Compartmental Epidemic Model Research Articles

Stability analysis of a [formula omitted] epidemic compartmental model with saturated incidence rate, vaccination and elimination strategies

In this study, we developed and analyzed a compartmental epidemic model and a system model that incorporate vaccination, elimination, and quarantine techniques, with a saturated incidence rate, by theoretical and numerical means. The model is described by a system of four nonlinear differential equations. Our analysis included determining the reproduction number Rq and examining equilibrium solutions. The outcomes of the disease are identified through the threshold Rq. When Rq<1, the disease-free equilibrium is globally asymptotically stable, as proved by the LaSalle invariance principle and disease extinction analysis. However, using the Routh–Hurwitz criterion, we have proved that the disease-free equilibrium is not stable, and when Rq>1, the unique endemic equilibrium is asymptotically stable locally. The stability of the endemic and disease-free equilibria globally has been examined using the Routh–Hurwitz and Dulac criteria. Subsequently, numerical simulations were utilized to illustrate the theoretical findings effectively.

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.

Instabilities and self–organization in spatiotemporal epidemic dynamics driven by nonlinearity and noise

Theoretical analysis of epidemic dynamics has attracted significant attention in the aftermath of the COVID–19 pandemic. In this article, we study dynamic instabilities in a spatiotemporal compartmental epidemic model represented by a stochastic system of coupled partial differential equations (SPDE). Saturation effects in infection spread–anchored in physical considerations–lead to strong nonlinearities in the SPDE. Our goal is to study the onset of dynamic, Turing–type instabilities, and the concomitant emergence of steady–state patterns under the interplay between three critical model parameters–the saturation parameter, the noise intensity, and the transmission rate. Employing a second–order perturbation analysis to investigate stability, we uncover both diffusion–driven and noise–induced instabilities and corresponding self–organized distinct patterns of infection spread in the steady state. We also analyze the effects of the saturation parameter and the transmission rate on the instabilities and the pattern formation. In summary, our results indicate that the nuanced interplay between the three parameters considered has a profound effect on the emergence of dynamical instabilities and therefore on pattern formation in the steady state. Moreover, due to the central role played by the Turing phenomenon in pattern formation in a variety of biological dynamic systems, the results are expected to have broader significance beyond epidemic dynamics.

Compartmental Nonlinear Epidemic Disease Model with Mixed Behavior

Compartmental Nonlinear Epidemic Disease Model with Mixed Behavior

Economic evaluation of two-Strain covid-19 compartmental epidemic model with pharmaceutical and non-pharmaceutical interventions and spatio-temporal patterns

This paper is a novel attempt to analyze different aspects of a two-strain epidemic model. The paper introduces a novel approach to analyzing a two-strain epidemic model, emphasizing the efficacy of combining non-pharmaceutical interventions and vaccination, particularly against new variants. Additionally, it presents a unique spatio-temporal model to assess spatial distribution of infections, offering fresh insights into how spatial factors influence disease transmission and control. The analysis involves stability analysis(local and global) and optimal control. Time-dependent social/non-pharmaceutical interventions coupled with vaccination of the susceptible class in the presence of both strains are analyzed using Pontryagin’s Maximum Principle. The numerical section shows the behavior of the infected class with and without control, the control intensity trend for the scenario when only non-pharmaceutical interventions (NPI) are practiced for the original strain, and when both NPI and vaccination are incorporated with the emergence of the new strain. The evaluation of the Incremental average ratio (IAR) and Incremental cost-effectiveness ratio (ICER) determines that NPI and vaccination as a combination is better and ideal in terms of costs incurred and effectiveness as a whole in averting infection in the presence of a new variant. Finally, we have also proposed a spatio-temporal pattern for the new strain model to analyze patterns using the finite element method by PDE Toolbox to show the effect of different initial conditions and geometry on the density of the infected population.

Task-oriented machine learning surrogates for tipping points of agent-based models

We present a machine learning framework bridging manifold learning, neural networks, Gaussian processes, and Equation-Free multiscale approach, for the construction of different types of effective reduced order models from detailed agent-based simulators and the systematic multiscale numerical analysis of their emergent dynamics. The specific tasks of interest here include the detection of tipping points, and the uncertainty quantification of rare events near them. Our illustrative examples are an event-driven, stochastic financial market model describing the mimetic behavior of traders, and a compartmental stochastic epidemic model on an Erdös-Rényi network. We contrast the pros and cons of the different types of surrogate models and the effort involved in learning them. Importantly, the proposed framework reveals that, around the tipping points, the emergent dynamics of both benchmark examples can be effectively described by a one-dimensional stochastic differential equation, thus revealing the intrinsic dimensionality of the normal form of the specific type of the tipping point. This allows a significant reduction in the computational cost of the tasks of interest.

Temporal logics for compartmental models

Abstract This paper introduces two logic frameworks for the study of SIR (Susceptible-Infected-Recovered) and SIRS compartmental epidemic models, one based on Linear Temporal Logic and the other on Computation Tree Logic. We provide a short literature overview on compartmental models and other related works using logics, and then define our logics with their respective axiomatizations, and demonstrate their soundness and completeness proofs.

Population dynamics and games of variable size

This work introduces the concept of Variable Size Game Theory (VSGT), in which the number of players in a game is a strategic decision made by the players themselves. We start by discussing the main examples in game theory: dominance, coexistence, and coordination. We show that the same set of pay-offs can result in coordination-like or coexistence-like games depending on the strategic decision of each player type. We also solve an inverse problem to find a d-player game that reproduces the same fixation pattern of the VSGT. In the sequel, we consider a game involving prosocial and antisocial players, i.e., individuals who tend to play with large groups and small groups, respectively. In this game, a certain task should be performed, that will benefit one of the participants at the expense of the other players. We show that individuals able to gather large groups to perform the task may prevail, even if this task is costly, providing a possible scenario for the evolution of eusociality. The next example shows that different strategies regarding game size may lead to spontaneous separation of different types, a possible scenario for speciation without physical separation (sympatric speciation). In the last example, we generalize to three types of populations from the previous analysis and study compartmental epidemic models: in particular, we recast the SIRS model into the VSGT framework: Susceptibles play 2-player games, while Infectious and Removed play a 1-player game. The SIRS epidemic model is then obtained as the replicator equation of the VSGT. We finish with possible applications of VSGT to be addressed in the future.

Global dynamics and optimal control of a nonlinear fractional-order cholera model

In this article, a fractional-order epidemic model for cholera is proposed and analyzed. Two transmission routes for cholera are considered to develop the compartmental epidemic model. The basic biological properties of the solutions of the fractional-order model are investigated. The global asymptotic stability of the equilibrium points have been established using appropriate Lyapunov functional. Moreover, a fractional-order control problem is presented, and its analytical solution is derived using Pontryagin’s maximum principle. Also, some graphical visualizations of the theoretical results are provided. It is found that the factional-order derivative only affect the time to reach the stationary states. Sensitivity analysis reveals that by reducing the rates of new recruitment and both the disease transmission rates, it may be possible to reduce the value of the basic reproduction number.

Global dynamics and optimal control of a nonlinear fractional-order cholera model

In this article, a fractional-order epidemic model for cholera is proposed and analyzed. Two transmission routes for cholera are considered to develop the compartmental epidemic model. The basic biological properties of the solutions of the fractional-order model are investigated. The global asymptotic stability of the equilibrium points have been established using appropriate Lyapunov functional. Moreover, a fractional-order control problem is presented, and its analytical solution is derived using Pontryagin’s maximum principle. Also, some graphical visualizations of the theoretical results are provided. It is found that the factional-order derivative only affect the time to reach the stationary states. Sensitivity analysis reveals that by reducing the rates of new recruitment and both the disease transmission rates, it may be possible to reduce the value of the basic reproduction number.

Estimation of Improvements in Mortality in Spectrum Among Adults With HIV Receiving Antiretroviral Therapy in High-Income Countries.

Mortality rates for people living with HIV (PLHIV) on antiretroviral therapy (ART) in high-income countries continue to decline. We compared mortality rates among PLHIV on ART in Europe for 2016-2020 with Spectrum's estimates. The AIDS Impact Module in Spectrum is a compartmental HIV epidemic model coupled with a demographic population projection model. We used national Spectrum projections developed for the 2022 HIV estimates round to calculate mortality rates among PLHIV on ART, adjusting to the age/country distribution of PLHIV starting ART from 1996 to 2020 in the Antiretroviral Therapy Cohort Collaboration (ART-CC)'s European cohorts. In the ART-CC, 11,504 of 162,835 PLHIV died. Between 1996-1999 and 2016-2020, AIDS-related mortality in the ART-CC decreased from 8.8 (95% CI: 7.6 to 10.1) to 1.0 (0.9-1.2) and from 5.9 (4.4-8.1) to 1.1 (0.9-1.4) deaths per 1000 person-years among men and women, respectively. Non-AIDS-related mortality decreased from 9.1 (7.9-10.5) to 6.1 (5.8-6.5) and from 7.0 (5.2-9.3) to 4.8 (4.3-5.2) deaths per 1000 person-years among men and women, respectively. Adjusted all-cause mortality rates in Spectrum among men were near ART-CC estimates for 2016-2020 (Spectrum: 7.02-7.47 deaths per 1000 person-years) but approximately 20% lower in women (Spectrum: 4.66-4.70). Adjusted excess mortality rates in Spectrum were 2.5-fold higher in women and 3.1-3.4-fold higher in men in comparison to the ART-CC's AIDS-specific mortality rates. Spectrum's all-cause mortality estimates among PLHIV are consistent with age/country-controlled mortality observed in ART-CC, with some underestimation of mortality among women. Comparing results suggest that 60%-70% of excess deaths among PLHIV on ART in Spectrum are from non-AIDS causes.

Drivers of social influence in the Twitter migration to Mastodon

The migration of Twitter users to Mastodon following Elon Musk’s acquisition presents a unique opportunity to study collective behavior and gain insights into the drivers of coordinated behavior in online media. We analyzed the social network and the public conversations of about 75,000 migrated users and observed that the temporal trace of their migrations is compatible with a phenomenon of social influence, as described by a compartmental epidemic model of information diffusion. Drawing from prior research on behavioral change, we delved into the factors that account for variations of the effectiveness of the influence process across different Twitter communities. Communities in which the influence process unfolded more rapidly exhibit lower density of social connections, higher levels of signaled commitment to migrating, and more emphasis on shared identity and exchange of factual knowledge in the community discussion. These factors account collectively for 57% of the variance in the observed data. Our results highlight the joint importance of network structure, commitment, and psycho-linguistic aspects of social interactions in characterizing grassroots collective action, and contribute to deepen our understanding of the mechanisms that drive processes of behavior change of online groups.

Modeling the Dynamic Effects of Human Mobility and Airborne Particulate Matter on the Spread of COVID-19

Identifying the relationship between human mobility, air pollution, and communicable disease poses a challenge for impact evaluation and public health planning. Specifically, Coronavirus disease 2019 (COVID-19) and air pollution from fine particulates (PM2.5), by which human mobility is mediated in a public health emergency. To describe the interplay between human mobility and PM2.5 during the spread of COVID-19, we proposed a nonlinear model of the time-dependent transmission rate as a function of these factors. A compartmental epidemic model, together with daily confirmed case data in Bangkok, Thailand during 2020–2021, was used to estimate the intrinsic parameters that can determine the impact on the transmission dynamic of the two earlier outbreaks. The results suggested a positive association between mobility and transmission, but this was strongly dependent on the context and the temporal characteristics of the data. For the ascending phase of an epidemic, the estimated coefficient of mobility variable in the second wave was greater than in the first wave, but the value of the mobility component in the transmission rate was smaller. Due to the influence of the baseline value and PM2.5, the estimated basic reproduction number of the second wave was higher than that of the first wave, even though mobility had a greater influence. For the descending phase, the value of the mobility component in the second wave was greater, due to the negative value of the estimated mobility coefficient. Despite this scaling effect, the results suggest a negative association between PM2.5 and the transmission rates. Although this conclusion agrees with some previous studies, the true effect of PM2.5 remains inconclusive and requires further investigation.

Modeling Study of the Effects of Ageratum conyzoides on the Transmission and Control of Citrus Huanglongbing.

Ageratum conyzoides (A. conyzoides) is commonly found or intentionally planted in citrus orchards due to its ability to provide habitat and breeding grounds for the natural enemies of citrus pests. This study aims to expand from a switching Huanglongbing model by incorporating the effects of A. conyzoides, vector preferences for settling, and pesticide application intervals on disease transmission. Additionally, we establish the basic reproduction number R0 and its calculation for a general switching compartmental epidemic model. Theoretical findings demonstrate that the basic reproduction number serves as a threshold parameter to characterize the dynamics of the models: if R0<1, the disease will disappear, whereas if R0>1, it will spread. Numerical results indicate that the recruitment rate of A. conyzoides not only affects the spread speed of Huanglongbing but also leads to paradoxical effects. Specifically, in cases of high infection rates, a low recruitment rate of A. conyzoides can result in a decrease, rather than an increase, in the basic reproduction number. Conversely, a high recruitment rate can accelerate the spread of Huanglongbing. Furthermore, we show how different vector bias and pesticide spraying periods affect the basic reproduction number.

Investigating the fractional dynamics and sensitivity of an epidemic model with nonlinear convex rate

The current study analyzes a fractional compartmental epidemic model for the transmission of infectious disease in a community. This study uses the Caputo derivative as well as Atangana–Baleanu derivative to consider a SIR fractional epidemic model with a convex incidence rate. We demonstrate the existence and uniqueness of solutions by using the idea of fixed-point theory. Comprehensive mathematical approaches show the physical properties of solutions, such as non-negativity and boundedness. The basic reproduction number is evaluated using the next-generation matrix method, determining whether the infection will spread in society. The stability analysis is performed with phase portraits, which show that the infection-free and infection-present steady states of the considered model are locally and globally asymptotically stable. The sensitivity of the basic reproduction number is carried out through 3D graphs, which indicate the most sensitive and impactful parameters. Finally, for the numerical solution of the underlying model, a numerical technique is applied and these solutions evaluate the theoretical results through numerical simulations. Numerical simulations that aim to decrease the fractional parameter and convex incidence rate can bring about a significant change in infected individuals.

Replicating superspreader dynamics with compartmental models

Infectious disease outbreaks often exhibit superspreader dynamics, where most infected people generate no, or few secondary cases, and only a small fraction of individuals are responsible for a large proportion of transmission. Although capturing this heterogeneity is critical for estimating outbreak risk and the effectiveness of group-specific interventions, it is typically neglected in compartmental models of infectious disease transmission—which constitute the most common transmission dynamic modeling framework. In this study we propose different classes of compartmental epidemic models that incorporate transmission heterogeneity, fit them to a number of real outbreak datasets, and benchmark their performance against the canonical superspreader model (i.e., the negative binomial branching process model). We find that properly constructed compartmental models can capably reproduce observed superspreader dynamics and we provide the pathogen-specific parameter settings required to do so. As a consequence, we also show that compartmental models parameterized according to a binary clinical classification have limited support.

Pathwise methods for the integration of a stochastic SVIR model

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed.

Wastewater-based modeling, reconstruction, and prediction for COVID-19 outbreaks in Hungary caused by highly immune evasive variants

(Motivation)Wastewater-based epidemiology (WBE) has emerged as a promising approach for monitoring the COVID-19 pandemic, since the measurement process is cost-effective and is exposed to fewer potential errors compared to other indicators like hospitalization data or the number of detected cases. Consequently, WBE was gradually becoming a key tool for epidemic surveillance and often the most reliable data source, as the intensity of clinical testing for COVID-19 drastically decreased by the third year of the pandemic. Recent results suggests that the model-based fusion of wastewater measurements with clinical data and other indicators is essential in future epidemic surveillance. (Method)In this work, we developed a wastewater-based compartmental epidemic model with a two-phase vaccination dynamics and immune evasion. We proposed a multi-step optimization-based data assimilation method for epidemic state reconstruction, parameter estimation, and prediction. The computations make use of the measured viral load in wastewater, the available clinical data (hospital occupancy, delivered vaccine doses, and deaths), the stringency index of the official social distancing rules, and other measures. The current state assessment and the estimation of the current transmission rate and immunity loss allow a plausible prediction of the future progression of the pandemic. (Results)Qualitative and quantitative evaluations revealed that the contribution of wastewater data in our computational epidemiological framework makes predictions more reliable. Predictions suggest that at least half of the Hungarian population has lost immunity during the epidemic outbreak caused by the BA.1 and BA.2 subvariants of Omicron in the first half of 2022. We obtained a similar result for the outbreaks caused by the subvariant BA.5 in the second half of 2022. (Applicability)The proposed approach has been used to support COVID management in Hungary and could be customized for other countries as well.

Simulating the Spread of Peste des Petits Ruminants in Kazakhstan Using the North American Animal Disease Spread Model

In this study, we simulated the potential spread of Peste des Petits Ruminants (PPR) between small ruminant (SR) farms in the Republic of Kazakhstan (RK) in case of the disease’s introduction into the country. The simulation was based on actual data on the location and population of SR farms in the RK using the North American Animal Disease Spread Model (NAADSM). The NAADSM employs the stochastic simulations of the between-farm disease spread predicated on the SIR compartmental epidemic model. The most important epidemiological indicators of PPR, demography of SR farms, and livestock management characteristics in the RK were used for model parameterization. This article considers several scenarios for the initial introduction of PPR into the territory of Kazakhstan, based on previously identified high-risk regions and varying sizes of initially infected farms. It is demonstrated that the duration and size of the outbreak do not depend on the size of initially infected farms but rather depend on the livestock concentration and number of farms in the affected area. This implies that the outbreak may affect the largest number of farms in the case of introduction of the disease into farms in southern Kazakhstan. However, even in the most unfavorable scenario, the total number of affected farms does not exceed 2.4% of all SR farms in the RK. The size of the affected area is, in most cases, no larger than an averaged 2-level administrative division’s size, which suggests the scale of a local epidemic. The chosen model provides ample opportunity to study the impact of different control and prevention measures on the spread of PPR as well as to assess the potential economic damage.

A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models

In this paper, we show how to modify a compartmental epidemic model, without changing the dimension, such that separable static heterogeneity is taken into account. The derivation is based on the Kermack–McKendrick renewal equation.