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SK, UD, and JD designed the model. SK wrote the code. SK, UD, SAL, and JD wrote the paper.

The authors have declared that no competing interests exist.

Many pathogens exist in phenotypically distinct strains that interact with each other through competition for hosts. General models that describe such multi-strain systems are extremely difficult to analyze because their state spaces are enormously large. Reduced models have been proposed, but so far all of them necessarily allow for coinfections and require that immunity be mediated solely by reduced infectivity, a potentially problematic assumption. Here, we suggest a new state-space reduction approach that allows immunity to be mediated by either reduced infectivity or reduced susceptibility and that can naturally be used for models with or without coinfections. Our approach utilizes the general framework of status-based models. The cornerstone of our method is the introduction of immunity variables, which describe multi-strain systems more naturally than the traditional tracking of susceptible and infected hosts. Models expressed in this way can be approximated in a natural way by a truncation method that is akin to moment closure, allowing us to sharply reduce the size of the state space, and thus to consider models with many strains in a tractable manner. Applying our method to the phenomenon of antigenic drift in influenza A, we propose a potentially general mechanism that could constrain viral evolution to a one-dimensional manifold in a two-dimensional trait space. Our framework broadens the class of multi-strain systems that can be adequately described by reduced models. It permits computational, and even analytical, investigation and thus serves as a useful tool for understanding the evolution and ecology of multi-strain pathogens.

Microbial pathogens are tremendously diverse. Pathogens that cause one and the same disease may differ remarkably in both their genotype and their phenotype, like in HIV/AIDS [

The first type of interaction may be referred to as ecological interference [

The second type of interaction, referred to as cross-immunity interference, is specific to different strains of the same pathogen: these can confer full or partial immunity to each other. This means that a host infected with one strain becomes substantially less susceptible to certain other strains of the pathogen for a prolonged period of time after the initial infection is cleared. Cross-immunity is highest between phenotypically similar strains. Since phenotypic similarity usually implies recent common ancestry, a pathogen's ecology is thus intrinsically entangled with its evolution.

Understanding the dynamics of multi-strain pathogens at a general theoretical level turns out to be extremely difficult. Numerous models have been proposed during the past twenty years (e.g., [

Virtually all equation-based models of disease dynamics can be traced back to the compartment model introduced by Kermack and McKendrick in 1927 [

Traditionally, full models have been developed based on the assumption of reduced susceptibility, which implies that immune hosts are able to block off an infection completely, with a certain probability [^{2} and 2^{n}

To illustrate the utility of our approach, and that of reduced models in general, we demonstrate its application to the phenomenon of drift in influenza A. Using reduced models we are able to simulate up to 400 strains. Influenza A is a multi-strain pathogen whose epidemiology and evolution display an intricate interaction pattern. Because the human immune system can produce protective antibodies against influenza's surface glycoprotein hemagglutinin, individuals gain lifelong immunity against each strain of the virus with which they have been infected [

A few recent studies have attempted to model, and thereby explain, the phenomenon of antigenic drift in influenza A. Apart from individual-based models, most of these studies consider a one-dimensional strain space in which some sort of traveling-wave behavior is observed [

In a recent study, Koelle et al. [

As mentioned above, SIR models are based on partitioning a host population into susceptible, infected, and recovered classes. Abundances in the resultant compartments are then tracked through time by means of ordinary differential equations. There are three major dichotomies according to which multi-strain SIR models can be classified. The first dichotomy refers to the treatment of individuals with respect to cross-immunity: there are history-based and status-based approaches.

In the history-based approach introduced by Castillo-Chavez et al. [

In the status-based approach introduced by Gog and Swinton [

The second dichotomy is the permission or prohibition of coinfections. Coinfection is an event through which an individual, while already being infected with one strain, gets simultaneously infected with a second strain. Unrestricted permission or complete prohibition of coinfections are, of course, mathematical abstractions. One or the other may be more plausible for any particular pathogen.

The third dichotomy refers to the way protective immunity works. As mentioned above, either the chance for an immune host to get infected or the infectivity of an immune host during a secondary infection is reduced. Within the history-based approach, models constructed under both assumptions were shown to behave qualitatively similar, at least in simple systems [

In general, infection of a host with a virus results in two effects: (a) the host becomes sick and transmits the virus, and (b) the immune system of the host develops protective antibodies against the infecting variant. If the host is already partially immune to the infecting strain, that is, protective antibodies had been developed prior to the infection, then (a) the severity and the duration of infection are reduced, and (b) the production of new types of antibodies is slowed down. In the limit case, when the host has full immune protection against the infecting strain, it is capable of completely fending off the infection with the existing arsenal of antibodies, resulting in no infectiousness and no production of new types of antibodies. In the simplest version of status-based models, individuals are either fully susceptible to a strain or fully immune against it, so this limit case should apply. The assumption of reduced infectivity within this framework correctly captures the first effect but neglects the second effect. In other words, hosts that are fully immune against a particular variant do not transmit it, but, paradoxically, still increase their repertoire of antibodies in exactly the same way as do susceptible individuals, upon an infection with this variant (see

Full history-based or status-based multi-strain SIR models are cumbersome in terms of their analytical treatment [

1. When attempting to infect a potential host, an infecting strain does not in any way “perceive” the host's entire immune status or disease history. What it does perceive is simply whether or not this potential host possesses any immunity against the focal strain. This consideration suggests a natural set of state variables: it is helpful to keep track of the proportions of a population that are immune to each strain, or combinations thereof. It turns out to be possible to reformulate any full status-based model in terms of such new immunity variables. Below we refer to this transformation as an “expansion in immunity variables.”

2. The utility of this transformation becomes clear when we recognize that, at any moment in time and for most diseases, many hosts will be immune to only a few of the strains currently circulating, while only a few hosts will be immune to many of these strains. Consequently, the immunity variables that describe the latter small proportions of the host population need not be tracked exactly and independently but can instead be approximated, without much disturbing the overall disease dynamics. Higher-order immunity variables can thus be approximated by functions of lower-order immunity variables. Below we refer to this approximation as truncating or “closing” the disease dynamics at a desired order.

The approach presented in this article introduces a general representation of status-based multi-strain models in terms of immunity variables. This representation is useful because it produces a hierarchical structure of equations describing the dynamics of a multi-strain system. The equations at any given order ℓ of this hierarchy are decoupled from equations at all orders above ℓ + 1, under the assumption of reduced susceptibility, or even above ℓ, under the assumption of reduced infectivity. Thus, this hierarchy can easily be truncated at any order, either by approximating higher-order immunity variables with functions of lower-order variables under the former assumption, or by simply ignoring higher-order variables under the latter assumption. The resultant truncated models provide either approximate or exact reduced descriptions of the original system.

We now proceed to the mathematical formulation of our framework. For the sake of clarity, we develop our reasoning for the model with coinfections; the model in which coinfections are prohibited is outlined in _{A}_{∅︀} thus denotes the class of naive individuals, i.e., individuals that have no immunity whatsoever; and _{i}_{A}_{A}_{⊂}_{K} _{A}

Notations Used Throughout This Study

Based on this notational framework, we can specify the disease dynamics of a multi-strain pathogen in three steps, (a) to (c) below, by considering the three processes that cause host individuals to enter and exit classes defined by their immune status.

(a) _{∅︀}. The birth rate into class _{A}_{A}_{,∅︀}, where

We further assume that infections do not alter the death rate of hosts and that the host population is at its demographic equilibrium. This implies a constant per capita death rate μ for all classes.

(b)

Any chosen set of functions

(c) _{i}_{i}I_{i}_{i}_{i} ⊂ K_{i}_{i}_{j}_{i}

Based on (a) to (c), we obtain the following system of equations [^{n}

To complete the definition of our multi-strain model, we further specify the process of immunity acquisition. We introduce the probability _{1},_{2},…,_{ℓ} (all different) after an infection with strain

We assume that (a) the chance of obtaining immunity against strain _{ki}_{kj}_{ki}_{kj}^{*}_{j}_{∈∅︀} σ_{kj}_{ij}_{ij}_{ii}

To derive the approximations of model (1)–(2), we rewrite that system in terms of the immunity variables,

Each immunity variable has a clear intuitive interpretation: _{1},_{2},…,_{ℓ} ∈ _{…i…i…} = ξ_{…i…}. By definition, the immunity variables satisfy monotonicity conditions, 1 ≥ _{1},_{2},…∈

Recalling that Σ_{A}_{⊂}_{ K} _{A}_{i}

Derivation of the equations for क़ƃ_{i}_{ij}_{i}_{i}

The first term in

The second term in

The last equality is satisfied due to the inclusion–exclusion principle [

Collecting the three results above, we obtain the equation for क़ƃ_{i}

The equations for क़ƃ_{i,j}

Fortunately, it is not necessary to rewrite the full system of _{ij}

1. Symmetry condition:

2. Monotonicity condition:

3. Redundancy condition: the approximate immunity variables with duplicate indices must be equal to the corresponding immunity variables of the previous order when the duplicate index is removed,

Below we introduce and discuss two simple closures of order 1 and one simple closure of order 2.

(a)

The symmetry and redundancy conditions are evidently met, and the monotonicity condition is satisfied because, by definition, ξ_{k}

(b)

Again, the symmetry and redundancy conditions are evidently met, and the monotonicity condition is satisfied because क़̂_{ij}_{i}_{j}_{i}_{j}_{ij}_{ji}_{ij}_{ij}_{i}_{j}

(c)

The motivation underlying this closure is analogous to that of the order-1 independence closure. All conditions are fulfilled.

Models truncated at first order have 2

It is interesting to compare the obtained first-order equations under, say, the independence closure, with the reduced model proposed by Gog and Grenfell [

Now, we define the fraction of susceptibles to strain _{i}_{i}

These equations are very similar to _{i}. As expected from the initial assumptions, immunization of the host population happens at a faster rate in Gog and Grenfell's model.

We have thus demonstrated how our framework of approximation—based on transformation to, and expansion in, immunity variables—provides simplified disease dynamics of multi-strain pathogens, in particular under the assumption of reduced susceptibility. In the next section we will compare, for a pathogen with four strains, the dynamics of the full system with that of the proposed approximations and Gog and Grenfell's model.

All numerical analyses were carried out using the MATLAB computing environment (The Mathworks, ^{−7}. Code is available upon request.

We consider a simple system of four strains along a line. The strain space is given by

Adjacent strains also confer cross-immunity to each other, resulting in a tridiagonal cross-immunity matrix,

The functions

With such a small number of strains, the behavior of the full SIR model is tractable and can be used as a baseline reference. We can also examine how Gog and Grenfell's reduced infectivity model performs if we consider it as an approximation to the full model with reduced susceptibility. We numerically solve the following equations for the time interval [0,_{ij}_{ij}_{ijk}

The following parameters are kept fixed: ν = 1, μ = 0, and _{i}_{0} for all _{0}, the cross-immunity coefficient _{0} ∈ {2,3,4,5}, ^{−8},10^{−6},10^{−4}}. Initially, 99% of the host population is fully susceptible to all strains, while 1% is infected with strain 1, is immune against it, and is fully susceptible to all other strains.

To assess how well the reduced models approximate the full model, we introduce one qualitative and one quantitative accuracy measure. We consider an epidemic detection threshold ɛ = 10^{−4}, describing the smallest proportion of infected hosts at which the disease can still be detected in the population. The results presented below are not particularly sensitive to the exact value of this parameter within three orders of magnitude (unpublished data). We denote by _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

To obtain the overall qualitative error, ρ, we average ρ_{i}_{i}

To obtain the overall quantitative accuracy measure, Δ, in a way that we could compare across models with different parameter values, we normalize Δ_{i}_{i}_{i}

The results of this comparison are shown in _{ij}

Qualitative accuracy measure (vertical axis) is shown in dependence on the cross-immunity coefficient _{0} (rows), and the mutation probability ^{−6}, _{0} = 5, and

Details as in

Solution of the full model, Gog and Grenfell's model, order-1 independence closure model, order-1 interpolation closure model, and order-2 independence closure model are denoted by F, GG, O1A, O1B, and O2, respectively. Parameter values: ^{−6}, _{0} = 5, and

We have conducted the same type of analysis for a circular four-strain system and for two six-strain systems (see

The application presented in this section illustrates how one could capture some aspects of the phenomenon of antigenic drift in influenza A's subtype H3N2 using models capable of tracing hundreds of variables. This is an attempt to extend the work by Gog and Grenfell [

We assume that the strain space of the considered influenza-like virus is a two-dimensional rectangular lattice, i.e., each variant of the virus is characterized by a pair of integers (^{2} and the phenotypic mutational neighborhood is given by the next-neighbor relation on the lattice, _{(i,j)} = {(^{2} characterizing a particular strain implies that epitope A is in its

By the same token, it is natural to incorporate another idea, introduced by Gupta and colleagues [_{(i,j)(k,ℓ)} = 1 if either ^{−1/2} = 60.7%,

Strains to which strain (0,0) confers full, 37%, 2%, or no cross-immunity are shown in white, light gray, dark gray, and black, respectively.

(A) Our model. (B) Gog and Grenfell's model [

The cross-immunity matrix σ thus specified is sufficient for generating a one-dimensional trajectory of strain evolution: variants along the diagonal of strain space cause epidemics, whereas other variants do not (see

In this simple model, one strain strictly dominates the host population at each epidemic season. In a slightly more general setting, several strains can coexist within an epidemic season [

Parameter values: μ = 0, ν = 1, ^{−4}, and _{(i,j)} were drawn from a normal distribution with mean 3 and standard deviation 0.5. The numerical solution for the time interval _{(1,1)}(0) = 0.01 and ξ_{(1,1)}(0) = 0.01, corresponding to a healthy and fully susceptible host population with 1% of hosts infected with strain (1,1).

(A) Strains whose maximum epidemic size exceeded 0.01 are shown. The gray shade indicates the maximum epidemic size; the number above each shaded square indicates the time when the maximum of the epidemic for that particular strain was reached. Circles indicate strains whose transmission coefficients are less than 3; crosses indicate strains with transmission coefficients greater than 3.

(B) The sum of all proportions of infectious hosts as a function of time.

To show that this result is a consequence of the immunity structure suggested here, rather than just a peculiarity of the considered equations, we have simulated exactly the same system for Gog and Grenfell's equations and for the model with no coinfections. We thus can show that the results reported above are qualitatively robust to model choice (see

The real influenza virus, which is likely to live in an approximately two-dimensional strain space, appears to experience a selection regime that, while allowing for the temporary coexistence of a small number of variants, constrains the long-term evolution of the virus to a single branch. Our model suggests a possible mechanism for explaining this surprising reduction. In particular, we conjecture that two ingredients are responsible for the associated evolutionary dynamics. 1) The first ingredient is the non-local nature of the immune response after an infection. This results from the fact that cross-immunity protects hosts not only against strains that are very similar to the infecting strain, but also against strains that are quite distant from it in strain space, as long as at least one of their epitope conformations resembles that of the infecting strain. Non-locality of the immune response prevents the virus from conquering the entire trait space. 2) The second ingredient, which enables temporary coexistence of several strains, is the heterogeneity of transmission coefficients in trait space. Paradoxically, this heterogeneity occasionally leads to temporary parity among the effective reproduction ratios of different strains. The effective reproduction ratio of a particular strain is the quantity that determines whether this strain takes off and causes an epidemic or dies out without ever reaching the epidemic threshold [_{(i,j)} of strain (_{(i,j)}_{(i,j)}/ν. When all strains possess the same transmission coefficient, effective reproduction ratios thus depend only on the fractions of susceptible individuals. Clearly, the pool of susceptibles to the diagonal strains is larger than the pool of susceptibles to nearby off-diagonal strains, because past infections have induced low cross-immunity against the former and high cross-immunity against the latter. Hence, the diagonal strains successively cause epidemics, while the off-diagonal strains do not—accordingly, no polymorphism can emerge, not even in the short term. By contrast, in a trait space that is heterogeneous with respect to the transmission coefficient, relatively small values of _{(i,j)} for off-diagonal strains may occasionally be compensated by high values of β_{(i,j)}. In this manner, the effective reproduction ratios for off-diagonal strains may become comparable to those for diagonal strains. If, in addition, strains with comparable effective reproduction ratios start from comparable initial conditions, they reach epidemic values around the same time. This leads to short-term polymorphisms.

The approach introduced here enables systematic reductions in the complexity of status-based models of multi-strain pathogens. It is applicable to models with or without coinfections, with reduced susceptibility or reduced infectivity. If coinfections are allowed and reduced infectivity is assumed, our approach coincides with that of Gog and Grenfell [

It would be interesting to perform a rigorous mathematical analysis of the behavior of our approximations and compare it with the behavior of the full system [

In this work we presented a state-space reduction approach that applies only to the class of status-based models. We focused on this class of models for two reasons. Apart from easier mathematical treatment, there is an important conceptual difference that favors status-based models, at least under the reduced susceptibility assumption. Consider a situation when a host is repeatedly challenged with a strain. In the history-based approach, the probability for a host to acquire an infection remains the same across successive challenges. Thus, a host that has successfully used cross-reacting antibodies to repel one or more challenges from a particular pathogen is just as likely to be infected at the next challenge as is a host with the same infection history that has never seen this particular pathogen. To us, the status-based assumption—that, if antibodies fend off the first challenge, subsequent challenges will fail too—seems more realistic. Nevertheless, it would be interesting to know whether a state-space reduction approach similar to the one presented here could also be applied to history-based models.

Using our framework, we have investigated a potentially general mechanism for constraining viral evolution to one-dimensional manifolds when the underlying strain space is two-dimensional. Based on general knowledge about the antigenic space of the real influenza virus, we considered a hypothetical influenza-like virus whose phenotype space was given by a regular lattice with the same dimensionality as the number of principal components of the virus' antigenic space. We associate the movement along the axes of the resultant phenotype space with changes in the conformation of the viral “effective epitopes.” We based our analysis on two plausible qualitative assumptions: (a) during an infection, immunity is independently generated against all effective epitopes, and (b) immunity against one effective epitope suffices for full protection against viruses with similar epitopes. The resultant cross-shaped cross-immunity neighborhood drives the evolution of the virus along the diagonal of the phenotype space. This observation offers a conceptually simple approach to understanding single-trunk phylogenies of infectious pathogens.

Qualitatively different hypotheses for explaining the single-trunk phylogeny of influenza were introduced earlier on by Ferguson et al. [

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The authors express thanks to Sarah Cobey, Arkadii Kryazhimskiy, and Pleuni Pennings for fruitful discussions during the initial stages of this work, to Burton Singer for helpful comments on an early version of this manuscript, and to Julia Gog and two anonymous reviewers for valuable comments. Miriam Dushoff also made comments on the manuscript. SK gratefully acknowledges financial support by the Burroughs Wellcome Fund Training Program in Biological Dynamics (1001782), by Defense Advanced Research Projects Agency grant HR0011-05-1–0057, and by the IIASA's US National Member Organization during his stay at the International Institute for Applied Systems Analysis (IIASA). UD gratefully acknowledges financial support by the Vienna Science and Technology Fund (WWTF). SAL and JD gratefully acknowledge financial support by the US National Institutes of Health grant P50 GM071508 and DARPA grant HR0011-05-1–0057.

susceptible–infected–recovered