Multi-strain Models Research Articles

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.

Competitive dual-strain SIS epidemiological models with awareness programs in heterogeneous networks: two modeling approaches.

Epidemic diseases and media campaigns are closely associated with each other. Considering most epidemics have multiple pathogenic strains, in this paper, we take the lead in proposing two multi-strain SIS epidemic models in heterogeneous networks incorporating awareness programs due to media. For the first model, we assume that the transmission rates for strain 1 and strain 2 depend on the level of awareness campaigns. For the second one, we further suppose that awareness divides susceptible population into two different subclasses. After defining the basic reproductive numbers for the whole model and each strain, we obtain the analytical conditions that determine the extinction, coexistence and absolute dominance of two strains. Moreover, we also formulate its optimal control problem and identify an optimal implementation pair of awareness campaigns using optimal control theory. Given the complexity of the second model, we use the numerical simulations to visualize its different types of dynamical behaviors. Through theoretical and numerical analysis of these two models, we discover some new phenomena. For example, during the persistence analysis of the first model, we find that the characteristic polynomials of two boundary equilibria may have a pair of pure imaginary roots, implying that Hopf bifurcation and periodic solutions may appear. Most strikingly, multistability occurs in the second model and the growth rate of awareness programs (triggered by the infection prevalence) has a multistage impact on the final size of two strains. The numerical results suggest that the spread of a two-strain epidemic can be controlled (even be eradicated) by taking the measures of enhancing awareness transmission, reducing memory fading of aware individuals and ensuring high-level influx and rapid growth of awareness programs appropriately.

A vigorous study of fractional order mathematical model for SARS-CoV-2 epidemic with Mittag-Leffler kernel

SARS-CoV-2 and its variants have been investigated using a variety of mathematical models. In contrast to multi-strain models, SARS-CoV-2 models exhibit a memory effect that is often overlooked and more realistic. Atangana-Baleanu’s fractional-order operator is discussed in this manuscript for the analysis of the transmission dynamics of SARS-CoV-2. We investigated the transmission mechanism of the SARS-CoV-2 virus using the non-local Atangana-Baleanu fractional-order approach taking into account the different phases of infection and transmission routes. Using conventional ordinary derivative operators, our first step will be to develop a model for the proposed study. As part of the extension, we will incorporate fractional order derivatives into the model where the used operator is the fractional order operator of order Ψ1. Additionally, some basic aspects of the proposed model are examined in addition to calculating the reproduction number and determining the possible equilibrium. Stability analysis of the model is conducted to determine the necessary equilibrium conditions as they are also required in developing a numerical setup. Utilizing the theory of nonlinear functional analysis, for the model, Ulam-Hyers’ stability is established. We present a numerical scheme based on Newton’s polynomial in order to set up an iterative algorithm for the proposed ABC system. The application of this scheme to a variety of values of Φ1 indicates that there is a relationship between infection dynamics and the derivative’s order. We present further simulations which demonstrate the importance and cruciality of different parameters, as well as their effect on the dynamics and administer the disease. Furthermore, this study will provide a better understanding of the mechanisms underlying contagious diseases, thus supporting the development of policies to control them.

Modeling secondary infections with temporary immunity and disease enhancement factor: Mechanisms for complex dynamics in simple epidemiological models

Modeling insights for epidemiological scenarios characterized by chaotic dynamics have been largely unexplored. A rigorous analysis of such systems are essential for a real predictive power and a more accurate disease control decision making. Motivated by dengue fever epidemiology, we study a basic SIR–SIR type model for the host population, capturing differences between primary and secondary infections. This model is the minimalistic version to previously suggested multi-strain models for dengue fever in which deterministic chaos was found in wider parameter regions. Without strain structure of pathogens, we consider temporary immunity after a primary infection and disease enhancement in a subsequent infection to identify to which extent these biological mechanisms can generate complex behavior in simple epidemiological models.Stability analysis of the system is performed using the classical linearization theory, and the qualitative behavior of the model is investigated with a detailed bifurcation analysis. Rich dynamical structures are identified, including the Bogdanov–Takens, cusp and Bautin bifurcations which has never been described in dengue fever epidemiology. Besides the conventional transcritical bifurcation, a backward bifurcation occurs for higher disease enhancement in secondary infections, exhibiting bi-stability when biological temporary immunity period is assumed. The backward bifurcation is formalized using the center manifold theory. While the Hopf and the global homoclinic bifurcation curves were computed numerically, analytical expressions for the transcritical and tangent bifurcations are obtained. The combination of temporary immunity and disease enhancement play a significant role in the complexity of the system dynamics, with chaotic behavior observed after including seasonal forcing.

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.

Bayesian inference for multistrain epidemics with application to ESCHERICHIA COLI O157:H7 in feedlot cattle

For most pathogens, testing procedures can be used to distinguish between different strains with which individuals are infected. Due to the growing availability of such data, multistrain models have increased in popularity over the past few years. Quantifying the interactions between different strains of a pathogen is crucial in order to obtain a more complete understanding of the transmission process, but statistical methods for this type of problem are still in the early stages of development. Motivated by this demand, we construct a stochastic epidemic model that incorporates additional strain information and propose a statistical algorithm for efficient inference. The model improves upon existing methods in the sense that it allows for both imperfect diagnostic test sensitivities and strain misclassification. Extensive simulation studies were conducted in order to assess the performance of our method, while the utility of the developed methodology is demonstrated on data obtained from a longitudinal study of Escherichia coli O157:H7 strains in feedlot cattle.

Threshold Dynamics and Competitive Exclusion in a Virus Infection Model with General Incidence Function and Density-Dependent Diffusion

In this paper, we investigate single-strain and multistrain viral infection models with general incidence function and density-dependent diffusion subject to the homogeneous Neumann boundary conditions. For the single-strain viral infection model, by using the linearization method and constructing appropriate Lyapunov functionals, we obtain that the global threshold dynamics of the model is determined by the reproductive numbers for viral infection ℛ0. For the multistrain viral infection model, we have discussed the competitive exclusion problem. If the reproduction number ℛi for strain i is maximal and larger than one, the steady state Ei corresponding to the strain i is globally stable. Thus, competitive exclusion happens and all other strains die out except strain i. Meanwhile, we can prove that the single-strain and multistrain viral infection models are well posed. Furthermore, numerical simulations are also carried out to illustrate the theoretical results, which is seldom seen in the relevant known literatures.

Modeling Antibiotic Use Strategies in Intensive Care Units: Comparing De-escalation and Continuation.

Antimicrobial de-escalation refers to the treatment mechanism of switching from empiric antibiotics with good coverage to alternatives based on laboratory susceptibility test results, with the aim of avoiding unnecessary use of broad-spectrum antibiotics. In a previous study, we have developed multi-strain and multi-drug models in an intensive care unit setting, to evaluate the benefits and trade-offs of de-escalation in comparison with the conventional strategy called antimicrobial continuation. Our simulation results indicated that for a large portion of credible parameter combinations, de-escalation reduces the use of the empiric antibiotic but increases the probabilities of colonization and infections. In this paper, we first simplify the previous models to compare the long-term dynamical behaviors between de-escalation and continuation systems under a two-strain scenario. The analytical results coincide with our previous findings in the complex models, indicating the benefits and unintended consequences of de-escalation strategy result from the nature of this treatment mechanism, not from the complexity of the high-dimensional systems. By extending the models to three-strain scenarios, we find that de-escalation is superior than continuation in preventing outbreaks of invading strains that are resistant to empiric antibiotics. Thus decisions on antibiotic use strategies should be made specifically according to ICU conditions and intervention objectives.

A workflow for generating multi-strain genome-scale metabolic models of prokaryotes.

Genome-scale models (GEMs) of bacterial strains' metabolism have been formulated and used over the past 20 years. Recently, with the number of genome sequences exponentially increasing, multi-strain GEMs have proved valuable to define the properties of a species. Here, through four major stages, we extend the original Protocol used to generate a GEM for a single strain to enable multi-strain GEMs: (i) obtain or generate a high-quality model of a reference strain; (ii) compare the genome sequence between a reference strain and target strains to generate a homology matrix; (iii) generate draft strain-specific models from the homology matrix; and (iv) manually curate draft models. These multi-strain GEMs can be used to study pan-metabolic capabilities and strain-specific differences across a species, thus providing insights into its range of lifestyles. Unlike the original Protocol, this procedure is scalable and can be partly automated with the Supplementary Jupyter notebook Tutorial. This Protocol Extension joins the ranks of other comparable methods for generating models such as CarveMe and KBase. This extension of the original Protocol takes on the order of weeks to multiple months to complete depending on the availability of a suitable reference model.

BiGG Models 2020: multi-strain genome-scale models and expansion across the phylogenetic tree.

The BiGG Models knowledge base (http://bigg.ucsd.edu) is a centralized repository for high-quality genome-scale metabolic models. For the past 12 years, the website has allowed users to browse and search metabolic models. Within this update, we detail new content and features in the repository, continuing the original effort to connect each model to genome annotations and external databases as well as standardization of reactions and metabolites. We describe the addition of 31 new models that expand the portion of the phylogenetic tree covered by BiGG Models. We also describe new functionality for hosting multi-strain models, which have proven to be insightful in a variety of studies centered on comparisons of related strains. Finally, the models in the knowledge base have been benchmarked using Memote, a new community-developed validator for genome-scale models to demonstrate the improving quality and transparency of model content in BiGG Models.

Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period

Coinfection of hosts with multiple strains or serotypes of the same agent, such as different influenza virus strains, different human papilloma virus strains, and different dengue virus serotypes, is not only a very serious public health issue but also a very challenging mathematical modeling problem. In this paper, we study a time-periodic two-strain SIS epidemic model with diffusion and latent period. We first define the basic reproduction number R0i and introduce the invasion number Rˆ0i for each strain i(i=1,2), which can determine the ability of each strain to invade the other single-strain. The main question that we investigate is the threshold dynamics of the model. It is shown that if R0i⩽1(i=1,2), then the disease-free periodic solution is globally attractive; if R0i>1⩾R0j(i≠j,i,j=1,2), then competitive exclusion, where the jth strain dies out and the ith strain persists, is a possible outcome; and if Rˆ0i>1(i=1,2), then the disease persists uniformly. Finally we present the basic framework of threshold dynamics of the system by using numerical simulations, some of which are different from that of the corresponding multi-strain SIS ODE models.

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.

A51 Dengue virus multi-strain models as hypotheses for serotype interaction.

A51 Dengue virus multi-strain models as hypotheses for serotype interaction.

Mathematical Models for the Epidemiology and Evolution of Mycobacterium tuberculosis.

This chapter reviews the use of mathematical and computational models to facilitate understanding of the epidemiology and evolution of Mycobacterium tuberculosis. First, we introduce general epidemiological models, and describe their use with respect to epidemiological dynamics of a single strain and of multiple strains of M. tuberculosis. In particular, we discuss multi-strain models that include drug sensitivity and drug resistance. Second, we describe models for the evolution of M. tuberculosiswithin and between hosts, and how the resulting diversity of strains can be assessed by considering the evolutionary relationships among different strains. Third, we discuss developments in integrating evolutionary and epidemiological models to analyse M. tuberculosisgenetic sequencing data. We conclude the chapter with a discussion of the practical implications of modelling - particularly modelling strain diversity - for controlling the spread of tuberculosis, and future directions for research in this area.

Lyapunov functionals for multistrain models with infinite delay

We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

Multistrain models predict sequential multidrug treatment strategies to result in less antimicrobial resistance than combination treatment.

BackgroundCombination treatment is increasingly used to fight infections caused by bacteria resistant to two or more antimicrobials. While multiple studies have evaluated treatment strategies to minimize the emergence of resistant strains for single antimicrobial treatment, fewer studies have considered combination treatments. The current study modeled bacterial growth in the intestine of pigs after intramuscular combination treatment (i.e. using two antibiotics simultaneously) and sequential treatments (i.e. alternating between two antibiotics) in order to identify the factors that favor the sensitive fraction of the commensal flora.Growth parameters for competing bacterial strains were estimated from the combined in vitro pharmacodynamic effect of two antimicrobials using the relationship between concentration and net bacterial growth rate. Predictions of in vivo bacterial growth were generated by a mathematical model of the competitive growth of multiple strains of Escherichia coli.ResultsSimulation studies showed that sequential use of tetracycline and ampicillin reduced the level of double resistance, when compared to the combination treatment. The effect of the cycling frequency (how frequently antibiotics are alternated in a sequential treatment) of the two drugs was dependent upon the order in which the two drugs were used.ConclusionSequential treatment was more effective in preventing the growth of resistant strains when compared to the combination treatment. The cycling frequency did not play a role in suppressing the growth of resistant strains, but the specific order of the two antimicrobials did. Predictions made from the study could be used to redesign multidrug treatment strategies not only for intramuscular treatment in pigs, but also for other dosing routes.Electronic supplementary materialThe online version of this article (doi:10.1186/s12866-016-0724-5) contains supplementary material, which is available to authorized users.

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.

Natural, Persistent Oscillations in a Spatial Multi-Strain Disease System with Application to Dengue

Many infectious diseases are not maintained in a state of equilibrium but exhibit significant fluctuations in prevalence over time. For pathogens that consist of multiple antigenic types or strains, such as influenza, malaria or dengue, these fluctuations often take on the form of regular or irregular epidemic outbreaks in addition to oscillatory prevalence levels of the constituent strains. To explain the observed temporal dynamics and structuring in pathogen populations, epidemiological multi-strain models have commonly evoked strong immune interactions between strains as the predominant driver. Here, with specific reference to dengue, we show how spatially explicit, multi-strain systems can exhibit all of the described epidemiological dynamics even in the absence of immune competition. Instead, amplification of natural stochastic differences in disease transmission, can give rise to persistent oscillations comprising semi-regular epidemic outbreaks and sequential dominance of dengue's four serotypes. Not only can this mechanism explain observed differences in serotype and disease distributions between neighbouring geographical areas, it also has important implications for inferring the nature and epidemiological consequences of immune mediated competition in multi-strain pathogen systems.