Coupled, multi-strain epidemic models of mutating pathogens
Introduction
The growing threat of antimicrobial drug resistance presents a significant challenge not only to the medical community, but the wider general population [1], [2]. Indeed, resistant strains of infectious diseases are already endemic in many communities — particularly in developing countries and lower socio-economic settings — with new strains, that enjoy even more extensive resistance, continually emerging (see e.g. [3]). For clarity, at the outset we define separate pathogen strains as micro-organisms whose phenotypic characteristics (e.g. reproductive capacity, pathogenicity, resistance profile) are measurably distinct.
Although resistance develops as a pathogen’s natural biological response to antimicrobial treatment [4], [5], [6], [7], the misuse and overuse of existing antimicrobials severely exacerbates the problem, rendering previously successful treatments ineffective. Consequently, we are now faced with a rapidly diminishing arsenal of effective therapies, with poor practice only accelerating us along this path [8]. As such, concern mounts that “superbugs” will emerge that are resistant to all available treatments, with many fearing that we are approaching the end of the antimicrobial era [9].
To examine this growing public health concern, we introduce a general mathematical model designed to simulate the emergence and dissemination of mutant strains of infectious diseases. The formal framework assumes the form of a coupled multi-strain SIS/SIR/SIRS epidemic model [10] in which individuals can transition between the various infectious compartments associated with each strain. This structure is intended to replicate the phenotypic phenomenon of amplification, whereby individuals infected with a particular pathogen strain develop a new, mutant strain that is possibly resistant to some combination of antimicrobials. The model can be applied to pathogens in which mutations resulting in important phenotypic changes tend to accrue clonally, e.g. viruses, tuberculosis. This work would be less applicable to bacterial pathogens for which horizontal gene transfer may occur (e.g. via plasmids or bacteriophages), unless the transferred genetic element included all genes relevant to bacterial fitness.
Several epidemiological investigations into the imminent threat of mutation and drug resistance, most often linked to a specific infection [11], [12], [13], [14] and geographic setting [15] have already appeared in the literature (see also [16], [17], [18]).1 These papers [12], [13], [22], which often utilize special cases of the general multi-strain network presented here, usually focus on specific epidemiological outcomes (e.g. interim infection burden, optimal intervention strategies [23]) whilst providing less mathematical detail on the dynamics of the model. Alternatively, detailed mathematical analyses of multi-strain epidemic models [24], [25] primarily treat the infectious compartments as being parallel (i.e. uncoupled) and do not consider the possibility of infected patients “amplifying” to an alternate strain. The present paper attempts to bridge this gap by formally examining the mathematical implications of linking the various infectious compartments in a sufficiently general manner.2
Specifically, this article analytically examines the mathematical and biological aspects of the proposed coupled epidemic models. As such we focus our attention on the functional form of the basic reproduction number of each strain, discuss the importance of these quantities for regulating the system dynamics and outline the necessary conditions for an epidemic outbreak. We also analyze the nature and structure of the asymptotic solutions of the system and explore how they relate to both the structure of the infectious compartment network and the relative magnitudes of the basic reproduction number of each strain. Numerical simulations of the model addressing infection-specific epidemiological issues will be the subject of future work.
To begin, we give a brief description of the coupled network of infectious compartments and their corresponding connectivity in Sections 2 and 3. Then, in Sections 4 and 5, we define our model parameters and introduce and analyze the set of differential equations governing the evolution of the SIS, SIR and SIRS systems. In Section 6 we derive the set of basic reproduction numbers associated with each pathogen strain and demonstrate that these quantities represent threshold parameters that govern the dynamics of each system and, in particular, the potential for an epidemic outbreak (see Sections 7 and 8).
We then determine the equilibrium states of each system in Section 9 and derive the necessary and sufficient conditions for the existence of nonnegative endemic solutions. In doing so we verify that the coupling term leads to the coexistence of several pathogen strains at the endemic equilibria and demonstrate that the strain with the greatest reproductive capacity is not necessarily the most prevalent.
Following this, in Section 10, we compare our findings with previous results derived within the context of uncoupled multi-strain epidemic models and examine how the coupling term modifies the system properties. Finally, in Section 11, we summarize our results, discuss their significance for public health policy and suggest directions for future work.
Section snippets
Coupled infectious compartments
In this article we wish to analyze the dynamics of systems (i.e. populations) with several co-circulating pathogen strains. In particular, we are interested in the case where the co-circulating strains are related by genetic mutation. In this manner we are naturally led to coupled, multi-strain epidemic models (for examples of uncoupled multi-strain models see e.g. [24], [25]).
To provide a rigorous framework, we will begin with a simple class of epidemic models in which the population is
Connectivity of the infectious compartment network
In this section we formalize several definitions that will be used throughout the text and derive important relationships pertaining to the connectivity of the network of Ii compartments (Fig. 1). These include the reachability of compartment Ij from compartment Ii and the definition of ancestor and descendant strains. Primarily, this discussion is designed to provide the reader a deeper appreciation of the structure of the next-generation matrix and endemic equilibrium solutions (derived in
Model description and system equations
Returning to our description of the structure of the SIS, SIR and SIRS epidemic models, we adopt the convention that italicized letters denote the number of individuals in each of the corresponding categories so that, for example, Ii represents the number of individuals in compartment Ii. Following this convention, the total population, N(t), is given by9
Demographically, we assume that individuals are recruited directly
System bounds
Given the system (6)–(8), we can show that the regionis a positively invariant and absorbing set that attracts all solutions of (6)–(8) in . To see this, we show that the system solutions are bounded and nonnegative, provided the state variables are nonnegative initially, i.e. S(0) ≥ 0, Ii(0) ≥ 0 and Ri(0) ≥ 0.
To begin, we use the variation of constants formula to express the solution of (7) as
Basic reproduction number
The basic reproduction number, R0, is defined as the average number of newly infected individuals generated by a single typical infectious individual introduced into a fully susceptible population. In the context of multi-strain epidemic models we anticipate a set of basic reproduction numbers R0i, with each member of the set being associated with the reproductive capacity of each pathogen strain i. In this case an epidemic may occur if any of the R0i > 1 (provided the susceptible population is
Epidemic outbreak
We now return to the statement made at the opening of the previous section: If each of the R0i ≤ 1, the infected population quickly fades out; however, if R0i > 1 for any i an epidemic occurs. To see this we simply rewrite Eq. (7) in terms of the basic reproduction numbers R0i:Taking as our initial condition (which is the most optimistic case from the infection’s perspective), we can clearly see that if R0i ≤ 1 for all i, Ii(t) decays exponentially
Asymptotic system dynamics
Having discussed the initial epidemic trajectory in terms of the basic reproduction numbers, R0i, we now turn our attention to the late-time dynamics of the system. First, we show (by induction) that if R0f ≤ 1 (⇔R0i ≤ 1 for all i) the infection is eradicated from the population. We will then consider the alternative case R0f > 1 and show that only the fittest strain f, along with each of its descendants j▷f, remain in circulation indefinitely. The general approach used to arrive at each of
Equilibrium solutions
To reinforce the asymptotic analysis presented in the previous section, here we determine the equilibrium states of the system (6)–(8). In particular, we find that in addition to the infection-free equilibrium point there are a set of n endemic equilibria, where i ∈ [1, n]. Explicitly, we will show that at the i-th endemic equilibrium point P*i, all strains are driven to extinction except for strain i and its di descendants.18
Comparison with uncoupled models
The expressions derived in the previous two sections for the coupled SIS, SIR and SIRS models are all generalizations of the familiar uncoupled results. In this section we demonstrate how the uncoupled expressions can be derived from the more general expressions given above to retrospectively reveal how the coupling term modifies the behaviour of uncoupled multi-strain models.
To begin, we remind the reader that the uncoupled case can be derived from the general case by taking the limit in
Conclusion
In this article we have introduced a general framework to study coupled, multi-strain epidemic models designed to simulate the emergence and dissemination of mutational (e.g. drug-resistant) variations of pathogens. In particular, we have analyzed the properties of the infectious compartment network in these models and how they influence the system dynamics and solutions.
First, we found that the introduction of a unidirectional coupling term (which we argue is the most general coupling
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