narrative review was developed from the experiences of COVID-19 by members of the Public Health Agency of Canada External Modelling Network for Infectious Diseases (PHAC EMN-ID), a national community of practice on mathematical modelling of infectious diseases for public health. Modelling can best support pandemic preparedness in two ways: 1) by modelling to support decisions on resource needs for likely future pandemics by estimating numbers of infections, hospitalized cases and cases needing intensive care, associated with epidemics of "hypothetical-yet-plausible" pandemic pathogens in Canada; and 2) by having ready-to-go modelling methods that can be readily adapted to the features of an emerging pandemic pathogen and used for long-range forecasting of the epidemic in Canada, as well as to explore scenarios to support public health decisions on the use of interventions. There is a need for modelling expertise within public health organizations in Canada, linked to modellers in academia in a community of practice, within which relationships built outside of times of crisis can be applied to enhance modelling during public health emergencies. Key challenges to modelling for pandemic preparedness include the availability of linked public health, hospital and genomic data in Canada.

On the use of reactive multiparticle collision dynamics to gather particulate level information from simulations of epidemic models

This paper discusses the application of reactive multiparticle collision (RMPC) dynamics, a particle-based method, to epidemic models. First, we consider a susceptible-infectious-recovered framework to obtain data on contacts of susceptibles with infectious people in a population. It is found that the number of contacts increases and the contact duration decreases with increases in the disease transmission rate and average population speed. Next, we obtain reinfection statistics for a general infectious disease from RMPC simulations of a susceptible-infectious-recovered-susceptible model. Finally, we simulate a susceptible-exposed-infectious-recovered model and gather the exposure, infection, and recovery time for the individuals in the population under consideration. It is worth mentioning that we can collect data in the form of average contact duration, average initial infection time, etc., from RMPC simulations of these models, which is not possible with population-based stochastic models, or deterministic systems. This study provides quantitative insights on the potential of RMPC to simulate epidemic models and motivates future efforts for its application in the field of mathematical epidemiology.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 < 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 < 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

A Study on the Institutional Context and Dynamics Surrounding the Emergence of SARS-CoV-2

This dissertation presents an in-depth analysis of the institutional context and dynamics surrounding the emergence of SARSCoV-2, the virus responsible for the COVID-19 pandemic. Focusing on the case of Brazil, the study employs a multi-disciplinary approach that integrates mathematical economics, epidemiology, and public health policy analysis. Key aspects include examining the initial outbreak in Wuhan, the virus's rapid global spread, and the specific impacts and responses in Brazil. The research employs mathematical models to understand the dynamics of virus transmission, the effectiveness of various public health interventions, and the socio-economic consequences of the pandemic. Special attention is given to Brazil's vaccination campaigns and their role in pandemic control. The findings offer valuable insights into the complexities of managing a public health crisis in a socio-economically diverse country like Brazil and underscore the importance of timely and coordinated responses to global health emergencies.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

Change of measure in a Heston–Hawkes stochastic volatility model

Abstract We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Global stability for SIR models with nonlinear incidence and removal functions

In this work, we provide some sufficient conditions to study the global asymptotic stability of the endemic equilibrium for certain models in mathematical epidemiology with nonlinear incidence and removal functions. We also present numerical examples in order to illustrate our results.

Evidence-based Decision Making: Infectious Disease Modeling Training for Policymakers in East Africa.

Mathematical modeling of infectious diseases is an important decision-making tool for outbreak control. However, in Africa, limited expertise reduces the use and impact of these tools on policy. Therefore, there is a need to build capacity in Africa for the use of mathematical modeling to inform policy. Here we describe our experience implementing a mathematical modeling training program for public health professionals in East Africa. We used a deliverable-driven and learning-by-doing model to introduce trainees to the mathematical modeling of infectious diseases. The training comprised two two-week in-person sessions and a practicum where trainees received intensive mentorship. Trainees evaluated the content and structure of the course at the end of each week, and this feedback informed the strategy for subsequent weeks. Out of 875 applications from 38 countries, we selected ten trainees from three countries - Rwanda (6), Kenya (2), and Uganda (2) - with guidance from an advisory committee. Nine trainees were based at government institutions and one at an academic organization. Participants gained skills in developing models to answer questions of interest and critically appraising modeling studies. At the end of the training, trainees prepared policy briefs summarizing their modeling study findings. These were presented at a dissemination event to policymakers, researchers, and program managers. All trainees indicated they would recommend the course to colleagues and rated the quality of the training with a median score of 9/10. Mathematical modeling training programs for public health professionals in Africa can be an effective tool for research capacity building and policy support to mitigate infectious disease burden and forecast resources. Overall, the course was successful, owing to a combination of factors, including institutional support, trainees' commitment, intensive mentorship, a diverse trainee pool, and regular evaluations.

On the Effects of Type II Left Censoring in Stable and Chaotic Compartmental Models for Infectious Diseases: Do Small Sample Estimates Survive Censoring?

In this paper, we discuss a selection of tools from dynamical systems and order statistics, which are most often utilized separately, and combine them into an algorithm to estimate the parameters of mathematical models for infectious diseases in the case of small sample sizes and left censoring, which is relevant in the case of rapidly evolving infectious diseases and remote populations. The proposed method relies on the analogy between survival functions and the dynamics of the susceptible compartment in SIR-type models, which are both monotone decreasing in time and are both determined by a dual variable: the hazard function in survival prediction and the number of infected people in SIR-type models. We illustrate the methodology in the case of a continuous model in the presence of noisy measurements with different distributions (Normal, Poisson, Negative Binomial) and in a discrete model, reminiscent of the Ricker map, which admits chaotic dynamics. This estimation procedure shows stable results in experiments based on a popular benchmark dataset for SIR-type models and small samples. This manuscript illustrates how classical theoretical statistical methods and dynamical systems can be merged in interesting ways to study problems ranging from more fundamental small sample situations to more complex infectious disease and survival models, with the potential that this tools can be applied in the presence of a large number of covariates and different types of censored data.

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.

Introduction to the special issue on mathematical economic epidemiology models

Introduction to the special issue on mathematical economic epidemiology models

Dynamic analysis and optimal control of co-infection system under different outbreak times of mutant strains

<p>In epidemic prevention efforts, the emergence of new virus strains due to mutations greatly complicates the prediction and management of epidemics. Most of the current mathematical models of infectious diseases assume that the mutant strain and the original strain have the same outbreak time, which is obviously an ideal situation. In order to make the study more practical, we consider the general situation of outbreaks of mutated strains. At the same time, the optimal control strategy under different emergence time of mutant strains was proposed by using the optimal control theory and numerical simulation. This study provides a new theoretical framework for the dual strain competition model with different outbreak times. The final theoretical results and numerical simulation showed that although the emergence time of the mutant did not affect the final trend of the epidemic, it would affect the cost of prevention and control during the control period.</p>

Mathematical Analysis of Malaria Epidemic: Asymptotic Stability With Cost‐Effectiveness Study

Malaria is an old, curable vector‐borne disease that is devastating in the tropics and subtropical regions of the world. The disease has unmatched complications in the human host, especially in children. Mathematical models of infectious diseases have been the steering wheel, driving scientists towards elucidation of the dynamic behaviour of epidemics and providing tailored strategic management of diseases. With the ongoing vaccination programs for vector‐borne diseases, the research proposes a nonlinear differential equation model for the malaria disease that provides public health with a shift from the classical understanding of nonpharmaceutical preventive malaria control to pharmaceutical measures of vaccines. The asymptotic dynamic behaviour of the model is studied at the model’s equilibria. The bifurcation type invoked at the disease‐free state is analysed, and the result revealed that the convention that is the condition for eradicating the disease is not always sufficient when the system undergoes backward bifurcation. Furthermore, sensitivity analysis was investigated to quantify the amount of influence each parameter has on . With the Latin hypercube sampling and partial rank correlation coefficient method, the uncertainty in is computed with a 95% confidence interval, with the mean, and 5th and 95th percentiles, respectively, simulated as 0.143788, 0.01545, and 0.41491. An intervention model was derived from the nonintervention model to experiment with and evaluate the respective effects of the various pairings of interventions on the dynamics of the disease. Lastly, an in‐depth cost analysis was studied to identify the most cost‐effective intervention regarding rewarding the desired outcome. From the analysis, we recommend that besides the nonpharmaceutical measure of bed nets and insecticide spray, public health should target the pharmaceutical intervention of vaccine as it can close the gap in malaria prevention.

Mathematical modeling of infectious diseases and the impact of vaccination strategies.

Mathematical modeling plays a crucial role in understanding and combating infectious diseases, offering predictive insights into disease spread and the impact of vaccination strategies. This paper explored the significance of mathematical modeling in epidemic control efforts, focusing on the interplay between vaccination strategies, disease transmission rates, and population immunity. To facilitate meaningful comparisons of vaccination strategies, we maintained a consistent framework by fixing the vaccination capacity to vary from 10 to 100% of the total population. As an example, at a 50% vaccination capacity, the pulse strategy averted approximately 45.61% of deaths, while continuous and hybrid strategies averted around 45.18 and 45.69%, respectively. Sensitivity analysis further indicated that continuous vaccination has a more direct impact on reducing the basic reproduction number $ R_0 $ compared to pulse vaccination. By analyzing key parameters such as $ R_0 $, pulse vaccination coefficients, and continuous vaccination parameters, the study underscores the value of mathematical modeling in shaping public health policies and guiding decision-making during disease outbreaks.

On mathematical model of infectious disease by using fractals fractional analysis

On mathematical model of infectious disease by using fractals fractional analysis

Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

The basic reproduction number R0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R0 is not a function of the original ODE model, unless we also include in it a certain (F,V) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (F,V) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (F,V) gradient decomposition (and hence of the resulting R0). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the (F,V) decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (F,V) decompositions are available as well.