Differential susceptibility epidemic models

Modelling the impact of precaution on disease dynamics and its evolution

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction numberR 0 < 1 . ForR 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics.

Characterization of differential susceptibility and differential infectivity epidemic models

Heterogeneity in susceptibility and infectivity is a central issue in epidemiology. Although the latter has received some attention recently, the former is often neglected in modeling of epidemic systems. Moreover, very few studies consider both of these heterogeneities. This paper is concerned with the characterization of epidemic models with differential susceptibility and differential infectivity under a general setup. Specifically, we investigate the global asymptotic behavior of equilibria of these systems when the network configuration of the Susceptible-Infectious interactions is strongly connected. These results prove two conjectures by Bonzi et al. (J Math Biol 62:39-64, 2011) and Hyman and Li (Math Biosci Eng 3:89-100, 2006). Moreover, we consider the scenario in which the strong connectivity hypothesis is dropped. In this case, the model exhibits a wider range of dynamical behavior, including the rise of boundary and interior equilibria, all based on the topology of network connectivity. Finally, a model with multidirectional transitions between infectious classes is presented and completely analyzed.

Effects of heterogeneous susceptibility on epidemiological models of reinfection

This paper studies an epidemic model with heterogeneous susceptibility which generalizes the SIS (susceptible-infected-susceptible), SIR (susceptible-infected-recovered) and SIRI (susceptible-infected-recovered-infected) models. The proposed model considers the case that some infected people are susceptible again after recovery, some infected people develop immunity after infection, and some infected people are reinfected after recovery. We perform a comprehensive theoretical analysis of the model, showing that under appropriate initial conditions, delayed outbreak phenomenon occurs that can give people false impressions. Moreover, compared with the SIRI model, the proposed model exists the delayed outbreak phenomenon under more probable conditions. Finally, we present a numerical example to illustrate the effectiveness of the theoretical results.

Stability behavior of a two-susceptibility SHIR epidemic model with time delay in complex networks

Taking two susceptible groups into account, we formulate a modified subhealthy-healthy-infected-recovered (SHIR) model with time delay and nonlinear incidence rate in networks with different topologies. Concretely, two dynamical systems are designed in homogeneous and heterogeneous networks by utilizing mean field equations. Based on the next-generation matrix and the existence of a positive equilibrium point, we derive the basic reproduction numbers R 0 1 and R 0 2 which depend on the model parameters and network structure. In virtue of linearized systems and Lyapunov functions, the local and global stabilities of the disease-free equilibrium points are, respectively, analyzed when R 0 1 < 1 in homogeneous networks and R 0 2 < 1 in heterogeneous networks. Besides, we demonstrate that the endemic equilibrium point is locally asymptotically stable in homogeneous networks in the condition of R 0 1 > 1 . Finally, numerical simulations are performed to conduct sensitivity analysis and confirm theoretical results. Moreover, some conjectures are proposed to complement dynamical behavior of two systems.

Cyber Physical Systems Dependability Using CPS-IOT Monitoring

Recently, vast investments have been made worldwide in developing Cyber-Physical Systems (CPS) as solutions to key socio-economic challenges. The Internet-of-Things (IoT) has also enjoyed widespread adoption, mostly for its ability to add “sensing” and “actuation” capabilities to existing CPS infrastructures. However, attention must be paid to the impact of IoT protocols on the dependability of CPS infrastructures. We address the issues of CPS dependability by using an epidemic model of the underlying dynamics within the CPS’ IoT subsystem (CPS-IoT) and an interference-aware routing reconfiguration. These help to efficiently monitor CPS infrastructure—avoiding routing oscillation, while improving its safety. The contributions of this paper are threefold. Firstly, a CPS orchestration model is proposed that relies upon: (i) Inbound surveillance and outbound actuation to improve dependability and (ii) a novel information diffusion model that uses epidemic states and diffusion sets to produce diffusion patterns across the CPS-IoT. Secondly, the proposed CPS orchestration model is numerically analysed to show its dependability for both sensitive and non-sensitive applications. Finally, a novel interference-aware clustering protocol called “INMP”, which enables network reconfiguration through migration of nodes across clusters, is proposed. It is then bench-marked against prominent IoT protocols to assess its impact on the dependability of the CPS.

Epidemic dynamics with a time-varying susceptibility due to repeated infections

In this paper, we consider the impact of heterogeneous susceptibility on disease transmission dynamics, using a simple mathematical model formulated by a system of ordinary differential equations. Our model describes a time-varying immunity level, which is enhanced by reinfection and diminished by waning immunity. Through mathematical analysis, we show that unexpected outbreaks, called delayed outbreaks, occur even when the basic reproduction number is less than one. A reallocation of susceptibility at the individual level, by repeated infections, in the host population induces delayed outbreaks.

Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis

The long and binding treatment of tuberculosis (TB) at least 6-8 months for the new cases, the partial immunity given by BCG vaccine, the loss of immunity after a few years doing that strategy of TB control via vaccination and treatment of infectious are not sufficient to eradicate TB. TB is an infectious disease caused by the bacillus Mycobacterium tuberculosis. Adults are principally attacked. In this work, we assess the impact of vaccination in the spread of TB via a deterministic epidemic model (SVELI) (Susceptible, Vaccinated, Early latent, Late latent, Infectious). Using the Lyapunov-Lasalle method, we analyse the stability of epidemic system  (SVELI) around the equilibriums (disease-free and endemic). The global asymptotic stability of the unique endemic equilibrium whenever [Formula: see text] is proved, where [Formula: see text] is the reproduction number. We prove also that when [Formula: see text] is less than 1, TB can be eradicated. Numerical simulations, using some TB data found in the literature in relation with Cameroon, are conducted to approve analytic results, and to show that vaccination coverage is not sufficient to control TB, effective contact rate has a high impact in the spread of TB.

Threshold dynamics of a time periodic and two--group epidemic model with distributed delay

In this paper, a time periodic and two--group reaction--diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number R0 for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of R0, that is, the disease is uniformly persistent if R0>1, while the disease goes to extinction if R0< 1. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio--temporal variables.

The asymptotic behavior of a stochastic multigroup SIS model

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

Traveling wave solutions in a two-group SIR epidemic model with constant recruitment

Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number [Formula: see text] More specifically, we prove that (i) when the basic reproduction number [Formula: see text] there exists a minimal wave speed [Formula: see text] such that for each [Formula: see text] the system admits a nontrivial traveling wave solution with wave speed c and for [Formula: see text] there exists no nontrivial traveling wave satisfying the system; (ii) when [Formula: see text] the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.

A TWO-STRAIN EPIDEMIC MODEL WITH DIFFERENTIAL SUSCEPTIBILITY AND MUTATION

A two-strain epidemic model with differential susceptibility and mutation is formulated and analyzed in this paper. The susceptible population is divided into two subgroups according to the vaccine that provides complete protection against one of the strains (strain two) but only partial against the other (strain one). The explicit formulae for the basic reproduction number and invasion reproduction number corresponding to each strain with and without mutation are derived, respectively. It is shown that there exist exclusive equilibria and coexistence equilibria, even if the reproduction number is below one. The stability of the disease-free equilibrium, strain dominance with or without mutation are investigated. The persistence of the disease is also briefly discussed. Numerical simulations are presented to illustrate the results.

GLOBAL DYNAMICS OF A DIFFERENTIAL SUSCEPTIBILITY MODEL

An SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals has been analyzed. We prove that all the solutions tend to the equilibria of the system. Then we use the Poincaré Index theorem to determine the number of the rest points and their stability properties. It has been shown that bistability occurs for suitable values of the involved parameters. We use the perturbations of the pitchfork bifurcation points to give examples of all possible dynamics of the system. Some numerical examples of bistability and hysteresis behavior of the system has been also provided.

DIFFERENTIAL SUSCEPTIBILITY TIME-DEPENDENT SIR EPIDEMIC MODEL

A non-autonomous epidemic dynamical system, in which we include variable susceptibility, is proposed. Some threshold conditions are derived which determine whether or not the disease will go to extinction. Some new threshold values, [Formula: see text], [Formula: see text] and [Formula: see text], are deduced for this general time-dependent system such that when [Formula: see text] is greater than 0, the disease is endemic in the sense of permanence and when one of the threshold values [Formula: see text] and [Formula: see text] is less than 0, the disease will die out. As an application of these results, the basic reproductive number ℛ0will be given if all the coefficients are periodic with common period. In addition, ℛ0< 1 implies the global stability of the disease-free periodic solution. Some corollaries are given for periodic and almost-periodic cases. The theoretical results are confirmed by a special example and numerical simulations.

The size of epidemics in populations with heterogeneous susceptibility

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R0≤1 and the existence and uniqueness of an endemic equilibrium when R0>1. We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when R0>1. Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626-644, 2005; Math Biosci Eng 3:89-100, 2006).

Heterogeneity in susceptibility to infection can explain high reinfection rates

Heterogeneity in susceptibility and infectivity is inherent to infectious disease transmission in nature. Here we are concerned with the formulation of mathematical models that capture the essence of heterogeneity while keeping a simple structure suitable of analytical treatment. We explore the consequences of host heterogeneity in the susceptibility to infection for epidemiological models for which immunity conferred by infection is partially protective, known as susceptible-infected-recovered-infected (SIRI) models. We analyze the impact of heterogeneity on disease prevalence and contrast the susceptibility profiles of the subpopulations at risk for primary infection and reinfection. We present a systematic study in the case of two frailty groups. We predict that the average rate of reinfection may be higher than the average rate of primary infection, which may seem paradoxical given that primary infection induces life-long partial protection. Infection generates a selection mechanism whereby fit individuals remain in S and frail individuals are transferred to R. If this effect is strong enough we have a scenario where, on average, the rate of reinfection is higher than the rate of primary infection even though each individual has a risk reduction following primary infection. This mechanism may explain high rates of tuberculosis reinfection recently reported. Finally, the enhanced benefits of vaccination strategies that target the high-risk groups are quantified.

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.