Multi-group Epidemic Model Research Articles

Stochastic multi-group epidemic SVIR models: Degenerate case

This work considers a multi-group epidemic SIR models with vaccination. The fluctuation of the environment is taken into account by introducing both color noise and white noise to a compartmental model. Unlike existing results on stochastic multigroup models, which were not successful in finding the reproduction numbers and fully classify the longtime behaviors of the models, we will provide a formula for the reproduction number R0 of our model and will show that the disease is persistent if R0>1 while the disease will be eradicated if R0≤1. We also provide the explicit formulae for R0 in some special cases. The formulae will be useful to determine the herb immunity threshold of the disease, which can be used to make right and timely policy to control a disease.

DECIPHERING DYNAMICS OF COVID-19 OUTBREAK IN INDIA: AN AGE-STRUCTURED MODEL

The dynamics of infectious disease transmission depends on social contact patterns and the precautions people take to minimize disease transmission. The social contact pattern varies depending on the community at large age distribution at work, school, and recreation. Consequently, knowing the age-specific prevalence and incidence of infectious diseases is critical for predicting future disease burden and the efficacy of interventions like vaccination. In this study, we use an SEIR age-structured multi-group epidemic model to understand how social contact affects disease control. We construct location-specific social contact matrices to observe that how social mixing affects disease spread. For mathematical analysis, we compute the basic reproduction number [Formula: see text] and exhibit the global behavior of the system in terms of [Formula: see text]. We also estimate the transmission rate using the empirical data of COVID-19 for India from 1 September 2020, to 31 December 2020. We simulate the lifting of non-pharmaceutical interventions by allowing people to return to work in phases and investigate its impact. Our findings imply that raising awareness in symptomatic infected individuals aged 20–49 years can help in reducing the number of infected individuals. When some schools are partially open, awareness in symptomatic infected individuals in school-age groups also plays a significant role in reducing disease cases. The simulation results also suggest that limiting contact at school and other gathering places could significantly lower the number of instances. Using the least square approach, we show that the time-dependent transmission rate is more realistic than the constant spread rate for COVID-19 in India. To reduce the COVID-19 burden, we obtain that gradually loosening control measures could flatten and lower other peaks. Our findings may help health policymakers to decide on timely age-based vaccination distribution strategies and control measures.

Multi-patch epidemic models with partial mobility, residency, and demography

The emergence and re-emergence of infectious diseases has been a global cause of concern in the past few decades. Previous research in the field has revealed that human connectivity and mobility behavior play a major role in the spreading of an infectious disease. In this work, we propose multi-patch models that take into account the effects of human mobility on the evolution of disease dynamics in a multi-population environment. In particular, we develop SEIRS multi-patch and multi-group epidemic models, extending the work of Bichara et al. (2015) and Bichara and Abderrahman (2018) to practically account for distinct epidemiological-status-dependent mobilities in each patch. We rigorously show that the disease free equilibria (DFE) for both models are stable when R0≤1. We also prove that the models have a unique endemic asymptotically stable equilibrium when R0>1. In addition, we introduce new local reproduction numbers from the point of view of the sub-populations, and establish some important relation between them and the global reproduction number. Various numerical simulations are conducted to study the effects of mobility and the residence time matrix on the evolution of the disease in individual patches and the overall environment.

Global stability of multi-group epidemic model with distributed delays and indirect transmission

Global stability of multi-group epidemic model with distributed delays and indirect transmission

Global stability of a multi-group delayed epidemic model with logistic growth

<abstract><p>In this work, we aim to investigate the mechanism of a multi-group epidemic model taking into account the influences of logistic growth and delay time distribution. Despite the importance of the logistic growth effect in such models, its consideration remains rare. We show that $ \mathcal{R}_0 $ has a crusher role in the global stability of a disease-free and endemic equilibria. That is, if $ \mathcal{R}_0 $ is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable, whereas, if $ \mathcal{R}_0 $ is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable. In addition, we construct suitable Lyapunov functions to investigate the global stability of disease-free and endemic equilibria. Finally, we introduce numerical simulations of the model.</p></abstract>

Matching theory and evidence on Covid-19 using a stochastic network SIR model.

SummaryThis paper develops an individual‐based stochastic network SIR model for the empirical analysis of the Covid‐19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate the identification and estimation of the transmission rates. The paper then proposes a method that jointly estimates the transmission rate and the magnitude of under‐reporting of infected cases. Empirical evidence on six European countries matches the simulated outcomes once the under‐reporting of infected cases is addressed. It is estimated that the number of actual cases could be between 4 to 10 times higher than the reported numbers in October 2020 and declined to 2 to 3 times in April 2021. The calibrated models are used in the counterfactual analyses of the impact of social distancing and vaccination on the epidemic evolution and the timing of early interventions in the United Kingdom and Germany.

Multi-patch multi-group epidemic model with varying infectivity

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $ L $ distinct patches (with migrations between them) and $ K $ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d.The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $ {\bf {D}} $, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.

Matching Theory and Evidence on COVID-19 Using a Stochastic Network SIR Model

This paper develops an individual-based stochastic network SIR model for the empirical analysis of the Covid-19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate the identification and estimation of the transmission rates. The paper then proposes a method that jointly estimates the transmission rate and the magnitude of under-reporting of infected cases. Empirical evidence on six European countries matches the simulated outcomes once the under-reporting of infected cases is addressed. It is estimated that the number of actual cases could be between 4 to 10 times higher than the reported numbers in October 2020 and declined to 2 to 3 times in April 2021. The calibrated models are used in the counterfactual analyses of the impact of social distancing and vaccination on the epidemic evolution, and the timing of early interventions in the UK and Germany.

Vaccination of a multi-group model of zoonotic diseases with direct and indirect transmission

Vaccination plays an important role in preventing or reducing the spread of zoonotic diseases. In this paper, we develop a susceptible-vaccinated-exposed-infectious-pathogen multi-group epidemic model of zoonotic diseases incorporating nonlinear direct and indirect incidence rates, nonlinear pathogen shedding rates, and common environmental contamination. Under certain assumptions, we first obtained the basic reproduction number of the model. Then, we utilized the comparison principle and global Lyapunov function method to prove global stability of dynamical equilibria. Finally, we analyzed optimal vaccination strategy. All the theoretical predictions were verified by numerical simulations.

Threshold dynamics of a multi-group SEAR alcoholism model with public health education

In this paper, we investigate the threshold dynamics of a multi-group SEAR alcoholism epidemic model with public health education. The multi-group model allows us to describe interactions both within-group and inter-group separately. We prove that the basic reproduction number [Formula: see text] plays the role of a threshold for the long-time behavior of the model. The alcohol-free equilibrium [Formula: see text] of the model is globally asymptotically stable if [Formula: see text], while the alcohol-present equilibrium [Formula: see text] exists uniquely and is globally asymptotically stable if [Formula: see text]. For the proofs of main results, we use the classical method of Lyapunov functions and apply subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory. Our results expand the previous results which have been obtained in single-group models. Sensitivity analysis and numerical simulations are also performed to illustrate our results.

Global stability of multi-group SIR epidemic model with group mixing and human movement.

In this paper, an SIR multi-group epidemic model with group mixing and human movement is investigated. The control reproduction number ℜυ is derived and the global dynamics of the model are completely determined by the value of ℜυ. By using the graph-theoretical approach, the results show that the disease-free equilibrium is globally asymptotically stable if ℜυ<1, and the unique endemic equilibrium is globally asymptotically stable if ℜυ>1. Two numerical examples are further presented to testify the validity of the theoretical results.

Control Strategies in Multigroup Models: The Case of the Star Network Topology.

In this paper, we propose control strategies for multigroup epidemic models. We use compartmental [Formula: see text] models to study the dynamics of n host groups sharing the same source of infection in addition to the transmission among members of the same group. In particular, we consider a model for infectious diseases with free-living pathogens in the environment and a metapopulation model with a central patch. We give the detailed derivation of the target reproduction number under three public health interventions and provide the corresponding biological insights. Moreover, using the next-generation approach, we calculate the basic reproduction numbers associated with subsystems of our models and determine algebraic connections to the target reproduction number of the complete model. The analysis presented here illustrates that understanding the topological structure of the infection process and partitioning it into simple cycles is useful to design and evaluate the control strategies.

A multigroup SEIR epidemic model with age‐dependent latency and relapse

Different multigroup epidemic models have been proposed, but few models include the latent class that becomes infectious at different rates and the fact that the removed class may relapse into an infectious class at different rates. In this paper, a multigroup SEIR epidemic model is constructed to study the transmission dynamics of infectious diseases with age‐dependent latency and relapse. The model is realistic for some infectious diseases, such as TB and herpes virus. The sharp threshold condition, which corresponds to the well‐known basic reproduction number , is derived, and it determines the global stability of each equilibrium. In particular, if , then the disease‐free equilibrium is globally asymptotically stable, whereas if , the endemic equilibrium exists uniquely and is globally asymptotically stable. We utilize appropriate Lyapunov functionals, graph‐theoretical results, and the LaSalle's invariance principle to prove these results. Two specific examples and their corresponding numerical simulations are provided to explain the obtained results.

Multi-patch and multi-group epidemic models: a new framework.

We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number [Formula: see text] is derived. The effects of heterogeneity in groups, patches and mobility patterns on [Formula: see text] and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.

A nonstandard finite difference scheme for a multi-group epidemic model with time delay

In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if mathfrak{R}_{0}leq1, then the disease-free equilibrium P_{0} is globally asymptotically stable; if mathfrak{R}_{0}>1, then the infection equilibrium P_{*} is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model.

Modelling and analysis of global resurgence of mumps: A multi-group epidemic model with asymptomatic infection, general vaccinated and exposed distributions

Based on the possible mechanism of the recent global resurgence of mumps, a novel multi-group SVEIAR epidemic model with infinite distributed delays of vaccination and latency, asymptomatic infection and nonlinear incidence is formulated, which is also equivalent to the multi-group version with ages of vaccination and latency. Here, it is assumed that the vaccine confers complete protection against infection. By constructing Lyapunov functionals, it is shown that the disease will die out if the vaccinated reproduction number Rv≤1 and the disease becomes endemic if Rv>1. Moreover, the threshold dynamics of the multi-group SVEIAR epidemic model of ordinary differential equations (ODEs) with imperfect vaccination is also established. Our main conclusions generalize and improve some existing results. In the end, it is found that the possible reasons for occurrence of backward bifurcation in epidemic models with vaccination may include partial immunity, standard incidence by comparing our results obtained with the existing literatures.

Stability preserving NSFD scheme for a multi-group SVIR epidemic model

A discrete multi-group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number R0vac by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

Asymptotic behavior of stochastic multi-group epidemic models with distributed delays

Abstract In this paper, we introduce stochasticity into multi-group epidemic models with distributed delays and general kernel functions. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by using the method of stochastic Lyapunov functions, we carry out a detailed analysis on the asymptotic behavior of the stochastic model regarding of the basic reproduction number R 0 . If R 0 ≤ 1 , the solution of the stochastic system oscillates around the disease-free equilibrium E 0 , while if R 0 > 1 , the solution of the stochastic model fluctuates around the endemic equilibrium E ∗ . Moreover, we also establish sufficient conditions of these results.

Modeling direct and indirect disease transmission using multi-group model

The survival of pathogens outside the host in the environment is an important factor for some diseases transmission. Environment-to-individuals is an indirect mode of transmission, which also plays a role in the dynamics of some disease. In this paper, we proposed a general multi-group epidemic model with nonlinear direct and indirect transmission incidence rates, nonlinear pathogen shedding rates, and common environmental contamination for indirect transmission. Under the certain assumptions, the basic reproduction number of the model is identified. We proved the global stability of the equilibria by using global Lyapunov functions with the specific coefficients and graph-theoretical approach theorem, which are determined by the basic reproduction number. The main result of our model is that we give the specific coefficients of global Lyapunov functions. From the discussion of our model, we conclude that our model contains earlier cholera models, brucellosis models, and other general disease multi-group models as special cases.