Multi-strain Epidemic Model Research Articles

Extinction of some strains and asymptotic profiles of coexistence endemic equilibria in a multi-strain epidemic model

This work studies the asymptotic dynamics of classical solutions of a multi-strain diffusive epidemic model in spatially heterogeneous environments. Results on the nonexistence of coexistence endemic equilibria are established. Moreover, explicit conditions on the parameters range which guarantee the extinctions of some strains of the disease are examined. In particular, we establish that the competition-exclusion principle occurs when the population diffuses very fast and one strain uniquely maximizes the ratio of the spatial average of transmission rate over that of the recovery rate. However, several strains of the disease may coexist when either: (i) the population diffuses very slowly, (ii) or a subgroup diffuses very slowly and the other subgroup diffuses very fast. In these two scenarios, the asymptotic profiles of the coexistence endemic equilibria with respect to the diffusion rates of the population are presented. In particular, we observe a spatial segregation of the infected populations when they diffuse very slowly.

Influence of waning immunity on vaccination decision-making: A multi-strain epidemic model with an evolutionary approach analyzing cost and efficacy

In this research, we introduce a comprehensive epidemiological model that accounts for multiple strains of an infectious disease and two distinct vaccination options. Vaccination stands out as the most effective means to prevent and manage infectious diseases. However, when there are various vaccines available, each with its costs and effectiveness, the decision-making process for individuals becomes paramount. Furthermore, the factor of waning immunity following vaccination also plays a significant role in influencing these choices. To understand how individuals make decisions in the context of multiple strains and waning immunity, we employ a behavioral model, allowing an epidemiological model to be coupled with the dynamics of a decision-making process. Individuals base their choice of vaccination on factors such as the total number of infected individuals and the cost-effectiveness of the vaccine. Our findings indicate that as waning immunity increases, people tend to prioritize vaccines with higher costs and greater efficacy. Moreover, when more contagious strains are present, the equilibrium in vaccine adoption is reached more rapidly. Finally, we delve into the social dilemma inherent in our model by quantifying the social efficiency deficit (SED) under various parameter combinations.

Multi-Strain Epidemic Models of Mutating Viruses with Airborne Transmission Based on Cellular Automata and Ordinary Differential Equations

The emergence of SARS-CoV-2 virus mutations is a pressing concern in the context of the COVID-19 pandemic. In this paper, a Susceptible–Infected–Recovered (SIR) compartmental model has been formulated in terms of cellular automata and ordinary differential equations to simulate the transmission dynamics of a virus. This model accounts for the potential of new virus mutations to create additional infections of a multi-strain disease while also considering the impact of vaccination on disease control within a population. The basic reproduction number of the disease is derived, and the effect of virus mutations and vaccination rates is evaluated on a population over five years. The results of numerical simulations demonstrate the significant role of maintaining high vaccination rates in controlling the spread of the virus, even when assuming that all variants have similar illness characteristics and that a single shot of vaccine provides complete and lifelong protection against all strains. The findings underscore the necessity for countries to implement a consistent and high-level vaccination plan as soon as vaccines become available in order to mitigate the impact of a pandemic effectively. In conclusion, this study highlights the importance of taking into account the potential impact of virus mutations for controlling the COVID-19 pandemic. Furthermore, it emphasizes the critical role of vaccination in limiting the spread of the virus, and emphasizes the need to implement and maintain high vaccination rates as part of a comprehensive approach to managing the ongoing pandemic.

Immunity-induced criticality of the genotype network of influenza A (H3N2) hemagglutinin.

Seasonal influenza kills hundreds of thousands every year, with multiple constantly changing strains in circulation at any given time. A high mutation rate enables the influenza virus to evade recognition by the human immune system, including immunity acquired through past infection and vaccination. Here, we capture the genetic similarity of influenza strains and their evolutionary dynamics with genotype networks. We show that the genotype networks of influenza A (H3N2) hemagglutinin are characterized by heavy-tailed distributions of module sizes and connectivity indicative of critical behavior. We argue that (i) genotype networks are driven by mutation and host immunity to explore a subspace of networks predictable in structure and (ii) genotype networks provide an underlying structure necessary to capture the rich dynamics of multistrain epidemic models. In particular, inclusion of strain-transcending immunity in epidemic models is dependent upon the structure of an underlying genotype network. This interplay is consistent with self-organized criticality where the epidemic dynamics of influenza locates critical regions of its genotype network. We conclude that this interplay between disease dynamics and network structure might be key for future network analysis of pathogen evolution and realistic multistrain epidemic models.

A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: Application to French data

Many SARS-CoV-2 variants have appeared over the last months, and many more will continue to appear. Understanding the competition between these different variants could help make future predictions on the evolution of epidemics. In this study we use a mathematical model to investigate the impact of three different SARS-CoV-2 variants on the spread of COVID-19 across France, between January-May 2021 (before vaccination was extended to the full population). To this end, we use the data from Geodes (produced by Public Health France) and a particle swarm optimisation algorithm, to estimate the model parameters and further calculate a value for the basic reproduction number R0. Sensitivity and uncertainty analysis is then used to better understand the impact of estimated parameter values on the number of infections leading to both symptomatic and asymptomatic individuals. The results confirmed that, as expected, the alpha, beta and gamma variants are more transmissible than the original viral strain. In addition, the sensitivity results showed that the beta/gamma variants could have lead to a larger number of infections in France (of both symptomatic and asymptomatic people).

Control Strategies for a Multi-strain Epidemic Model.

This article studies a multi-strain epidemic model with diffusion and environmental heterogeneity. We address the question of a control strategy for multiple strains of the infectious disease by investigating how the local distributions of the transmission and recovery rates affect the dynamics of the disease. Our study covers both full model (in which case the diffusion rates for all subgroups of the population are positive) and the ODE–PDE case (in which case we require a total lock-down of the susceptible subgroup and allow the infected subgroups to have positive diffusion rates). In each case, a basic reproduction number of the epidemic model is defined and it is shown that if this reproduction number is less than one then the disease will be eradicated in the long run. On the other hand, if the reproduction number is greater than one, then the disease will become permanent. Moreover, we show that when the disease is permanent, creating a common safety area against all strains and lowering the diffusion rate of the susceptible subgroup will result in reducing the number of infected populations. Numerical simulations are presented to support our theoretical findings.

Near-optimal control for a stochastic multi-strain epidemic model with age structure and Markovian switching

ABSTRACT Stochastic noise can affect the dynamics of infectious diseases. In this paper, a new stochastic multi-strain epidemic model with white noise, Markovian switching, vaccination control and age structure is established. Some priori estimates of the susceptible, infected and recovered are obtained. Under the condition of the objective function is minimised, we formulate sufficient and necessary conditions for the near-optimal control problems of the system, which is using a nearly maximum condition of the Hamiltonian function. The great influence of age structure, white noise, Markovian switching and vaccination control on the dynamical behaviour of infectious diseases and the existence of near-optimal control are demonstrated by numerical simulations.

Coupled, multi-strain epidemic models of mutating pathogens

We introduce and analyze coupled, multi-strain epidemic models designed to simulate the emergence and dissemination of mutant (e.g. drug-resistant) pathogen strains. In particular, we investigate the mathematical and biological properties of a general class of multi-strain epidemic models in which the infectious compartments of each strain are coupled together in a general manner. We derive explicit expressions for the basic reproduction number of each strain and highlight their importance in regulating the system dynamics (e.g. the potential for an epidemic outbreak) and the existence of nonnegative endemic solutions. Importantly, we find that the basic reproduction number of each strain is independent of the mutation rates between the strains — even under quite general assumptions for the form of the infectious compartment coupling. Moreover, we verify that the coupling term promotes strain coexistence (as an extension of the competitive exclusion principle) and demonstrate that the strain with the greatest reproductive capacity is not necessarily the most prevalent. Finally, we briefly discuss the implications of our results for public health policy and planning.

Introduction to Focus Issue: Time-delay dynamics.

The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to time-delayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal test-beds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of mode-locked laser systems. Moreover, two original theoretical approaches are explored for the so-called Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30 years. Current hot problems such as the synchronization properties of networks of delay-coupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multi-strain epidemic models and the interaction of glucose and insulin for more effective treatment.

Chimera states in multi-strain epidemic models with temporary immunity.

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

An age-structured multi-strain epidemic model for antigenically diverse infectious diseases: A multi-locus framework

The establishment of cross-protective responses and development of immunity within a host exert pressure on pathogens through cross-immunity mediated competition between antigenic forms. In this paper, we incorporate age-specificity in the multi-locus epidemic model used to study the pathogen-specific dynamic behaviours for infectious diseases with diverse co-circulating antigenic types. We establish results on the existence of a unique mild solution, and on the necessary conditions for local stability of the steady-states. In particular, we find that, when the reproductive number R0<1, all strains go to extinction. When R0>1, we show that there exist additional conditions which determine the stability of different types of endemic equilibria, namely weak and strong endemicity, where the weak endemic equilibria correspond to the existence of principle of competitive exclusions of pathogen-specific clusters, while strong endemicity represents the co-existence of all strains. Using numerical simulations, we also show that weak endemic equilibria yield dynamic features in which only one of the clusters containing discrete strain structures (e.g., of minimally, or non-overlapping antigenic types) persists while others go to extinction. For unique strong endemicity, we observe no strain structure, where antigenic types co-exist or exhibit cyclical strain structure with diverse dynamical behaviours (e.g., (quasi-)periodicity, intermittency, chaos). This demonstrates that pathogenic-specific dynamic features are ubiquitous and shows how cross-immunity between antigenic variants shape the maintenance and evolution of strain structures.

A Framework for Inferring Unobserved Multistrain Epidemic Subpopulations Using Synchronization Dynamics.

A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations.

Regression-based ranking of pathogen strains with respect to their contribution to natural epidemics.

Genetic variation in pathogen populations may be an important factor driving heterogeneity in disease dynamics within their host populations. However, to date, we understand poorly how genetic diversity in diseases impact on epidemiological dynamics because data and tools required to answer this questions are lacking. Here, we combine pathogen genetic data with epidemiological monitoring of disease progression, and introduce a statistical exploratory method to investigate differences among pathogen strains in their performance in the field. The method exploits epidemiological data providing a measure of disease progress in time and space, and genetic data indicating the relative spatial patterns of the sampled pathogen strains. Applying this method allows to assign ranks to the pathogen strains with respect to their contributions to natural epidemics and to assess the significance of the ranking. This method was first tested on simulated data, including data obtained from an original, stochastic, multi-strain epidemic model. It was then applied to epidemiological and genetic data collected during one natural epidemic of powdery mildew occurring in its wild host population. Based on the simulation study, we conclude that the method can achieve its aim of ranking pathogen strains if the sampling effort is sufficient. For powdery mildew data, the method indicated that one of the sampled strains tends to have a higher fitness than the four other sampled strains, highlighting the importance of strain diversity for disease dynamics. Our approach allowing the comparison of pathogen strains in natural epidemic is complementary to the classical practice of using experimental infections in controlled conditions to estimate fitness of different pathogen strains. Our statistical tool, implemented in the R package StrainRanking, is mainly based on regression and does not rely on mechanistic assumptions on the pathogen dynamics. Thus, the method can be applied to a wide range of pathogens.

Incorporating demographic stochasticity into multi-strain epidemic models: application to influenza A

We develop mathematical models of the transmission and evolution of multi-strain pathogens that incorporate strain extinction and the stochastic generation of new strains via mutation. The dynamics resulting from these models is then examined with the applied aim of understanding the mechanisms underpinning the evolution and dynamics of rapidly mutating pathogens, such as human influenza viruses. Our approach, while analytically relatively simple, gives results that are qualitatively similar to those obtained from much more complex individually based simulation models. We examine strain dynamics as a function of cross-immunity and key transmission parameters, and show that introducing strain extinction and modelling mutation as a stochastic process significantly changes the model dynamics, leading to lower strain diversity, reduced infection prevalence and shorter strain lifetimes. Finally, we incorporate transient strain-transcending immunity in the model and demonstrate that it reduces strain diversity further, giving patterns of sequential strain replacement similar to that seen in human influenza A viruses.

The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model

We study dynamical behavior and bifurcation structure of a multi-strain SIR epidemiological model with seasonal forcing in transmission rate. The conventional single strain SIR dynamics with seasonal forcing is known to show a cascade of bifurcations as the strength of seasonality increases. In this study with an extension to multiple strains of pathogens, we first investigate the bifurcation patterns of the two strain SIR model. In the two strain SIR model, a new parameter called cross-immunity between strains plays a key role in the dynamical behavior. As analogous to the single strain model, we found that the period doubling bifurcation occurs as the strength of seasonal forcing is increased. However, bifurcation patterns differ greatly both in their subharmonic periods and the relative phase of cycles between strains, depending mostly on the degree of the cross-immunity between strains. With strong and weak cross-immunities, the period doubling cascade proceeds gradually toward chaotic cycles. On the other hand, with intermediate cross-immunity, the loss of stability of annual cycle attractor is immediately followed by chaotic cycles. Asynchronous cycles of two strains are robust outcome when annual cycles lose stability, but the population can converge to perfectly synchronized biennial cycles if two strains are antigenically distant from each other. Second, we found that there are simultaneously stable multiple attractors in two strain SIR model. Biennial and chaotic attractors, e.g., occasionally coexist at the same degree of seasonal forcing, and the trajectories converge to one of them depending on the initial conditions. Such multiple attractors are widely seen between biennial and annual cycles, or among the different types of biennial cycles, at the same strength of the seasonality. Finally, we found that the population may switch from one attractor to another by introducing a small random noise in seasonally varying transmission rate.