Consequences of host heterogeneity, epitope immunodominance, and immune breadth for strain competition

Consumer-resource dynamics of hosts with their pathogens are modulated by complex interactions between various branches of hosts’ immune systems and the imperfectly perceived pathogen. Multistrain SIR models tend to sweep competitive interaction terms between different pathogen strains into a single parameter representing cross-immunity. After reviewing several hypotheses about the generation of immune responses, we look into the consequences of assuming that hosts with identical immune repertoires respond to new pathogens identically. In particular, we vary the breadth of the typical immune response, or the average number of pathogen epitopes a host perceives, and the probability of perceiving a particular epitope. The latter quantity in our model is equivalent both to the degree of diversity in host responses at the population level and the relative immunodominance of different epitopes. We find that a sharp transition to strain coexistence occurs as host responses become narrow or skewed toward one epitope. Increasing the breadth of the immune response and the immunogenicity of different epitopes typically increases the range of cross-immunity values in which chaotic strain dynamics and competitive exclusion occur. Models attempting to predict the outcomes of strain competition should thus consider the potential diversity and specificity of hosts’ responses to infection.     

The dynamics of cocirculating influenza strains conferring partial cross-immunity

 We develop a model that describes the dynamics of a finite number of strains that confer partial cross-protection among strains. The immunity structure of the host population is captured by an index-set notation where the index specifies the set of strains to which the host has been exposed. This notation allows us to derive threshold conditions for the invasion of a new strain and to show the existence of an endemic multi-strain equilibrium in a special case. The dynamics of systems consisting of more than two strains can exhibit sustained oscillations caused by an overshoot in the immunity to a specific strain if cross-protection is sufficiently strong. Received 15 January 1996; received in revised form 24 June 1996     

Leishmania tropica and Leishmania mexicana: cross-immunity in mice

The effect of a previous or concurrent Leishmania tropica major infection on a L. mexicana infection was studied. Mice which were recovering from or had recovered from a L. tropica infection were found to be totally resistant to L. mexicana. Infection of mice already carrying a L. mexicana infection with L. tropica resulted in subsequent ulceration and eventual healing of the lesions caused by both Leishmania species. Mice infected with L. mexicana were found normally to be no more susceptible to L. tropica than untreated mice: Only when L. tropica infections were located in the region of a draining lymph node already serving a L. mexicana infection did lesions of the former parasite persist.     

Ascaris suum and Toxocara canis: Egg extract antigens in guinea pigs and the macrophage migration inhibition test

Guinea pigs immunized with 500 μg of either Ascaris suum or Toxocara canis egg extracts, emulsified with Freund's complete adjuvant, were skin tested with both the homologous and heterologous antigens. Cross-reactions were observed in both groups. Migration of macrophages from sensitized animals was more inhibited by homologous than by heterologous antigens. Lymph-node lymphocytes from sensitized animals were stimulated to incorporate [⁸H]thymidine similarly by both antigens.     

Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity

We investigate the population dynamics of host-pathogen systems in which the pathogen has a potentially arbitrary number of antigenically distinct strains interacting via cross-immunity. The interior equilibrium configuration of the symmetric multiple strain SIR model with cross-immunity is characterized. We develop an efficient iterative method for numerically solving the equilibrium equation together with a number of informative analytical approximations to the full solution. Equilibrium properties are studied as a function of the number of strains, reproduction number, infectious period, and cross immunity profile. We establish that the prevalence in the system increases monotonically with the number of strains and the reduction in cross immunity. Moreover, we demonstrate the existence of a phase transition separating high prevalence and low prevalence parameter regions, with the critical point being defined by R₀1, where is the level of cross-immunity and R₀ is the reproduction number. Above the threshold, prevalence saturates with increasing numbers of strains as a result of the inclusion of prohibition of co-infection in the model. Below the threshold, prevalence saturates much more rapidly as the number of strains increases - indicating that when cross-protection is sufficiently intense, the selective advantage for a pathogen to increase its diversity is substantially less than in the threshold region. Similarly, there is limited benefit to increased transmissibility (or decreased cross-immunity) both for the high and low diversity pathogen systems compared with systems at the threshold R₀1 where small increase in transmissibility can result in significant increase in prevalence.     

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.     

Competition of pathogen strains leading to infection with variable infectivity and the effect of treatment

A model for the spread of two strains of a pathogen leading to an infection with variable infectivity is considered. The course of infection is described by two stages with different infectivity levels. The model is extended to account for treatment by including a third stage with different infectivity and survival for those treated. The contribution of each stage to incidence and prevalence is investigated and the effect of infectivity and survival on the basic reproduction ratio is examined. Standard equilibrium analysis is performed for both models, revealing that the successful strain is the one with highest reproduction ratio. If therapy, however, is more effective against the strain that wins in the absence of treatment and its reproduction ratio is sufficiently reduced, it might be outcompeted by the other strain after treatment becomes widely available. In this case, early introduction of treatment can prevent a major outbreak.     

Influenza drift and epidemic size: the race between generating and escaping immunity

Influenza in humans is characterised by strongly annual dynamics and antigenic evolution leading to partial escape from prior host immunity. The variability of new epidemic strains depends on the amount of virus currently circulating. In this paper, the amount of antigenic variation produced each year is dependent on the epidemic size. Our model reduces to a one-dimensional map and a full mathematical analysis is presented. This simple system suggests some basic principles which may be more generally applicable. In particular, for diseases with antigenic drift, vaccination may be doubly beneficial. Not only does it protect the population through classical herd immunity, but the overall case reduction reduces the chance of new variants being produced; hence, subsequent epidemics may be milder as a result of this positive feedback. Also, a disease with a high innate rate of antigenic variation will always be able to invade a susceptible population, whereas a disease with less potential for variation may require several introduction events to become endemic.     

Virus evolution within patients increases pathogenicity

Viruses like the human immunodeficiency virus (HIV), the hepatitis B virus (HBV), the hepatitis C virus (HCV) and many others undergo numerous rounds of inaccurate reproduction within an infected host. The resulting viral quasispecies is heterogeneous and sensitive to any selection pressure. Here we extend earlier work by showing that for a wide class of models describing the interaction between the virus population and the immune system, virus evolution has a well-defined direction toward increased pathogenicity. In particular, we study virus-induced impairment of the immune response and certain cross-reactive stimulation of specific immune responses. For eight different mathematical models, we show that virus evolution reduces the equilibrium abundance of uninfected cells and increases the rate at which uninfected cells are infected. Thus, in general, virus evolution makes things worse. An idea for combating HIV infection, however, is constructing a virus mutant that could outcompete the existing infection without being pathogenic itself.     

Age profile of immunity to influenza: effect of original antigenic sin

When multiple infections are possible during an individual’s lifetime, as with influenza, a host’s history of infection and immunity will determine the result of future exposures. In turn, the suite of varying individual infection histories will shape the population level dynamics of the disease. Exploring the consequences of precisely how immunity is acquired using mathematical models has proven challenging though: if n strains have circulated previously, there are 2ⁿ combinations of past infection to consider. However, by using an age-structured mathematical model of a disease with multiple strains, we can examine the population immune profile without explicitly keeping track of all possible infection histories. This framework allows previously unknown consequences of assumptions about immune acquisition to be observed. In particular, we see that ‘original antigenic sin’ can reduce immunity in some age groups: these immune blind spots could be responsible for the unexpectedly high severity of certain past influenza epidemics.