Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

Subthreshold and superthreshold coexistence of pathogen variants: the impact of host age-structure

It is well known that in the most general epidemic models with multiple pathogen variants a competitive exclusion principle is valid, such that the variant with the highest reproduction number eliminates the rest. Mechanisms such as super-infection, coinfection, and cross-immunity can lead to pathogen polymorphism where multiple strains coexist. It is also known that variability of infectivity with host age can destabilize the endemic equilibrium and cause oscillations. In this article we show that the hosts' chronological age can itself lead to coexistence of microparasites in the most basic model where competitive exclusion will occur without the age structure. Moreover, the host age-structure leads to multiple subthreshold dominance equilibria, and both weakly and strongly subthreshold coexistence. We find that the two pathogens cannot cooperate to persist subthreshold if neither one of them can persist subthreshold by itself. If, however, one of them can persist subthreshold by itself, it can cause the two pathogens to coexist in a strongly subthreshold equilibrium. The second strain that persists subthreshold through the mediation of the first always has a lower virulence. Our results show that age structure in infectivity can permit the coexistence of competing pathogens when the incidence is of proportionate mixing type (frequency-dependent transmission) and at least one of the strains is virulent.     

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

Parasite strain coexistence in a heterogeneous host population

Heterogeneity in host susceptibility and transmissibility to parasite attack allows a lower transmission rate to sustain an epidemic than is required in homogeneous host populations. However, this heterogeneity can leave some hosts with little susceptibility to disease, and at high transmission rates, epidemic size can be smaller than for diseases where the host population is homogeneous. In a heterogeneous host population, we model natural selection in a parasite population where host heterogeneity is exploited by different strains to varying degrees. This partitioning of the host population allows coexistence of competing parasite strains, with the heterogeneity-exploiting strains infecting the more susceptible hosts, in the absence of physiological tradeoffs and spatial heterogeneity, and even for markedly different transmission rates. In our model, intermediate-strategy parasites were selected against: should coexistence occur, an equilibrium is reached where strains occupied only the extreme ends of trait space, under appropriate conditions selecting for lower R₀.     

Linking immunological and epidemiological dynamics of HIV: the case of super-infection

In this paper, a two-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the two strains eliminate each other with the strain with the larger immunological reproduction persisting. However, on the population scale superinfection is possible, with the strain with larger immunological reproduction number super-infecting the strain with the smaller immunological reproduction number. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition, the between-host transmission and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers, the epidemiological reproduction numbers and invasion reproduction numbers are computed. Besides the disease-free equilibrium, there are two population-level strain one and strain two isolated equilibria, as well as a population-level coexistence equilibrium when both invasion reproduction numbers are greater than one. The single-strain population-level equilibria are locally asymptotically stable suggesting that in the absence of superinfection oscillations do not occur, a result contrasting previous studies of HIV age-since-infection structured models. Simulations suggest that the epidemiological reproduction number and HIV population prevalence are monotone functions of the within-host parameters with reciprocal trends. In particular, HIV medications that decrease within-host viral load also increase overall population prevalence. The effect of the immunological parameters on the population reproduction number and prevalence is more pronounced when the initial viral load is lower.     

Malaria Drug Resistance: The Impact of Human Movement and Spatial Heterogeneity

Human habitat connectivity, movement rates, and spatial heterogeneity have tremendous impact on malaria transmission. In this paper, a deterministic system of differential equations for malaria transmission incorporating human movements and the development of drug resistance malaria in an \(n\) patch system is presented. The disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. For a two patch case, the boundary equilibria (drug sensitive-only and drug resistance-only boundary equilibria) when there is no movement between the patches are shown to be locally asymptotically stable when they exist; the co-existence equilibrium is locally asymptotically stable whenever the reproduction number for the drug sensitive malaria is greater than the reproduction number for the resistance malaria. Furthermore, numerical simulations of the connected two patch model (when there is movement between the patches) suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity. With slow movement (or low migration) between the patches, the drug sensitive strain dominates the drug resistance strain. However, with fast movement (or high migration) between the patches, the drug resistance strain dominates the drug sensitive strain.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.     

On the stability of polymorphic host-pathogen populations

The stability of populations of hosts and micro-parasites is investigated where each consists of n varieties that are equal in every respect except that each strain of parasites can infect only one specific strain of hosts and none of the others. Collectively the host strains are limited by a carrying capacity and through this limitation the host populations interact with each other. Hosts are assumed to reproduce asexually or such that different strains do not mate or are not fertile if they do. When the excess death rate caused by the pathogenic parasites is sufficiently large, then the host population is regulated to an equilibrium below the carrying capacity of the environment. This polymorphic equilibrium is shown to be locally asymptotically stable. When one of the parasite strains is absent, then all the other strains die out asymptotically. However, if host resistance to all infectious strains of parasites is achieved at the cost of a lower birthrate of the resistant host strain, then, if a certain condition for the various parameters is satisfied, stable coexistence between infected and resistant hosts is possible. There are many examples where susceptibility and resistance of hosts depends upon the conformation of specific proteins that are involved in host-parasite interactions and hence upon alleles at genetic loci that code for these proteins. We propose that polymorphism in wildtype populations which has been the subject of much theorizing in mathematical genetics may be due to host-pathogen interactions. Our model suggests how a polymorphic population, once established, can remain polymorphic indefinitely.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

An evolutionary model of influenza A with drift and shift

Influenza A virus evolves through two types of evolutionary mechanisms - drift and shift. These two evolutionary mechanisms allow the pathogen to infect us repeatedly, as well as occasionally create pandemics with large morbidity and mortality. Here we introduce a novel model that incorporates both evolutionary mechanisms. This necessitates the modelling of three types of strains - seasonal human strains, bird-to-human transmittable H5N1 strains and evolved pandemic H5N1 strain. We define reproduction and invasion reproduction numbers and use them to establish the presence of dominant and coexistence equilibria. We find that the amino acid substitution structure of human influenza can destabilize the human influenza equilibrium and sustained oscillations are possible. We find that for low levels of infection in domestic birds, these oscillations persist, inducing oscillations in the number of humans infected with the avian flu strain. The oscillations have a period of 365 days, similar to the one that can be observed in the cumulative number of human H5N1 cases reported by the World Health Organization (WHO). Furthermore, we establish some partial global results on the competition of the strains.     

Spatially structured superinfection and the evolution of disease virulence

When pathogen strains differing in virulence compete for hosts, spatial structuring of disease transmission can govern both evolved levels of virulence and patterns in strain coexistence. We develop a spatially detailed model of superinfection, a form of contest competition between pathogen strains; the probability of superinfection depends explicitly on the difference in levels of virulence. We apply methods of adaptive dynamics to address the interplay of spatial dynamics and evolution. The mean-field approximation predicts evolution to criticality; any small increase in virulence capable of dynamical persistence is favored. Both pair approximation and simulation of the detailed model indicate that spatial structure constrains disease virulence. Increased spatial clustering reduces the maximal virulence capable of single-strain persistence and, more importantly, reduces the convergent-stable virulence level under strain competition. The spatially detailed model predicts that increasing the probability of superinfection, for given difference in virulence, increases the likelihood of between-strain coexistence. When strains differing in virulence can coexist ecologically, our results may suggest policies for managing diseases with localized transmission. Comparing equilibrium densities from the pair approximation, we find that introducing a more virulent strain into a host population infected by a less virulent strain can sometimes reduce total host mortality and increase global host density.     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.     

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

The effect of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains on competitive exclusion

We investigate the in-hospital transmission dynamics of two methicillin-resistant Staphylococcus aureus (MRSA) strains: hospital-acquired methicillin resistant S. aureus (HA-MRSA) and community-acquired methicillin-resistant S. aureus (CA-MRSA). Under the assumption that patients can only be colonized with one strain of MRSA at a time, global results show that competitive exclusion occurs between HA-MRSA and CA-MRSA strains; the strain with the larger basic reproduction ratio will become endemic while the other is extinguished due to competition. Because new studies suggest that patients can be concurrently colonized with multiple strains of MRSA, we extend the model to allow patients to be co-colonized with HA-MRSA and CA-MRSA. Using the extended model, we explore the effect of co-colonization on competitive exclusion by determining the invasion reproduction ratios of the boundary equilibria. In contrast to results derived from the assumption that co-colonization does not occur, the extended model rarely exhibits competitive exclusion. More commonly, both strains become endemic in the hospital. When transmission rates are assumed equal and decolonization measures act equally on all strains, competitive exclusion never occurs. Other interesting phenomena are exhibited. For example, solutions can tend toward a co-existence equilibrium, even when the basic reproduction ratio of one of the strains is less than one.     

Multi-stage Vector-Borne Zoonoses Models: A Global Analysis

A class of models that describes the interactions between multiple host species and an arthropod vector is formulated and its dynamics investigated. A host-vector disease model where the host’s infection is structured into n stages is formulated and a complete global dynamics analysis is provided. The basic reproduction number acts as a sharp threshold, that is, the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}_0^2\le 1\) and that a unique interior endemic equilibrium exists and is GAS if \({\mathcal {R}}_0^2>1\). We proceed to extend this model with m host species, capturing a class of zoonoses where the cross-species bridge is an arthropod vector. The basic reproduction number of the multi-host-vector, \({\mathcal {R}}_0^2(m)\), is derived and shown to be the sum of basic reproduction numbers of the model when each host is isolated with an arthropod vector. It is shown that the disease will persist in all hosts as long as it persists in one host. Moreover, the overall basic reproduction number increases with respect to the host and that bringing the basic reproduction number of each isolated host below unity in each host is not sufficient to eradicate the disease in all hosts. This is a type of “amplification effect,” that is, for the considered vector-borne zoonoses, the increase in host diversity increases the basic reproduction number and therefore the disease burden.     

Mathematical model for Zika virus dynamics with sexual transmission route

Zika virus is a flavivirus transmitted to humans primarily through the bite of infected Aedes mosquitoes. In addition to vector-borne spread, however, the virus can also be transmitted through sexual contact. In this paper, we formulate and analyze a new system of ordinary differential equations which incorporates both vector and sexual transmission routes. Theoretical analysis of this model when there is no disease induced mortality shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the associated reproduction number is less than unity and unstable otherwise. However, when we extend this same model to include Zika induced mortality, which have been documented in Latin America, we find that the model exhibits a backward bifurcation. Specifically, a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. To further explore model predictions, we use numerical simulations to assess the importance of sexual transmission to disease dynamics. This analysis shows that risky behavior involving multiple sexual partners, particularly among male populations, substantially increases the number of infected individuals in the population, contributing significantly to the disease burden in the community.     

Space and the persistence of male-killing endosymbionts in insect populations

Male-killing bacteria are bacteria that are transmitted vertically through the females of their insect hosts. They can distort the sex ratio of their hosts by killing infected male offspring. In nature, male-killing endosymbionts (male killers) often have a 100% efficient vertical transmission, and multiple male-killing bacteria infecting a single population are observed. We use different model formalisms to study these observations. In mean-field models a male killer with perfect transmission drives the host population to extinction, and coexistence between multiple male killers within one population is impossible; however, in spatially explicit models, both phenomena are readily observed. We show how the spatial pattern formation underlies these results. In the case of high transmission efficiencies, waves with a high density of male killers alternate with waves of mainly wild-type hosts. The male killers cause local extinction, but this creates an opportunity for uninfected hosts to re-invade these areas. Spatial pattern formation also creates an opportunity for two male killers to coexist within one population: different strains create spatial regions that are qualitatively different; these areas then serve as different niches, making coexistence possible.     

Latent Coinfection and the Maintenance of Strain Diversity

Technologies for strain differentiation and typing have made it possible to detect genetic diversity of pathogens, both within individual hosts and within communities. Coinfection of a host by more than one pathogen strain may affect the relative frequency of these strains at the population level through complex within- and between-host interactions; in infectious diseases that have a long latent period, interstrain competition during latency is likely to play an important role in disease dynamics. We show that SEIR models that include a class of latently coinfected individuals can have markedly different long-term dynamics than models without coinfection, and that coinfection can greatly facilitate the stable coexistence of strains. We demonstrate these dynamics using a model relevant to tuberculosis in which people may experience latent coinfection with both drug sensitive and drug resistant strains. Using this model, we show that the existence of a latent coinfected state allows the possibility that disease control interventions that target latency may facilitate the emergence of drug resistance.     

On the role of cross-immunity and vaccines on the survival of less fit flu-strains

A pathogen's route to survival involves various mechanisms including its ability to invade (host's susceptibility) and its reproductive success within an invaded host ("infectiousness"). The immunological history of an individual often plays an important role in reducing host susceptibility or it helps the host mount a faster immunological response de facto reducing infectiousness. The cross-immunity generated by prior infections to influenza A strains from the same subtype provide a significant example. The results of this paper are based on the analytical study of a two-strain epidemic model that incorporates host isolation (during primary infection) and cross-immunity to study the role of invasion mediated cross-immunity in a population where a precursor related strain (within the same subtype, i.e. H3N2, H1N1) has already become established. An uncertainty and sensitivity analysis is carried out on the ability of the invading strain to survive for given cross-immunity levels. Our findings indicate that it is possible to support coexistence even in the case when invading strains are "unfit", that is, when the basic reproduction number of the invading strain is less than one. However, such scenarios are possible only in the presence of isolation. That is, appropriate increments in isolation rates and weak cross-immunity can facilitate the survival of less fit strains. The development of "flu" vaccines that minimally enhance herd cross-immunity levels may, by increasing genotype diversity, help facilitate the generation and survival of novel strains.