Host-multipathogen infection model

We consider a two-pathogen compartmental model that tracks hosts according to their pathogen-specific infection status. The infection by each strain is described by a susceptible-infectious-recovered (SIR) dynamics [55]. We assume full cross-immunity, so that after infection by one strain the host is found to be fully immune to the other strain. We also assume that no other interaction among strains occurs besides full cross-immunity, therefore neglecting coinfection or superinfection events, a plausible assumption for rapid acute infections such as influenza. The case of partial cross-immunity among strains is also presented in the Text S1, however the full exploration of this case and of the resulting phase space of the system will be the object of further studies.

The SIR dynamics for strain , with , is ruled by the transition rates, and , representing the disease transmissibility rate (for the transition from susceptible to infectious) and the recovery rate (for the transition from infectious to recovered), respectively. The dynamics is characterized by the basic reproductive number , defined as the expected number of secondary infections that one infectious host can produce during its lifetime as an infectious host placed in an entirely susceptible population, leading to the threshold condition for an epidemic outbreak in the population, [55]. In our study, we assume that the two strains have the same basic reproductive number, , but different infectious periods. This represents a case in which the pathogens have the same transmission potential and generate outbreaks characterized by the same impact on the population expressed in terms of attack rates. The epidemic waves are however different, unfolding with different timescales, the faster being the one characterized by the shorter generation time. In our case the generation time is uniquely determined by the infectious period [55] (see Figure 1) and without loss of generality we consider a fast strain with infectious period and a slow strain with infectious period . The parameter quantifies the timescale separation.

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Figure 1. Schematic representation of the host-multipathogen metapopulation system.

At the macroscopic level the system is composed by a network of subpopulations connected via communication links that allow individuals to migrate from one subpopulation to the other. Inside each subpopulation the epidemic process take place. Susceptible individuals can be infected by the slow (fast) strain and change their status to ; infected individuals enter into the recovered class at rate and , for the slow and fast strain, respectively. Different epidemic waves are produced by the two strains when unfolding independently in a population, as shown by the number of new cases (incidence) over time.

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The infection transmission is modeled by dividing the population of individuals into four compartments: susceptible , infected by the fast strain , infected by the slow strain and recovered , i.e. immune to both strains. Each susceptible individual can contract either strain with the corresponding force of infection, or , where is imposed by the equivalence of the basic reproductive numbers of the two strains. The two infection events are independent and mutually exclusive because of the assumptions considered.

Host metapopulation network

The multipathogen disease dynamics affects a spatially structured population of hosts modeled through a metapopulation system. This theoretical framework was first used in population ecology, genetics and adaptive evolution to describe population dynamics whenever the spatial structure of populations is known to play a key role in the system's evolution [56]–[59] and later applied to understand the epidemic dynamics on such substrates [60]–[64]. For the case of epidemic modelling, the infectious disease spreads in an environment characterized by a non-continuous spatial distribution of susceptible hosts and the pathogen diffusion depends on the ability of hosts to move from one region of the system to another one, connecting otherwise isolated communities [58], [59]. Hosts mix homogeneously within the local communities (also called subpopulations or patches or nodes of the metapopulation network), whereas at the global, system-wide level, patches are coupled through the migration of hosts (represented in terms of mobility connections between patches), as schematically shown in Figure 1.

Here we consider a metapopulation network with subpopulations. To each node , we assign an initial number of individuals, , and a degree denoting the number of connections the node has with other subpopulations in terms of mobility processes. The degrees of the nodes are distributed according to a given probability distribution , which we choose to represent the two most abundant situations in real systems – namely, a Poisson distribution accounting for homogeneous networks of contacts and a power-law functional form which represents the case in which mobility patterns are highly heterogeneous. To compare the effects of changes in the structural pattern of the subpopulations with no variations of the corresponding average values, we set the average degree of both networks, , to be the same. Homogeneous networks are generated following the Erdős-Rényi algorithm [65], which consists of assigning a link between each pair of nodes with probability . It models a fairly homogenous system with low degree fluctuations. On the other hand, heterogeneous networks characterized by a power-law degree distribution, where we consider , are generated using an uncorrelated configuration model [66], [67]. In this second case, the probability distribution is characterized by a second moment that is much larger than the first, which makes it critical to explicitly take into account degree fluctuations. This feature is distinctive of a vast majority of social and demographic systems that have been empirically characterized [19]–[25], [68].

Host mobility pattern

In case of homogenous traveling probability, mobility fluxes are modeled by assigning to each individual in subpopulation a probability per unit of time to travel to another neighboring subpopulation . We assume that such probability is constant across nodes, namely , and that individuals leaving a subpopulation choose at random one of the available links [36], so that the probability of traveling from to is given by . According to the value of , different mobility scenarios emerge: high values of yield large mobility fluxes resulting in a well mixed metapopulation system where individuals easily move from one patch to another; on the contrary small probability values result in a dynamically fragmented scenario in which patches are fairly isolated. The mobility process is described by the following diffusion equation:(1)where the sum runs over the set of the nearest neighbors. According to this equation the population distribution at equilibrium is given by(2)where is the average population size. The main variables used in the model and the corresponding ranges of values considered are reported in Table 1. We also tested in the Text S1 more realistic definitions of host mobility, following empirical findings.

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Table 1. Model variables and their corresponding values.

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Computational modeling of competing pathogens

To simulate the spread of the two strains on the metapopulation system of susceptible hosts, we initialize the number of individuals of each subpopulation at the equilibrium value given by Eq. (2). We then seed randomly chosen subpopulations for each strain by setting a proportion equal to of the local population size in the corresponding infectious class. These conditions ensure the start of the outbreak for each strain for the values of the basic reproductive number considered, and at the same time they aim to avoid competition at the initial stage of the multistrain epidemic. Values of the number of initially infected nodes different from 50 where tested in order to check that this does not alter the simulation results. Once the system is initialized, the transmission dynamics of the two strains is reproduced by means of Monte Carlo numerical simulations at the discrete individual level. We consider hosts as integer units and we explicitly simulate both their mobility among different subpopulations and the infection transmission within each subpopulation as discrete-time stochastic processes, with fixed time step representing the unitary time scale of the process. To this end, at each time step, the number of hosts traveling along any connection of the system belonging to any compartment and the number of new infectious and recovered hosts for each subpopulation are extracted randomly from binomial and multinomial distributions to consider all possible outcomes of these events. Further details on the algorithm used for the simulations, as well as initial conditions and parameters, are described in Section 1 of the Text S1.

For each set of parameters we simulate 2,000 stochastic realizations of the spatial epidemic spreading averaging over different initial conditions, and over different instances of the metapopulation network that defines the spatial structure of the system population. For each scenario, we collect statistics of epidemiological quantities, including the number of subpopulations affected by each strain, the outbreak probability, and the incidence and attack rates of each strain, both at the global level and within each subpopulation. This allows to monitor the evolution of the two epidemics, their impact on the system, and the result of the competition process.

Invasion threshold of a pathogen in metapopulation models

Several works have recently studied the global spreading of a single strain SIR-like epidemic in metapopulation models [34]–[41], [69], [70]. The threshold condition is sufficient for an epidemic outbreak to occur in a given subpopulation, but it does not guarantee the disease is able to spread globally. Low diffusion rates may indeed hinder a pathogen to disperse to other patches before it goes extinct locally, thus preventing the persistence of the virus and its spatial spread in the host population. The global spreading of an infectious disease in a metapopulation model is captured by the definition of an additional predictor of the disease dynamics, , regulating the number of subpopulations that become infected from a single initially infected subpopulation, analogously to the reproductive number at the individual level [71]–[73]. The parameter defines a global invasion threshold: the condition guarantees that the epidemic taking place in the seeding subpopulation is able to spread at the global scale reaching a non-infinitesimal fraction of the metapopulation system. depends on several factors, including disease parameters, demography, metapopulation network structure, travel fluxes and mobility timescales.

Theoretical studies in [34]–[41], [69], [70] have addressed the impact of empirically observed features on , thus providing a better understanding about how mobility patterns and demography affect the invasion threshold of an infection. The current analytical framework allows to get an expression for , in which the impact of several sources of heterogeneities in the topology of the metapopulation system, traffic fluxes [34]–[39], [69], [70] and time scales [40] can be quantitatively assessed. In order to provide an understanding of the mechanisms shaping the global invasion condition of a multistrain epidemic, we review here the derivation of for the simplest case, of a single strain on a homogeneous metapopulation network with uniform mobility pattern.

In a homogenous system, in which topological fluctuations can be neglected, all nodes can be assumed to have the same degree . If the mobility dynamics is described by Eq. (1), an expression for can be obtained by formalizing the seeding process of infected hosts through their migration from one patch to another. The probability that an infected patch will seed the epidemic in a disease-free patch is given by [74], where is the number of infectious hosts who traveled from to during the entire duration of the outbreak, while infectious. The latter quantity can be estimated as follows. The total number of individuals that experience the disease during the epidemic unfolding within the subpopulation will be , where is the attack rate given by the SIR equations and is equal to for all nodes – as recovered by Eq. (2) in the case . Each infected individual stays in the infectious state for an average time equal to the inverse of the recovery rate, during which it can travel to the neighboring subpopulation at rate . To a first approximation we can therefore consider that the number of seeds sent from to during the duration of the outbreak is given by . If we model the invasion from one patch to another in terms of a branching process, we obtain that an infected subpopulation infects on average subpopulations, where is the number of connections along which the disease can spread. This leads to the following expression for in the homogeneous assumption(3)

As discussed, the global invasion threshold quantifies the spreading potential of an epidemic at the global level. For any set of parameters values characterizing the infection dynamics, the threshold condition defines a critical value of the host mobility below which the epidemic is not able to spread globally. It is worth remarking that this transition cannot be uncovered by continuous deterministic models because of the stochastic features of the contagion process and the discrete nature of circulating hosts.

Let us now consider the case of two competing strains – one slow and another fast. Even if both strains have the same transmission potential at the local level, namely the same , their large scale spreading potential, encoded in , would be different. As shown by Eq. (3), is indeed an increasing function of the infectious period, therefore . This indicates that, in a metapopulation system of fully susceptible hosts, the slow strain would be able to infect on average a larger number of subpopulations than the fast strain, although at a much slower pace. As we will see in the following section the trade-off between transmission potential and spreading time-scale crucially impacts the population level competition among the two epidemics.

Two-strain competition

We consider two strains with relatively high transmission potential, i.e. , and infectious rates given by and . As an indicator of the outcome of the competition between the two strains we consider the final number of subpopulations and affected by each strain during the outbreak. We say that a patch has been affected by a strain if at least a fraction of the population within the patch has contracted the disease. We set equal to and we checked that the results are not sensitive to the value of this parameter. By looking at the average of and when varies, we inspect several competition scenarios that are determined by mobility regimes.

Figure 2A shows the results of the multistrain epidemic simulations assuming a homogeneous metapopulation structure. Different mobility regimes give rise to different coexistence and dominance patterns. For large values of the fast strain dominates affecting the vast majority of subpopulations infected in the system, the slow strain being constrained to roughly of the patches. As the value of decreases, the system-wide spreading potential of the slow strain progressively grows at the expense of the fast one, until a cross-over takes place at diffusion rate . This intermediate regime is characterized by the codominance of the two strains [75], each one affecting approximately the same portion () of infected subpopulations. Below this point, the slow strain becomes dominant, whereas the fast one only induces local outbreaks propagating through a small number of subpopulations. Eventually, for very small values of none of the strains is able to spread geographically and no global outbreak occurs. Figure 2b further illustrates this phenomenology by plotting the average value of the ratio as a function of . Values of the ratio much larger than 1 indicate the dominance of the slow strain, and values corresponding to to the opposite scenario in which the fast strain dominates. The codominance phase is obtained for values of the ratio close to 1, with the cross-over diffusion rate given by the intersect of the curve with the horizontal line . The figure also compares heterogeneous and homogeneous metapopulation systems. The results show that the behavior is qualitatively the same for both network structures, the main quantitative difference being given by a lower value in the heterogeneous case. Similar results are also recovered with a different model for the mobility fluxes as detailed in Section 2 of the .

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Figure 2. Competition between strains.

(A) Fraction of subpopulations infected by the fast and slow strains as a function of . The quantity plotted is the median and the confidence interval over 2000 stochastic runs. Simulations were performed on a random homogeneous network. (B) Ratio as a function of for both homogenous and heterogenous networks. The inter-quartile range is not displayed for the sake of visualization. In both panels the networks have average degree . Both strains have . Other parameters are and .

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The observed behavior can be understood according to the following intuitive explanation. After the two epidemics are seeded in their initial locations they evolve independently at the beginning, until one of the two strains reaches a subpopulation that has already been infected by the other strain, thus finding part of the population immune, i.e. a reduced pool of susceptible hosts to infect. This may prevent the strain to widely spread within the patch and diffuse further along the mobility connections towards other neighboring nodes. This competing mechanism favors the strain that spreads more rapidly and more efficiently from one patch to another, features that change depending on the mobility regime. When the traveling rate is high, the whole system is at risk of a major epidemic because of the large rate of mixing across different patches. Both and are much greater than 1, implying that the two epidemics would successfully reach the global invasion of the system, in absence of competition. When the two strains are competing on the same metapopulation system, the relevant factor for dominating the spread is given by the spreading speed; the shorter the infectious period and the more rapidly the strain reaches a large fraction of the system patches that will thus not be invaded by the slow strain. However by decreasing the value of , of each strain also decreases. In the proximity of the invasion threshold the condition becomes relevant for the spreading dynamics and favours the slow strain which percolates more efficiently through the network. Indeed, the global epidemic time scale is not anymore dominated by the local velocity of transmission but rather by the mobility time scale of individuals. Hosts contracting the slow strain remain infectious for a longer time and thus have more chances to migrate while infectious. The low mobility rate, coupled with a short infectious period, hinders the movement of infectious hosts, resulting in a lower probability of infecting a neighboring patch. We provide a more quantitative understanding of the crossover behavior in the section dedicated to the analytical discussion of the results.

The same argument applies to explain the difference between the two network topologies observed in Figure 2B. The topological fluctuations that characterize the heterogeneous topology induce larger values of the parameter with respect to the corresponding homogeneous network (provided that the rest of parameters is kept the same) [34]–[36]. Therefore the invasion threshold becomes larger than one for the two strains for smaller values of the mobility rate in the heterogenous case, which results in a shift of the cross-over diffusion rate towards lower values.

In the case of partial cross-immunity presented in the Text S1, we find that the main results reported for the full cross-immunity scenario still hold. Specifically, we have simulated situations in which recovered individuals from one strain may have up to cross-immunity to the other strain, which roughly correspond to estimates for diverse degrees of antigenic drift of influenza [76].

Within-patch coexistence and spreading pattern

We now focus on characterizing the coexistence of both strains at the within-patch level and their spread at the global spatial level. We define the coexistence probability as the probability that within the same subpopulation both strains produce at least 1% of the population infected. For both heterogeneous and homogeneous mobility networks is an increasing function of the traveling rate (Figure 3), therefore mobility favors the coexistence of the two strains within the same subpopulation. Coexistence is however generally unlikely to occur in a vast fraction of subpopulations, given the relatively small values of the probability obtained, showing that the two strains rarely coexist within the same subpopulation and the competition takes place at the metapopulation level.

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Figure 3. Coexistence probability within the same patch.

is defined as the probability that within the same subpopulation both strains produce at least of the population infected. The quantity plotted is the average and the standard deviation over 2000 runs. The parameters used for the simulations are the same as in Figure 2.

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To further characterize the two strain coexistence within a patch, we measure for each patch the attack rate at the end of the outbreak, defined by a two-dimensional variable , where is the fraction of hosts affected by the fast (slow) strain within the patch during the outbreak. In all mobility regimes explored, the interaction between the two strains can be mapped to a small region of the space. Specifically, the strains always produce attack rates with a strictly linear dependence (Figure 4, panels A,B,C), characterized by probability distributions centered around and and with different proportions in the three diffusion regimes considered (Figure 4, panels D,E,F). Moreover configurations in which only one strain is present in a subpopulation have a frequency of occurrence much higher than configurations where the two strains co-exist, further confirming the results of Figure 3.

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Figure 4. Within-patch coexistence and strain-specific attack rates.

A,B,C: heatmaps showing the frequency of occurrence of a given epidemic outcome within the patches, expressed in as obtained by numerical simulations. D,E,F: histogram of the within-patch attack rate (in ) for the slow and fast strains. From left to right, three different mobility regimes are displayed: in which the fast strain dominates (A,D), corresponding to the cross-over point (B,E), and in which the slow strain dominates (C,F).

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We explore whether the coexistence of the two strains at the local level may carry a spatial signature. In absence of georeferenced data in our model that is based on an abstract spatial network, we consider the topological properties of the patches as possible spatial indicators. Noticeable differences arise when the probability of within-patch coexistence is measures by degree classes (Figure 5). In both homogenous (panel A) and heterogenous (panel B) cases, is an increasing function of and it can vary over more than two orders of magnitude from poorly connected subpopulations to most connected ones. This behavior, although expected because highly connected patches are more likely to collect individuals from other subpopulations, highlights two different levels for strains competition in the system. On one side, in highly connected nodes the two strains compete at the single subpopulation level and the predominance of one of the two strains is dictated only by their epidemic parameters. Such behavior is mostly due to the fact that highly connected nodes are almost surely reached by infected individuals of both strains at the early stage of the spreading process. Thus, both strains are likely to infect a non-vanishing fraction of the node population at the same time, leading to higher probability of coexistence. On the other hand, as low connected nodes are harder to reach, the competition is mostly driven by the time at which one strain reaches the subpopulation. The first strain to disseminate to the low connected patch has likely enough time to infect a large fraction of the susceptible hosts before the arrival of the other strain. In this case the competition between the two strains acts at the metapopulation level as coexistence between the strains is almost zero.

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Figure 5. Probability of coexistence within a patch as a function of the patch connectivity .

Homogenous (A) and heterogeneous (B) cases are shown. Different traveling regimes are compared: they correspond to the scenarios in which the fast strain dominates (the highest value of considered in the two plots), the two strains coexist (intermediate value of ) and the slow strain dominates (smallest value of ). The quantity plotted is the average over 2000 runs; error bars are not displayed for the sake of visualization.

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To conclude our analysis of the system at the patches' degree level in Figure 6 we present the fraction of infected subpopulations with degree for the two strains as a function of the degree in the cross-over mobility region. In both homogenous and heterogenous networks the slow strain shows a higher incidence for low-degree nodes, whereas for intermediate and higher connectivities, the fast strain dominates the spreading process.

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Figure 6. Competition between strains per connectivity class in the cross-over regime.

Fraction of subpopulations infected by each strain within the degree class , , in the homogeneous (A) and heterogenous (B) networks. The two plots depict the behavior in the cross-over mobility regime ( in panel (A) and in panel (B)). The quantity plotted is the average over 2000 runs; error bars are not displayed for the sake of visualization.

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Impact of and

Finally, we focus on the two parameters that mainly affect the spreading of the strains and their interaction – namely, the reproductive number of both strains and the ratio between the infectious periods of the two strains. Variations of from 1.1 to 4 induce a variation of the cross-over mobility rate of more than two orders of magnitude (Figure 7A), with higher values of the basic reproductive number leading to smaller values of . The decrease observed in the cross-over rate is very rapid for , followed then by an almost constant value for larger values of in both network types, indicating the presence of a critical beyond which the interaction dynamics of the two strains is dominated only by the disease parameters and not by the mobility rate.

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Figure 7. Dependence of the cross-over diffusion rate on the epidemiological parameters.

Cross-over diffusion rate along with estimation error as a function of the reproductive number (A) and of (B) in the homogeneous and heterogeneous cases. The networks have average degree . Other parameters are , (A) and (B).

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Differently from , variations of do not strongly alter the value of the cross-over mobility rate, with a change of of one order of magnitude inducing variations in of less than (Figure 7B). Moreover, the initial fall off observed for at fixed (panel a) is not seen anymore. In both plots we note that the critical diffusion rate is smaller in the heterogeneous networks with respect to the homogenous ones for the whole range of parameters explored, confirming a favoring effect in the spatial spread of both strains as previously discussed.

To provide a specific example, we applied this framework to the case of two influenza-like strains spatially circulating on the real worldwide aviation network (assuming full cross-immunity and epidemiological parameters as in Figure 7B), we obtain that the air-transportation mobility scenario falls in the regime in which the fast strain is dominant for all the values of tested (more details are reported in the Text S1).

Analytical discussion

Here we focus on the case of homogeneous networks and propose a simplified analytical description of the dynamics to gain theoretical insights to further support the observed numerical behavior. We consider a continuous time approximation and assume that two strains do not interact at the early stage of the spreading process, in order to provide an estimation of the critical diffusion rate below which we have the dominance of the slow strain and above which we have the dominance of the fast strain. The basic approach is to treat the dynamics at the system level in terms of the usual SIR model in a well mixed population, considering the subpopulations as the elementary ingredients of the spreading process. Under this assumption, the number of infected subpopulations grows exponentially in time, and we can write(4)where is the duration of the outbreak in a single population and is the estimator of the invasion potential, as described in the Methods section, i.e. is the analogous of the basic reproductive number at the metapopulation level.

If we consider the case of two epidemics starting at different seeded subpopulations, by neglecting possible interactions among the two strains, we obtain that the ratio between the number of subpopulations infected by the slow strain and the number infected by the fast one is given by:(5)Our goal is to derive the cross-over diffusion rate at which we have that both strains cocirculate, which is given by the condition . Hence, from Eq. (5), we get(6)In the case of equal size populations, and for the same , it is possible to show that the timescale defining the epidemic unfolding, for instance the maximum of the removal rate, is well approximated by a linear dependence on [74], [77]. We can therefore assume that and substituting Eq. (3) into Eq. (6) we explicitly arrive to the crossover condition as(7)where simplifies and disappears from the equation.

Finally, denoting:(8)we have:(9)Eq. (9) can always be solved for numerically and, in some cases, analytically. The comparison between theoretical predictions and numerical simulation results shows a good agreement in the behavior of the cross-over diffusion rate as a function of (Figure 8), confirming that the analytical approximation is able to capture the fundamental mechanisms for competition between the two strains.

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Figure 8. Theoretical predictions.

Comparison between the numerical and theoretical cross-over diffusion rate as a function of the reproductive number for the case of homogeneous network. Numerical results are the average over stochastic runs, whereas theoretical values are obtained solving Eq. (9). The networks have average degree . Other parameters are and .

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