## Abstract

RNA viruses exist as genetically diverse populations displaying a range of virulence degrees. The evolution of virulence in viral populations is, however, poorly understood. On the basis of the experimental observation of an RNA virus clone in cell culture diversifying into two subpopulations of different virulence, we study the dynamics of mutating virus populations with varying virulence. We introduce a competition–colonization trade-off into standard mathematical models of intra-host viral infection. Colonizers are fast-spreading virulent strains, whereas the competitors are less-virulent variants but more successful within co-infected cells. We observe a two-step dynamics of the population. Early in the infection, the population is dominated by colonizers, which later are outcompeted by competitors. Our simulations suggest the existence of steady state in which all virulence classes coexist but are dominated by the most competitive ones. This equilibrium implies collective virulence attenuation in the population, in contrast to previous models predicting evolution of the population towards increased virulence.

## 1. Introduction

The replication cycle of a particular viral strain can be described by different life-history traits or fitness components, such as the stability of viral particles, burst size or virulence, among others [1–4]. Variation of these traits affects viral fitness in different ways, and fitness components can be traded off against each other such that variation of one trait affects the other. Virulence is a phenotypic property of particular biomedical interest. In analogy with the virulence concept of epidemiology, we regard here the cytopathogenicity of the virus as its virulence. Accordingly, viruses with a higher cell-killing rate are considered to be more virulent.

RNA virus populations are exceptionally diverse owing to the low fidelity of their replication process [5,6]. The intra-host ensembles of strains, termed viral quasi-species, consist of mutant clouds of closely related, but non-identical genomes [7]. The composition of a quasi-species is largely determined by the competitive fitness of its individual viruses [8]. Quasispecies diversity is the result of a balance between mutation and selection [9,10]. The role of virulence in this intra-species competition is, however, unclear.

Several mathematical models have been designed to study the evolution of virulence under specific fitness trade-offs [1,2,11]. For example, the trade-off between virulence and transmission derives from the assumption that the longer a virus exploits its host, the higher the chances that it infects a new host [12,13]. Under this assumption, it is predicted that if transmission is limited, virulence decreases and infections tend to attenuate over time [14].

However, the transmission–virulence trade-off, as postulated in epidemiological models, might not always operate in host–pathogen systems [15,16]. Mutants of different RNA viruses, such as foot-and-mouth disease virus (FMDV) or influenza, with a large difference in their cell-killing capacity produce similar levels of progeny [17–20]. Moreover, fitness and virulence are not necessarily correlated traits [21,22], suggesting that the trade-off between virulence and virus production does, in general, not hold at the cellular or intra-host level.

In a recent experiment with FMDV, two different phenotypes within the quasispecies were derived from a single purified clone. Each of these strains had adapted the ecological strategies of competition and colonization, respectively [19,23]. Highly virulent viral strains play the role of colonizers, because they kill cells faster and thus replicate faster, which allows faster spread and colonization of new cells. Local competition arises when two or more different viruses infect the same cell and compete for intracellular resources. Competitors manage to produce more offspring in a cell co-infected together with a colonizer and, at the same time, extend the cell-killing time characteristic of a colonizer, a phenomenon known as viral interference. A mixed competitor–colonizer population is subject to a density-dependent selection. Under high density of viruses, competitors have an advantage because of the frequent occurrence of co-infections. Under low-density conditions, the virulent colonizers are selected because of their faster spreading through unoccupied cells. Density-dependent selection has been described for different RNA viruses [24–28], suggesting that competition and colonization might be general strategies of RNA viruses.

In the present study, we aim to understand how a competition–colonization trade-off shapes the evolution of virulence during intra-host infections of mutating viral populations. We employ deterministic models of virus population dynamics, which arise as suitable adaptations of well-established mathematical models previously introduced by Wei *et al*. [29], Nowak *et al*. [30,31] and Ho *et al*. [32]. Moreover, we model mutations using a matrix of probabilities of transition between the different variants of the population defined by their virulence values. We compare the competition–colonization trade-off with the opposite assumption that more virulent variants are also more competitive, as previously suggested [33–35]. The major consequence of the competition–colonization trade-off is stable coexistence of multiple strains of reduced virulence that precludes a transient domination of virulent variants.

## 2. The model

The dynamics of intra-host viral infections have been studied using mathematical models [29–32,36]. We make use of this well-established methodology while capturing the competition–colonization dynamics by representing multiple infections (i.e. co-infections) in the model.

Although, in principle, the same cell could be sequentially infected by many strains, a virus infecting an already infected cell with sufficient delay after the initial infection will have a replicative disadvantage, because the late strain would need to synthesize its own materials in a partially or totally saturated cell. Accordingly, and to avoid a combinatorial explosion in the number of differential equations required, we limit the number of different virus types within co-infected cells to two, i.e. we consider only singly infected and doubly infected cells.

As shown in figure 1*a*, we assume a renewed cell pool that can be infected by different viral strains. Competition between viral strains takes place at two different levels: viruses compete for the cell pool and inside co-infected cells. These dynamics are described by the multiple-strain susceptible-infected-recovered-type model defined by the upcoming equations (2.1) and given in full generality in the appendix (equations (A 1)). We define *M*_{ij} as the relative frequency by which strain *j* arises from strain *i* owing to mutations during the replication of strain *i* within a mono- or a co-infected cell. In order to assign realistic values to the entries of this matrix, we explored the scarce experimental literature investigating the effect of mutation on virulence. Concrete numbers could only be derived from the work of Carrasco *et al*. [22], in which viral strains are distinguished in terms of their fitness rather than their virulence. Nevertheless, based on the measurements obtained in Carrasco *et al*. [22], we chose the parameters of a Dirichlet distribution such that, on average, during the error prone replication of virus *i* within an infected cell, 41 per cent of mutations are lethal, 23 per cent are neutral or lead to mutants with a virulence immediately close to the virulence of *i*, and 36 per cent are mutants with a virulence more distant from *i*'s (figure 1*b*). The latter proportion is evenly distributed among all possible viable mutants with a virulence value not adjacent to *i*'s. By sampling from this Dirichlet distribution, we obtained the rows of the transition probabilities matrix depicted in figure 1*b*. We use this matrix as our model of mutations, which, with relatively high probability (on average 0.41) produces unviable mutants, favours transitions among close virulence values, and allows for sporadic jumps between far distant values. Unless otherwise stated, this matrix is kept constant for the simulations we performed in this study.

In summary, our model, including mutations, can be written in the case of three viral strains as follows: 2.1

This ordinary differential equations (ODE) system (see (A 1) in the appendix for the model's equations in full generality, i.e. for an arbitrary number *n* > 1 of viral strains) describes the abundance of uninfected cells, *x*, that are replenished from an external supply at constant rate *λ* and die at rate *d*. Cells are infected by a variable pool of viruses *v*_{i}, characterized individually by the index according to their cell-killing rate *a*_{i}. The infection takes place with efficiency *β*. Singly infected cells, *y*_{i}, and co-infected cells, *y*_{jk}, die and release viral offspring at rate *a*_{i} and *a*_{jk}, the virulence of the respective strains (more on *a*_{jk} below). Free virus, *v*_{i}, is produced subject to mutations at rate *M*_{ji}, *j* = 1, … , *n*, and lytic bursts of average size *K*. Free virus is inactivated at rate *u*. Typical values of the parameters based on previous experiments with FMDV [19,37] are *a*_{1} = 0.15 *h*^{−1}, *a*_{2} = 0.25 *h*^{−1}, *a*_{3} = 0.35 *h*^{−1}, *β* = 5 × 10^{−8} *h*^{−1}, *K* = 150 viruses, *u* = 0.15 *h*^{−1}, *d* = 0.05 *h*^{−1} and *λ* = 10^{5} *h*^{−1}.

The parameters *c*_{i,jk} denote the proportion by which a cell co-infected with viruses of types *j* and *k* produce viral offspring of type *i*, where *i* ∈ {*j*,*k*}. We implement the competition–colonization trade-off by assuming intracellular competitiveness to be proportional to the reciprocal of virulence and set *c*_{i,jk} = *a*_{i}^{−1}/(*a*_{j}^{−1} + *a*_{k}^{−1}). Furthermore, co-infected cells die at the minimum rate of the two co-infecting strains, *a*_{jk} = min (*a*_{j}, *a*_{k}). For the alternative assumption of no intracellular viral interference, we set *c*_{i,jk} = *a*_{i} /(*a*_{j} + *a*_{k}) and *a*_{jk} = max(*a*_{j}, *a*_{k}).

We investigate the *n*-viral-strains model (A 1) (see appendix) for a value of *n* large enough to model realistic populations with a broad spectrum of viral variants. The following initial conditions were used in all simulations: *x*(0) = *λ*/*d*, *y*_{i}(0) = 0, *y*_{ij}(0) = 0 for all *i*,*j* = 1, … , *n*. The values of *v*_{i}(0) are set according to initial distributions of virulence further specified later in the text.

## 3. Results

### 3.1. Competition–colonization dynamics

On the basis of the experimental data presented in Ojosnegros *et al*. [19], we have simulated viral co-infection dynamics using 60 different viral variants and their pairwise interactions under the competition–colonization trade-off *c*_{i,jk} ∝ 1/*a*_{i}. According to this trade-off, the higher the virulence *a*_{i} of a virus, the lower the proportion *c*_{i,jk} of the progeny produced in co-infected cells.

The range of virulence chosen was [*d*,0.5], where *d* is the natural death rate of uninfected cells. The upper bound of this interval is taken from the maximum cell-killing rate described for FMDV, a highly pathogenic virus [37]. The choice of the lower bound *d* is based on the assumption that a viral infection significantly modifies the biology of the cell and increases its death rate to the virulence of the infecting strain. Decreased cell death rates owing to infection, as might occur with oncogenic viruses, are not considered here.

Every viral strain is defined by its virulence value and mutation allows for transition between classes. The model is, therefore, conceived in such a way that all populations of—in terms of virulence—different mutants are modelled and mutations give rise to only such mutants. In other words, mutations do not generate new viral variants not contemplated *a priori* in the model.

The interval of virulence was equidistantly sampled, yielding 60 different viable viral variants with a difference of virulence equal to *h*: = (0.5 − *d*)/60 between adjacent strains. The lowest virulence value of *d* was assigned to all non-viable mutants that arise as a result of mutational processes. This assignment is not to be interpreted as the non-viable mutants having a very low virulence, because non-viable mutants are uncapable of infecting cells (*β* = 0) and thus the concept of virulence no longer applies. Rather, this assignment was made in order to preserve the structure of the implemented model for simulation purposes. The number 60 was chosen as a compromise to get sufficient coverage of the interval of virulence while keeping the computational cost of simulations in a reasonable range.

Competitor variants have low virulence *a* and high competitiveness *c*, while colonizer variants have high virulence and low intracellular competitiveness. According to the results of Ojosnegros *et al*. [19], the life span of co-infected cells cannot be statistically distinguished from the life span of cells singly infected with the least virulent strain. This observation suggested the use of the smallest virulence of the two co-infecting viruses as the *per capita* death rate of co-infected cells in our model.

The dynamics of this model are shown in figure 2*a*. Uninfected cells become infected and produce progeny viruses during cell lysis. This process leads to a peak of viremia after about 10–20 h. Afterwards, viremia slightly declines to an equilibrium value as a result of the balance between external supply of cells and virus-induced cell death. At early stages of the infection, virulent variants dominate the population. As the infection progresses, competitor variants (higher *c*) increase their relative abundances in the population. At equilibrium, competitors and colonizers coexist. The succession of competitors by colonizers eventually leads to attenuation, i.e. reduction of average virulence, of the whole viral population.

In order to assess the robustness of these findings with respect to variation of the model parameters and the initial conditions, we conducted many simulations with perturbed parameter values. The population size was fixed to 10 000 viruses, and the proportion of each variant was randomly chosen in each simulation run. Among other instances, the simulations included initial excess of either colonizers or competitors. Additional simulations were run including an initial amount of viruses above the steady-state viral load. All simulations indicated that the equilibrium state, where the 60 variants coexist, remains invariant (data not shown), stressing the robustness of the model predictions regarding both, the qualitative dynamics and the steady state.

Additional random variations of the remaining parameters only slightly affected the dynamics. The simulations were carried out using a Gaussian distribution of each parameter with mean equal to the typical value specified above and variance one half of the mean. Variations in burst size *K*, the external supply of cells *λ* and the stability of viruses *u* produced similar effects. The total viral load increased or decreased accordingly with variations of the parameters, but the relative abundance of the strains at equilibrium remained constant. If *β* was varied, the dynamics run faster or slower, but the equilibrium was not affected. Variations in the natural death rate of uninfected cells, *d*, had little or no effect at all on the dynamics or the equilibrium. When considering the higher cell death rate for co-infected cells (under the current assumption the less-virulent virus imposes its killing rate on co-infected cells), the dynamics of the infection progress faster, but the equilibrium abundance of viruses is not affected. For the effects of sampling different mutation matrices *M*_{ij} from the Dirichlet distribution elucidated earlier, see later text.

In summary, the simulation results suggest that the model with 60 viable and one pool of unviable viruses has an asymptotically stable fixed point with a large basin of attraction.

### 3.2. Competition without intracellular interference

Many mathematical models for the evolution of virulence in viruses do not take co-infections into account [12,14,38]. The amount of co-infected cells, however, has been proposed to vary linearly with the number of singly infected cells [39].

When co-infections are considered, it is often assumed that parasites with higher virulence outcompete less-virulent strains also when co-infecting the same host, i.e. colonizers are also the better competitors [33,34,40]. This assumption is in contrast to our observations with FMDV [19] and it neglects intracellular interference during replication in host cells coinfected with different variants [5,24–27,41,42].

For comparison with the competition–colonization assumption, we analysed the model of no intracellular interference by setting *c*_{i,jk} = *a*_{i}/(*a*_{j} + *a*_{k}) and *a*_{kj} = max (*a*_{k}, *a*_{j}) in equation (A 1). The population dynamics of the two models are qualitatively different (figure 2*a*,*b*). At early stages of infection, highly virulent strains have an advantage in both models. However, without intracellular interference, competitors never dominate in the population. The advantage of colonizers at the end of the infection is however slightly smaller than at the initial stages of the infection (see figure 2*b*).

### 3.3. Virulence evolution

The virulence of the whole population depends on the relative proportions of competitors and colonizers and their respective virulence levels. As a measure of population virulence, we consider the average virulence . We have analysed the time course of the population virulence for the two models discussed earlier (figure 2*c*,*d*).

Under the competition–colonization trade-off, at early stages of the infection when the viral load reaches a maximum, the average virulence is maximal and the population is dominated by colonizers. Afterwards, both viral load and the average virulence decrease substantially. This final attenuation of the population is owing to the dominance of competitors.

In the absence of intracellular interference, the population virulence dynamics shows a less-pronounced qualitative change (figure 2*d*). After the initial increase in virulence, the average virulence barely drops and stays high during the entire infection. Colonizers are always the dominant species in this type of competition.

In order to assess the strength of the attenuation effect in the competition–colonization dynamics, we have performed multiple simulations, starting with a clonal population consisting of a single viral strain. For each such simulation, we chose a different initial strain and performed different simulations until the whole virulence spectrum considered in the model was covered (figure 2*e*,*f*). The virulence of the initial variant defines the initial average virulence of the population. During the infection, the mutations broaden the virulence spectra, attenuating initially highly virulent populations, or increasing the virulence of less-virulent ones. In all cases, under the effect of the competition–colonization dynamics, the average virulence reaches a lower value at equilibrium than that achieved without intracellular interference.

Thus, two different phenomena modulate the average virulence in the simulations, namely the competition process and the diversity resulting from erroneous replication. While the competition tends to favour competitor variants and attenuate the phenotype of the population, the mutation effect is conditional on the initial diversity of the population. The different trajectories of virulence shown in figure 2*e* collapse to the same value of average virulence at a steady state, suggesting the presence of an absorbing state.

### 3.4. Evolution of virulence distributions

In order to investigate the time evolution of virulence in a diverse viral quasispecies under the competition–colonization trade-off, we need to keep track of the distribution of virulence during an infection. This population-level perspective on virulence is not revealed by summary statistics or consensus measures, nor is it easily accessible from the time trajectories of figure 2.

Figure 3*a* shows the time evolution of a uniform initial virulence distribution of 60 different viral strains (see also the electronic supplementary material, video S1). The other parameter values are the same as in the virus population simulations (figure 2).

In this simulation, we can observe the key qualitative features of the process. The time evolution displays a two-step behaviour. During the initial phase, the more virulent strains are amplified and the virulence distribution is in favour of colonizers. Then a qualitative change occurs and the distribution becomes more uniform, without a bias towards extreme virulence values. Finally, the distribution changes again to give advantage to less-virulent competitors. This distribution becomes stationary.

Figure 3*b* shows the time evolution of a less-idealized initial virulence distribution (see also the electronic supplementary material, video S2). The initial abundances of the 60 viral strains (figure 3*b*, black crosses at *t* = 0) were obtained from a mixture distribution of seven Gaussian distributions with different means, variances and weights (figure 3*b*, solid line at *t* = 0). The time trajectory of this simulation is the one displayed in figure 2*a*. In this simulation, we again observe the two-step behaviour. A steady state is reached where all viruses coexist but competitors dominate.

Both simulations exemplify the mixing effect of mutations that eliminates the initial structure imposed by the initial distribution. Once the distribution has been mixed and enters a phase without a bias towards extreme virulence values, both simulations display similar dynamics and converge to the same stationary distribution. These results strongly suggest again the presence of a globally attracting fixed point with a seemingly large basin of attraction.

Furthermore, we explored the effect of sampling different mutation matrices *M*_{ij} (from the Dirichlet distribution elucidated above) on the stationary state distribution. To this end, we ran multiple simulations, each one using a different sampled matrix, and averaged the abundances of each viral strain at steady state. We compared this averaged distribution of virulence with the (stationary) distribution observed at steady state when the simulation was run using a matrix *M*_{ij}, with entries being equal to the expected values of the Dirichlet distribution. That is, 41 per cent of mutations are lethal, 23 per cent are neutral or lead to immediately close virulence values and 36 per cent generate mutants with a virulence more distant from the strain considered. The result is depicted in figure 4 and clearly demonstrates the tendency of the stationary distributions to be located around the stationary distribution that arises when the matrix of average probabilities is used.

## 4. Discussion

Experiments describing the molecular evolution of viral virulence are scarce and our knowledge about the mechanisms underlying this process is thus very limited. In a previous study, the diversification in cell culture of a clonal population into competitor and colonizer strategies was described in detail [19,23]. In the present study, we adapted well-established mathematical models to assess the evolution of these two host exploitation strategies during intra-host infections.

We have assumed that two strategies are traded off against each other. One important difference between intra-host and cell culture infections is the presence of a replenished pool of susceptible cells *in vivo*. The constant supply of new cells gives continuity to the system and allows assessment of the long-term behaviour of the population composition. We have simulated mutating, virulence-heterogeneous populations composed of 60 variants. All simulations predicted the same two basic features: sequential dominance of colonizers followed by competitors and the existence of a steady state of coexistence dominated by low-virulence competitors.

Competition and colonization strategies are subject to strong density-dependent selection [19,23]. This type of selection can account for the observed sequential domination of the infection. Early in the infection, the density of viruses is very low owing to the high availability of susceptible cells. The low density of viruses allows colonizers to spread faster in the initial stages of the infection. Progressively, the density of viruses increases along with the number of co-infections. Because competitors are more efficient in intracellular replication, during later stages of the infection, competitors take over and dominate in the population. The two-step behaviour is maintained after a perturbation of the initial conditions. Even when competitors are initially dominant, they will again be replaced by colonizers.

Infections of the gypsy moth (*Lymantria dispar*) with nuclear polyhedrosis virus resemble the two-step dynamics described here [43]. During sequential passages (infections) of the virus from moth to moth, the virulence of the virus sampled at initial or later stages of the infection oscillates from high to low, respectively. The lack of an adaptive immune system, which is also not studied in our current model, may have contributed to the good fit of the results from both studies. Further experimentation along similar lines would be of great interest to understand the evolution of viral virulence in real infections, beyond cell culture experiments. A good test of our model would be to perform serial infections of animals with an RNA virus, taking samples of the virus for the next infection during the peak of viremia or at steady state. Measuring the virulence of viruses obtained from each line of experiments would shed light on the evolution of virulence *in vivo*.

The switch of the favoured strategy, from colonization to competition, meets the replication requirements of the virus at each stage of the infection. Early in the infection, the virus benefits from colonizing the organism as fast as possible, before the immune response is mounted. However, the less-virulent variants have been predicted to maximize the viral load and the amount of infected cells. For a single virus model, if the basic reproductive number *R*_{0}≫1, then the equilibrium abundance of viruses and infected cells is approximately given by *v** ≈(*λ**k*)/(*au*) and *y** ≈*λ*/*a* [3]. These expressions imply that the equilibrium abundance of viruses and infected cells will be higher in organisms infected by low-virulence strains (low *a*). For this reason, once the organism is colonized, the viral population can benefit from the imposition of competitors.

The sequential replacement of colonizers by competitors during the infection of an organism has an interesting parallel in ecology successions [44], where empty habitats are typically populated initially by fast-spreading plants with shorter life cycles. Stronger competitors will successively replace the faster colonizers until the ecosystem reaches the climax.

### 4.1. Variability

Our simulations suggest that the infection eventually reaches a steady state where all variants coexist. We have carried out a rigorous mathematical analysis of the two-virus model in the absence of mutations [36]. Under conditions that allow for viral spread (i.e. *R*_{0} >1), there is a local, asymptotically stable equilibrium in which both viral strains coexist. The equilibrium abundances of viruses at the steady state satisfy *v*_{1}^{*} /*v*_{2}^{*} =*a*_{2}/*a*_{1} . This expression implies an advantage of strains of lower virulence in agreement with the observations derived from the simulations of the present work, although the effect is more moderate owing to the smoothing effect of mutations.

In theoretical ecology, a trade-off between the ability of each individual to colonize unoccupied territory and to compete with others for the same habitat patch has been suggested as a potential explanation for coexistence and species diversity in patchy habitats [33,45–48]. This trade-off has indeed been observed in plant and insect populations [49–51]. Our model is a space-implicit model, where the viruses replicate in patches defined by individual cells. The generation of new mutants by mutation and its fixation in the population are coupled in time in our error-prone replication model, unlike in classical ecology models. This fast dynamics prevents the complete extinction of any virulence class that can be beneficial when adapting to a different environment, for example, by increasing the invasive fitness when infecting a host that may be already colonized by another viral strain.

Variability at a population-level is a fundamental trait in the life cycle of RNA viruses, because they need to adapt to extremely changing environments. Pathology [52], fitness [53], evasion of antiviral drugs [54,55] and immune response [6] are critically linked to population diversity.

### 4.2. Virulence attenuation

Our simulations indicate that two mechanisms, mutation and the competition–colonization trade-off, modulate the composition of the population, sometimes in a complementary way and sometimes in an opposite way. Mutation tends to broaden the distribution of viral variants in the population. In those populations, where the initial distribution of virulence values is highly skewed towards the domination of the more virulent variants, the low-virulence variants will increase their proportion in the population simply by the effect of mutation. Conversely, virulence distributions that are initially highly skewed towards the presence of low-virulence competitors will increase the amount of virulent colonizers over time. In addition to the impact of mutations on the steady state, the competition–colonization mechanism always favours the increment of the relative abundance of competitors at steady state. If simulations are compared between populations with the same initial distributions of virulence values, but one subject to the rule *c*_{i,jk} ∝ 1/*a*_{i} (competition–colonization trade-off) and the other subject to *c*_{i,jk} ∝ *a*_{i} (replication without interference), then the amount of competitors at equilibrium will always be higher with the trade-off. This result suggests an attenuating role for the competition–colonization mechanism in viral replication.

Attenuation has been documented for several infections, both at the intra-host level and as a trend during epidemics [42,56–58]. Our model has been derived from observations of real experiments carried out with different RNA viruses. The rationale for the trade-off between competition and colonization is that, during the replication of RNA viruses, negative-dominant mutants that can benefit from the replication of other mutants in co-infected cells arise. When co-infections occur, the population is enriched for these mutants, called here competitors, which act as defectors in the sense of evolutionary game theory [24]. Cell culture infections carried out at a high density of viruses tend to select competitor strains that dominate over strains adapted to replicate without co-infections, as demonstrated for FMDV, vesicular stomatitis virus and bacteriophage *Φ*6, among other viruses [24–28,59]. In the extreme case, such defective mutants harbour internal deletions or lethal mutations and they require the co-infection of a helper virus to complete their replication cycle. It has been documented that defective viruses play a key role in the attenuation of several diseases [42]. This link between co-infection and disease attenuation is worthy of further investigation as co-infections are frequent during virus–host infections [60,61].

Despite the earlier-mentioned experimental evidence, the evolution of virulence has been classically studied under the contrary assumption of virulent strains being also more competitive. This assumption may hold for some parasites, such as bacteria or protozoa. These parasites do not necessarily exchange genetic products among individuals, which can result in limited interference [62,63]. We have compared the competition–colonization trade-off with the situation where there is no interference between mutants and the more virulent strain is also more efficient in co-infected cells. Our simulations suggest the existence of steady state where different variants coexist dominated by virulent colonizers, in agreement with previous work where the same assumption was made [34,35]. Hence, this model would imply constantly increasing levels of virulence, in contrast to many experimental and clinical observations.

In conclusion, we have presented a model to study the evolution of virulence during virus–host interaction, which is based on experimental observations. Our results indicate that virulence is a dynamic feature of the entire population and the interaction between its components.

## Acknowledgments

We are indebted to Moritz Lang for expert assistance with Matlab. The authors thank the anonymous referees for their valuable suggestions.

## Appendix: mathematical models

All models discussed in this paper are specializations of the following general multi-strain model:

A1

where *w*_{ℓ}(*j*,*k*) = 1 if *j* = ℓ or *k* = ℓ , and otherwise *w*_{ℓ}(*j*,*k*) = 0. The model does not explicitly account for the order of infection. The three-virus model (2.1) is a special case of this ODE system, obtained by setting *n* = 3. The competition–colonization model is derived from (A 1) by setting *a*_{jk} = min(*a*_{j}, *a*_{k}) and *c*_{i,jk} = *a*_{i}^{−1}/(*a*_{j}^{−1}+*a*_{k}^{−1}). The lack of intracellular interference is modelled by (A 1) with *a*_{jk} = max(*a*_{j}, *a*_{k}) and *c*_{i,jk} = *a*_{i}/(*a*_{j} + *a*_{k}).

- Received February 29, 2012.
- Accepted March 26, 2012.

- This journal is © 2012 The Royal Society