Compartmental Models in Epidemiology

In this chapter, the compartmental models for disease transmission are described and analyzed. At the outset, models for epidemics are illustrated, showing how to calculate the basic reproduction number and the final size of the epidemic. The authors also study models with multiple compartments, including treatment or isolation of infectives. Subsequently, they consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. They conclude by studying the age of infection models which give a unifying framework for more complicated compartmental models.

Robust optimal control of compartmental models in epidemiology: Application to the COVID-19 pandemic

Compartmental models in epidemiology characterize the spread of an infectious disease by formulating ordinary differential equations to quantify the rate of disease progression through subpopulations defined by the Susceptible-Infectious-Removed (SIR) scheme. The classic rate law central to the SIR compartmental models assumes that the rate of transmission is first order regarding the infectious agent. The current study demonstrates that this assumption does not always hold and provides a theoretical rationale for a more general rate law, inspired by mixed-order chemical reaction kinetics, leading to a modified mathematical model for non-first-order kinetics. Using observed data from 127 countries during the initial phase of the COVID-19 pandemic, we demonstrated that the modified epidemic model is more realistic than the classic, first-order-kinetics based model. We discuss two coefficients associated with the modified epidemic model: transmission rate constant k and transmission reaction order n. While k finds utility in evaluating the effectiveness of control measures due to its responsiveness to external factors, n is more closely related to the intrinsic properties of the epidemic agent, including reproductive ability. The rate law for the modified compartmental SIR model is generally applicable to mixed-kinetics disease transmission with heterogeneous transmission mechanisms. By analyzing early-stage epidemic data, this modified epidemic model may be instrumental in providing timely insight into a new epidemic and developing control measures at the beginning of an outbreak.

Compartmental models in epidemiology are widely used as a means to model disease spread mechanisms and understand how one can best control the disease in case an outbreak of a widespread epidemic occurs. However, a significant challenge within the community is in the development of approaches that can be used to rigorously verify and validate these models. In this paper, we present an approach to quantify and verify the behavioral properties of compartmental epidemiological models under several common modeling scenarios including: birth/death rates and multi-host/pathogen species. We build a workflow that uses metamorphic testing, novel visualization tools and model checking to gain insights into the functionality of compartmental epidemiological models. Our initial results indicate that metamorphic testing can be used to verify the implementation of these models and provide insights into special conditions where these mathematical models may fail. The visualization front-end allows the end-user to scan through a variety of parameters commonly used in these models to elucidate the conditions under which an epidemic can occur. Furthermore, specifying these models using a process algebra allows one to automatically construct behavioral properties that can be rigorously verified using model checking. Together, our approach allows for detecting implementation errors as well as handling conditions under which compartmental epidemiological models may fail to provide insights into disease spread dynamics.

Compartmental models have gained significant attention not only in public health studies but also in fields such as Operations Research (OR), social sciences, and logistics, particularly following the COVID-19 pandemic. Their broad applicability in epidemiology and their utility in understanding, predicting, and controlling the global spread of infectious diseases have made them indispensable across various disciplines. The appeal of these models lies in their simplicity yet effectiveness in capturing the essential dynamics of disease transmission. This paper provides a comprehensive review of compartmental models, focusing on the Susceptible-Infectious-Recovered (SIR) models and the key aspects of their structure. The primary objective of this review is to enhance the ability of researchers and practitioners to understand and manage infectious disease outbreaks through a twofold approach: (1) an evaluation of the assumptions, equations, and methodologies used for estimating critical parameters in SIR models, and (2) an exploration of the relationship between SIR models and optimization models. Additionally, a systematic micro-level review has identified the most significant research gaps in the literature on compartmental models, leading to recommendations for future research. A key finding emphasizes the need to revisit various assumptions to clarify the connection between SIR models and optimization approaches, which is expected to offer valuable insights for epidemic disease modeling.

Cybersecurity protects computer data, programs, systems, and networks from unauthorized access, attacks, or theft. By studying cyberattacks, cybersecurity professionals gain insights into attackers’ tactics, techniques, and methods, which are crucial for developing effective defense strategies and preventing future attacks. This paper introduces SERDUX-MARCIM, a model for simulation, modeling, and analyzing cyberattacks’ propagation in maritime infrastructure, considering network-specific characteristics and target and attacker capabilities. This proposal is supported by a simulation environment in Matlab and Netlogo, considering some of the most accepted cyber risk assessment methodologies and compartmental models in epidemiology. Considering the complexities of the maritime sector. SERDUX-MARCIM is also validated through extensive experimentation in different attack scenarios that represent real-world cyber campaigns in the maritime sector, showing the effectiveness of our proposal.

Hamiltonian structure of compartmental epidemiological models

Compartmental epidemic models have been widely used for predicting the course of epidemics, from estimating the basic reproduction number to guiding intervention policies. Studies commonly acknowledge these models’ assumptions but less often justify their validity in the specific context in which they are being used. Our purpose is not to argue for specific alternatives or modifications to compartmental models, but rather to show how assumptions can constrain model outcomes to a narrow portion of the wide landscape of potential epidemic behaviors. This concrete examination of well‐known models also serves to illustrate general principles of modeling that can be applied in other contexts.

Epidemiological models play a vital role in understanding the spread and severity of a pandemic of infectious disease, such as the COVID-19 global pandemic. The mathematical modeling of infectious diseases in the form of compartmental models are often employed in studying the probable outbreak growth. Such models heavily rely on a good estimation of the epidemiological parameters for simulating the outbreak trajectory. In this paper, the parameter estimation is formulated as an optimization problem and a metaheuristic algorithm is applied, namely Harmony Search (HS), in order to obtain the optimized epidemiological parameters. The application of HS in epidemiological modeling is demonstrated by implementing ten variants of HS algorithm on five COVID-19 data sets that were calibrated with the prototypical Susceptible-Infectious-Removed (SIR) compartmental model. Computational experiments indicated the ability of HS to be successfully applied to epidemiological modeling and as an efficacious estimator for the model parameters. In essence, HS is proposed as a potential alternative estimation tool for parameters of interest in compartmental epidemiological models.

Replica exchange (RE, or called parallel tempering) method can be used as a super simulated annealing. This chapter presents an effective global search algorithm in the use of replica exchange strategy refined by SA. Markov chain Monte Carlo (MCMC) (Andrieu et al., Mach Learn 50(1–2):5–43, 2003; Baldi and Brunak, Bioinformatics: the machine learning approach, 2nd edn. MIT, Cambridge, 2001; Bootsma and Ferguson, Proc Natl Acad Sci U S A 104(18):7588–7593, 2007; Iba, Int J Mod Phys C 12(5):623–656, 2001) algorithms are sampling from probability distributions based on constructing a Markov chain (Ross, Introduction to probability models, 9th edn. Elsevier Science & Technology Books Publisher, 2006) that has the desired distribution as its equilibrium distribution (Wikipedia, the free encyclopedia (en.wikipedia.org/wiki/): Epidemic model, Compartmental models in epidemiology, Mathematical modelling of infectious disease, Markov chain Monte Carlo, Parallel tempering, Metropolis-Hastings algorithm, etc. (and references therein)). The sampling strategy is very critical for a successful MCMC algorithm. However, in practice, the MCMC sampling methods such as Gibbs sampling (Baldi and Brunak, Bioinformatics: the machine learning approach, 2nd edn. MIT, Cambridge, 2001, Chapter 4.5) (from this reference we may know that Gibbs sampling can be rewritten as a Metropolis algorithm), Metropolis-Hastings (MH) algorithm (Baldi and Brunak, Bioinformatics: the machine learning approach, 2nd edn. MIT, Cambridge, 2001, Chapter 4.5), Multiple-try Metropolis (MM) algorithm sometimes just randomly walk (Ross, Introduction to probability models, 9th edn. Elsevier Science & Technology Books Publisher, 2006) and take a long time to explore all the solution space, will often double back and cover ground already covered, and usually own a slow algorithm convergence. In this chapter a more efficient sampling strategy of simulated annealing (Kirkpatrick et al., Science 220(4598):671–680, 1983)-refined RE (Earl and Deem, Phys Chem Chem Phys 7:3910–3916, 2005; Li et al., App Math Comput 212(1):216–228, 2009; Li et al., Parallel Comput 35(5):269–283, 2009; Swendsen and Wang, Phys Rev Lett 57(21):2607–2609, 1986; Thachuk et al., BMC Bioinformatics 8:342–362, 2007) is enclosed into the MCMC simulation.

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.

In this chapter, we discuss the basics of compartmental models in epidemiology and requisite analysis.

We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the τ -leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples.

We propose a novel use of generative adversarial networks (GANs) (i) to make predictions in time (PredGAN) and (ii) to assimilate measurements (DA-PredGAN). In the latter case, we take advantage of the natural adjoint-like properties of generative models and the ability to simulate forwards and backwards in time. GANs have received much attention recently, after achieving excellent results for their generation of realistic-looking images. We wish to explore how this property translates to new applications in computational modelling and to exploit the adjoint-like properties for efficient data assimilation. We apply these methods to a compartmental model in epidemiology that is able to model space and time variations, and that mimics the spread of COVID-19 in an idealised town. To do this, the GAN is set within a reduced-order model, which uses a low-dimensional space for the spatial distribution of the simulation states. Then the GAN learns the evolution of the low-dimensional states over time. The results show that the proposed methods can accurately predict the evolution of the high-fidelity numerical simulation, and can efficiently assimilate observed data and determine the corresponding model parameters.

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading.