Cluster formation for multi-strain infections with cross-immunity

A mathematical model of marine bacteriophage evolution

To explore how particularities of a host cell-virus system, and in particular host cell replication, affect viral evolution, in this paper we formulate a mathematical model of marine bacteriophage evolution. The intrinsic simplicity of real-life phage-bacteria systems, and in particular aquatic systems, for which the assumption of homogeneous mixing is well justified, allows for a reasonably simple model. The model constructed in this paper is based upon the Beretta-Kuang model of bacteria-phage interaction in an aquatic environment (Beretta & Kuang 1998 Math. Biosci.149, 57-76. (doi:10.1016/S0025-5564(97)10015-3)). Compared to the original Beretta-Kuang model, the model assumes the existence of a multitude of viral variants which correspond to continuously distributed phenotypes. It is noteworthy that the model is mechanistic (at least as far as the Beretta-Kuang model is mechanistic). Moreover, this model does not include any explicit law or mechanism of evolution; instead it is assumed, in agreement with the principles of Darwinian evolution, that evolution in this system can occur as a result of random mutations and natural selection. Simulations with a simplistic linear fitness landscape (which is chosen for the convenience of demonstration only and is not related to any real-life system) show that a pulse-type travelling wave moving towards increasing Darwinian fitness appears in the phenotype space. This implies that the overall fitness of a viral quasi-species steadily increases with time. That is, the simulations demonstrate that for an uneven fitness landscape random mutations combined with a mechanism of natural selection (for this particular system this is given by the conspecific competition for the resource) lead to the Darwinian evolution. It is noteworthy that in this system the speed of propagation of this wave (and hence the rate of evolution) is not constant but varies, depending on the current viral fitness and the abundance of susceptible bacteria. A specific feature of the original Beretta-Kuang model is that this model exhibits a supercritical Hopf bifurcation, leading to the loss of stability and the rise of self-sustained oscillations in the system. This phenomenon corresponds to the paradox of enrichment in the system. It is remarkable that under the conditions that ensure the bifurcation in the Beretta-Kuang model, the viral evolution model formulated in this paper also exhibits a rise in self-sustained oscillations of the abundance of all interacting populations. The propagation of the travelling wave, however, remains stable under these conditions. The only visible impact of the oscillations on viral evolution is a lower speed of the evolution.

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.

Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

Many important pathogens such as HIV/AIDS, influenza, malaria, dengue and meningitis generally exist in phenotypically distinct serotypes that compete for hosts. Models used to study these diseases appear as meta-population systems. Herein, we revisit one of the multiple strain models that have been used to investigate the dynamics of infectious diseases with co-circulating serotypes or strains, and provide analytical results underlying the numerical investigations. In particular, we establish the necessary conditions for the local asymptotic stability of the steady states and for the existence of oscillatory behaviors via Hopf bifurcation. In addition, we show that the existence of discrete antigenic forms among pathogens can either fully or partially self-organize, where (i) strains exhibit no strain structures and coexist or (ii) antigenic variants sort into non-overlapping or minimally overlapping clusters that either undergo the principle of competitive exclusion exhibiting discrete strain structures, or co-exist cyclically.

Role of stochastic processes in maintaining discrete strain structure in antigenically diverse pathogen populations

Many highly diverse pathogen populations appear to exist stably as discrete antigenic types despite evidence of genetic exchange. It has been shown that this may arise as a consequence of immune selection on pathogen populations, causing them to segregate permanently into discrete nonoverlapping subsets of antigenic variants to minimize competition for available hosts. However, discrete antigenic strain structure tends to break down under conditions where there are unequal numbers of allelic variants at each locus. Here, we show that the inclusion of stochastic processes can lead to the stable recovery of discrete strain structure through loss of certain alleles. This explains how pathogen populations may continue to behave as independently transmitted strains despite inevitable asymmetries in allelic diversity of major antigens. We present evidence for this type of structuring across global meningococcal isolates in three diverse antigens that are currently being developed as vaccine components.

Cross-immunity, invasion and coexistence of pathogen strains in epidemiological models with one-dimensional antigenic space

Several epidemic models with many co-circulating strains have shown that partial cross-immunity between otherwise identical strains of a pathogen can lead to exclusion of a subset of the strains. Here we examine the mechanisms behind these solutions by considering a host population in which two strains are endemic and ask when it can be invaded by a third strain. If the function relating antigenic distance to cross-immunity is strictly concave or linear invasion is always possible. If the function is strictly convex and has an initial gradient of zero invasion depends on the degree of antigenic similarity between strains and the basic reproductive number. Examining specific concave and convex functions shows that the shape of the cross-immunity function affects the role of secondary infections in invasion. The basic reproductive number affects the importance of tertiary infections. Thus the form of the relationship between antigenic distance and cross-immunity determines whether the pathogen population will consist of an unstructured cloud of strains or a limited number of strains with strong antigenic structuring. In the latter case the basic reproductive number determines the maximum number of strains that can coexist. Analysis of the evolutionary trajectory shows that attaining the maximum diversity requires large spontaneous changes in antigenic structure and cannot result from a sequence of small point mutations alone.