Metapopulation epidemic models with heterogeneous mixing and travel behaviour
- Andrea Apolloni^{1},
- Chiara Poletto^{2, 3, 4},
- José J Ramasco^{5},
- Pablo Jensen^{6} and
- Vittoria Colizza^{3, 4, 7}Email author
https://doi.org/10.1186/1742-4682-11-3
© Apolloni et al.; licensee BioMed Central Ltd. 2014
Received: 3 December 2013
Accepted: 6 January 2014
Published: 13 January 2014
Abstract
Background
Determining the pandemic potential of an emerging infectious disease and how it depends on the various epidemic and population aspects is critical for the preparation of an adequate response aimed at its control. The complex interplay between population movements in space and non-homogeneous mixing patterns have so far hindered the fundamental understanding of the conditions for spatial invasion through a general theoretical framework. To address this issue, we present an analytical modelling approach taking into account such interplay under general conditions of mobility and interactions, in the simplifying assumption of two population classes.
Methods
We describe a spatially structured population with non-homogeneous mixing and travel behaviour through a multi-host stochastic epidemic metapopulation model. Different population partitions, mixing patterns and mobility structures are considered, along with a specific application for the study of the role of age partition in the early spread of the 2009 H1N1 pandemic influenza.
Results
We provide a complete mathematical formulation of the model and derive a semi-analytical expression of the threshold condition for global invasion of an emerging infectious disease in the metapopulation system. A rich solution space is found that depends on the social partition of the population, the pattern of contacts across groups and their relative social activity, the travel attitude of each class, and the topological and traffic features of the mobility network. Reducing the activity of the less social group and reducing the cross-group mixing are predicted to be the most efficient strategies for controlling the pandemic potential in the case the less active group constitutes the majority of travellers. If instead traveling is dominated by the more social class, our model predicts the existence of an optimal across-groups mixing that maximises the pandemic potential of the disease, whereas the impact of variations in the activity of each group is less important.
Conclusions
The proposed modelling approach introduces a theoretical framework for the study of infectious diseases spread in a population with two layers of heterogeneity relevant for the local transmission and the spatial propagation of the disease. It can be used for pandemic preparedness studies to identify adequate interventions and quantitatively estimate the corresponding required effort, as well as in an emerging epidemic situation to assess the pandemic potential of the pathogen from population and early outbreak data.
Keywords
Metapopulation models Epidemic spreading Complex networks Mobility Mixing patterns Travel behaviourBackground
The spatial spread of directly transmitted infectious diseases depends on the interplay between local interactions among hosts, along which transmission can occur, and dissemination opportunities presented by the movements of hosts among different communities. The availability of increasingly large and detailed datasets describing contacts, mixing patterns, distribution in space and mobility of hosts have enabled a quantitative understanding of these two factors[1–11] and led to the development of data-driven mechanistic models to capture the epidemic dynamics of infectious diseases[7, 12–14].
Although numerical simulations have crucially contributed to our current ability to explain observed spatial epidemic patterns, predict future epidemic outcomes and evaluate strategies for their control, analytical methods offer an alternative valuable avenue for the assessment of an epidemic scenario that is able to clearly identify the key mechanisms at play and shed light on some of the complexity inherent in data-driven approaches. In the context of models for spatially transmitted infectious diseases, the metapopulation approach offers a theoretical framework that explicitly maps the spatial distribution of host population and mobility[15–18], while offering a tractable system under certain approximations[19, 20]. Originally introduced in the field of ecology and evolution[15], it considers a population subdivided into discrete local communities, where the infection transmission dynamics is described through standard compartmental schemes, coupled by connections representing the movements of hosts. Despite the mathematical complexity of explicitly considering the spatial dimension and non-trivial topologies connecting local communities, epidemic metapopulation approaches have shown their ability to analytically explain the failure of feasible mobility restriction measures[19–21], alert on the possible negative impact that adaptive travel behaviour of individuals may have on epidemic control[22], and interpret pathogen competition in space[23].
Based on network theory and reaction-diffusion approaches, these studies have quantified the potential for a global epidemic to occur in terms of a mathematical indicator, R_{∗}[19, 20], measuring the average number of subpopulations that an infected subpopulation may transmit the disease to, through mobility of infectious individuals during the outbreak duration. Values larger than 1 indicate that transmission can spatially propagate in the metapopulation system and reach global dimension, whereas epidemics with R_{∗} < 1 are contained at the source. Different mobility modes, traffic dynamics and path choices have been explored so far within the metapopulation framework[19, 20, 22, 24–27], however all these properties have been considered at aggregated fluxes level, implicitly assuming that all individuals resident in the same location are indistinguishable and equivalent. Therefore individuals are also considered homogeneous in their mixing pattern.
Empirical studies of social and contact networks relevant for disease transmission have however identified several heterogeneities in specific features at the individual or group level – including, e.g., the number of contacts, their frequency and duration, contacts’ clustering, assortativity, and their structure into communities – that affect the dynamics and control of infectious diseases[6, 8, 9, 28–39]. A particularly efficient theoretical framework that takes into account variations in population features is the transmission matrix approach that divides the population into groups and considers inter-group heterogeneities[40–42]. Individuals within the same group are assumed to be homogeneous with respect to their ability to contract and transmit the disease, and this approach can be used when variations at the individual level are considered to be negligible within the same group. Its advantage is to allow for a full parameterization of the model with the information available from empirical studies and for a mathematical formulation for the analytical computation of important epidemic parameters and observables, such as the basic reproductive number (measuring the average number of secondary cases per primary case)[41], the final size of the epidemic[42] and its extinction probability[43].
Although interactions between individuals of different types and at different scales through mobility have been included in numerical approaches, and each of them has been separately addressed in mathematical approaches, their joint integration into a general theoretical framework has yet to be developed. A clear example of the importance of both aspects acting together on the dynamics of an epidemic spreading through a population was recently put forward by the 2009 H1N1 pandemic outbreak, where age was observed to be a relevant factor differentiating between local community outbreaks (mainly driven by children) and case importation into unaffected regions (mainly driven by adults)[44–46]. Broken down to the basic mechanisms at play, the observed pattern could be explained through the interplay between two classes of individuals – children and adults – having different mixing behaviours[6, 47] and travel habits[46]. Other classifications of the population may be also relevant for the spatial spread of an infectious disease and the risk of an epidemic invasion, as prompted by the empirically observed dependence of travel frequency and contact patterns on different features of the population[10, 48].
In the present study, we present a general theoretical framework for the assessment of the pandemic risk for an infectious disease spreading through a spatially structured population characterized by contact and mobility heterogeneities. We integrate the metapopulation framework with the transmission matrix approach using a parsimonious model based on the subdivision of the population into two groups for each local community. We consider different types of mixing patterns across classes to provide a fundamental analytical understanding of the dependence of the global invasion parameter R_{∗} on epidemiological parameters and population features. By restricting to two classes, it is possible to provide a complete mathematical formulation of the model and recover an equation for R_{∗} that can be solved numerically, with approximate analytical solutions being possible under limit conditions on the parameters. These theoretical results are further tested against mechanistic Monte Carlo simulations of the infection dynamics in the metapopulation system individually tracking hosts in time and space. The framework is completely general and can be applied to different social settings, where host partition may depend on demographic or socio-economic factors, or to roles/conditions of individuals in specific settings (e.g. health-care workers and patients in hospitals[10], students classified by gender or class and teachers in schools).
Model description
Social layer and infectious disease transmission model
where the parameter ε here defined quantifies the degree of mixing in the way links are established across classes. It is defined in the range 0 < ε < min{α,η (1 - α)}, where values of ε close to zero indicate assortativity of the system (i.e. a tendency of individuals in a given class to preferably interact with individuals of the same class), whereas the upper bound of the range describes a scenario where individuals tend to avoid making contacts within their group. Far from the extremes we have a random or proportionate mixing where individuals distribute randomly their contacts in the population.
Population groups variables
Variable | Definition | Range |
---|---|---|
α | group 1 fraction of the population | ]0;1[ |
q_{1},q_{2} | average number of contacts established by individuals in group 1 and 2 | |
$\eta =\frac{{q}_{2}}{q1}$ | ratio of the average number of contacts | ]0;1] |
ε | total fraction of contacts across groups | ]0;min(α,η(1 - α))] |
r | group 1 fraction of traveling population | ]0;1] |
where the matrix Γ, is a diagonal matrix whose entries correspond to the relative sizes of the groups. The basic reproductive number R_{0} is calculated as the largest eigenvalue of the matrix R[41] and it provides a threshold condition for a local outbreak in the community; if R_{0} > 1 the epidemic will occur and will affect a finite fraction of the local population, otherwise the disease will die out.
Spatial layer and mobility model
Metapopulation model variables
Value used | ||
---|---|---|
in numerical | ||
Variable | Definition | simulations |
k | degree of a subpopulation, i.e. number of connectionsto other subpopulations | $[1;\sqrt{V}]$ |
P(k) = k^{-γ};γ | subpopulation degree distribution; power-law exponent | γ = 2.3,3 |
V;V_{ k } | total number of subpopulations; number of subpopulationswith degree k | V = 10^{4} |
average population of a node, population of a node; | ||
$\stackrel{\u0304}{N},{N}_{k}=\frac{\stackrel{\u0304}{N}{k}^{\varphi}}{\u3008{k}^{\varphi}\u3009};$ | with degree k | $\stackrel{\u0304}{N}=1{0}^{4}$ |
ϕ; | power-law exponent; | ϕ = 3/4 |
w _{0} | mobility scale | w_{0} = 0.05 |
number of travelers from a subpopulation with degree k_{ l } | ||
w_{ lm } = w_{0}(k_{ l }k_{ m })^{ θ }; | to a subpopulation with degree k_{ m }; | |
θ | power-law exponent | θ = 0.5 |
Analytical treatment and results
Identifying and understanding the conditions for the spatial invasion of an infectious disease, once it emerges in a given population or community of individuals, requires the consideration of all scales at play in the system. At the local scale, the reproductive number R_{0} provides a threshold condition for the occurrence of an outbreak locally. At the global scale, however, additional mechanisms need to be considered that may impede the spatial propagation of the disease from the seed of the epidemic to other regions of the system. Even in the case the condition R_{0} > 1 is satisfied, the epidemic may indeed fail to spread spatially if the mobility rate is not large enough to ensure the travel of infected individuals to other subpopulations before the end of the local outbreak, or if the amount of seeding cases is not large enough to ensure the start of an outbreak in the reached subpopulation counterbalancing local extinction events. It is then possible to identify at the metapopulation scale an additional predictor of the disease dynamics, R_{∗}, that defines the condition for spatial (or global) invasion, R_{∗} > 1[19, 20, 50, 51], analogously to the reproductive number R_{0} at the individual level. An analytical expression for R_{∗} has been found in metapopulation models characterized by homogeneous or heterogenous mobility structures and different types of mobility processes: markovian mobility[19, 20], adaptive traveling behaviour in response to the pandemic alert[22], time varying mobility patterns[26], non-markovian mobility with uniform return rates (i.e. commuting-type of mobility)[24, 25], or with heterogeneous length of stay at destination[27, 52]. In all cases, the analytical expression of R_{∗} is obtained with a mean-field approximation assuming that all subpopulations with the same degree are statistically equivalent (degree-block approximation)[19, 20, 29]. This translates in assuming that all features characterising the metapopulation systems (e.g. population size, traveling flux between two subpopulations, in/out traffic of a subpopulation) can be expressed as functions of the degree of the considered subpopulations. While disregarding more specific properties of each subpopulation that may be related for instance to local, geographical or cultural aspects, such assumption is grounded on a large body of empirical evidence obtained from different transportation infrastructures and mobility systems at a variety of scales, pointing to a degree-dependence of average quantities characterising the system[2, 20]. In addition, this simplifying assumption enables an analytical treatment of the problem while accounting for the large degree fluctuations empirically observed in the data[19, 20].
Here we consider the same analytical approach adopted in previous works with the aim of exploring the effects of contact and travel heterogeneities in the host population on the invasion potential of an epidemic. We first define the general theoretical framework and present its analytical treatment, and then focus on different cases representing different interaction types between social groups.
General framework
where the index i refers to the two types of individuals (i = 1,2) and R_{ ij } are the terms of the next generation matrix of Eq. (4). If the infection is not able to produce an outbreak in a single population (R_{0} < 1), the only solution is π_{1} = π_{2} = 1, that is, the epidemic dies out. Otherwise, Eq. (11) have solutions in the domain of values (0,1) for π_{1} and π_{2}, yielding a non zero probability of global outbreaks. Notice that in the case the system is socially homogenous and there is only one type of individuals the two probabilities reduce to 1/R_{0}.
If we assume that the parameters characterising social interactions and travel behaviour are uniform across all subpopulations, the social and spatial layers of the system factorize. R_{∗} can be then evaluated by computing the combination of moments χ, and solving numerically Eq. (5) and Eq. (11) for the epidemic sizes z_{1,2} and the probabilities π_{1,2} respectively. Differently from previous works focusing on homogeneous populations of hosts, an explicit analytical solution of R_{∗} cannot be recovered in the general case, due to the z_{1,2} and π_{1,2} terms, however special cases can be solved through series expansion as discussed in the following subsections.
The global invasion parameter R_{∗} quantifies the potential for the spreading at the spatial level of a specific infectious disease in a given social, demographic and mobility setting and it can thus be used to provide an estimate of the pandemic risk associated to an emerging epidemic. As an example, we address in Section Application to the 2009 H1N1 pandemic influenza the case of the 2009 H1N1 influenza pandemic in Europe, highlighting the important role of age classes in determining local transmission and spatial spread of the disease.
The role of local contact structure is investigated in Figure2B. Given a reproductive number R_{0} > 1 ensuring the occurrence of a local outbreak in the seeding region, our results show that there exist a region of values of the parameters η and ε for which containment at the source is predicted (grey area). Low enough values of the social activity of group 2 vs. group 1 (measured by η) coupled with large enough assortativity (i.e. low enough values of ε) do not provide the conditions for the spatial invasion of the disease.
A more extensive characterisation of the global invasion threshold can be obtained for two specific social systems for which approximate analytical expression of Eq. (15) can be obtained. We discuss these systems in the following subsections.
Proportionate mixing
In the case r = 0, when only individuals of the type 2 travel, the threshold R_{∗} converges rapidly to zero (the order being η^{2}), implying that the epidemic remains local and no global spread is possible. On the other hand, if only individuals of type 1 travel (r = 1), R_{∗} approaches rapidly${R}_{\ast}^{h}=\frac{2{\left({R}_{0}-1\right)}^{2}}{{R}_{0}^{2}}\phantom{\rule{0.3em}{0ex}}\frac{{w}_{0}}{\mu}\chi $, that is the expression of the homogenous case where no partition of the population is considered[20]. This indicates that individuals of group 2 play a negligible role on the spread of the epidemic.
Panels C and D of Figure3 summarise the impact of the socio-demographic parameters α and η on the invasion condition for the two cases r = 0 and r = 1, respectively, and for different values of R_{0}. The curves represent the invasion threshold condition R_{∗}(η,α) = 1, with the invasion regions located above the curves of panel C, and to the left side of the curves of panel D. If r = 0, the curve η(α) corresponding to the global invasion condition is an increasing function of α, indicating that if the fraction of individuals belonging to group 2 is increased, the smaller need to be the associated social activity to reach the outbreak invasion, given that they represent the seeders of the epidemic. If r = 1, the functional relationship between η and α associated with the threshold condition displays a richer behaviour (panel D). In the limits η → 0 and η → 1, we recover the homogenous mixing regime where, for the two values of R_{0} considered in the figure, the epidemic is not able to spread globally. If we move from these boundary values to intermediate values of η, activating the social heterogeneities of the population in the model, we observe an increase in R_{∗} until the invasion threshold is crossed, and global invasion is reached. Differently from the case r = 0, if r = 1, i.e. only more active individuals (group 1) travel, the condition R_{∗} = 1 is not an increasing fraction of α. For values of α smaller than a critical value depending on R_{0}, the system experiences invasion for an entire range of η values, [η_{c,min}(α),η_{c,max}(α)] (panel D). The upper value of this range, η_{c,max}, becomes larger as the fraction of individuals in group 1 decreases, indicating that even if group 1 is relatively smaller (α decreasing) and less active (η increasing), its exclusive dominance on mobility is enough to ensure invasion. Proportionate mixing is then responsible to limit invasion to η ≥ η_{c,min}(α), so that no invasion is obtained by further increasing the social activity of travelers η < η_{c,min}(α).
Assortative mixing
for the limit η → 1.
In panels B and C of Figure4 we show the comparison between the approximate analytical solution and the numerical one by reporting the absolute difference between the corresponding results. The series expansion in Eq. (25) for the limit η → 0 yields a quadratic dependence on ε as the first non-constant term, with η disappearing from the first two terms of the equation. The approximated value of R_{∗} so obtained well approaches the numerical results for the case η → 0 as shown in panel B where absolute differences are of the order of magnitude of at most 10^{-4}, and relative differences of at most ∼43% in the displayed range. For the limit η → 1 we recover instead a linear dependence on the two parameters ε and η. Panel C of Figure4 shows an absolute difference in R_{∗} below 0.7 between the numerical value and the approximated one, corresponding to a relative difference of ∼36%.
Proportionate vs. assortative mixing
Numerical simulations
The theoretical framework described so far is based on the combination of continuous differential equations for the transmission dynamics within each subpopulation, with mathematical tools of complex network theory for describing the spatial invasion of the epidemic. In this section we validate the theoretical approach by presenting the comparison between the results recovered so far and the output of stochastic numerical simulations, where all processes are simulated explicitly. The system evolves following a stochastic microscopic dynamics where hosts are individually tracked and at each time step it is possible to monitor several quantities, as for example the number of infectious individuals within each subpopulation and for each group, or the number of subpopulations reached by the disease. Given the stochastic nature of the dynamics, the experiment can be repeated with different realisations of the noise, different underlying graphs and different initial conditions.
The mobility network consists of V = 10^{4} subpopulations and is generated by the uncorrelated configuration model[56] that allows building a network with a preassigned degree distribution. In agreement with the analytical calculations we choose a power-law degree distribution, P(k) ∝ k^{-γ} with exponent γ = 2.3. Once the mobility network is constructed, a number of inhabitants is assigned to each subpopulation according to the degree of the node. Specifically, for each node l, we assume a power-law relation between the population N_{ l } and its degree k_{ l },${N}_{l}=\frac{\stackrel{\u0304}{N}}{\u3008{k}^{\varphi}\u3009}{k}_{l}^{\varphi}$, where the$\stackrel{\u0304}{N}$ is the average population of the nodes, set to 10^{4}, and$\u3008{k}^{\varphi}\u3009={\sum}_{k}{k}^{\varphi}P(k)$. This relation was shown to reproduce the behaviour of empirical systems, with an estimate for ϕ of approximately 3/4[57]. Fluxes along each mobility link also follow a power-law relation with the degrees of the connected nodes, as described in Section Spatial layer and mobility model,${w}_{{k}_{l}{k}_{m}}={w}_{0}{({k}_{l}{k}_{m})}^{\theta}$, with θ = 0.5 and w_{0} = 0.05. With this definition, fluxes are symmetric and do not alter the occupancy number of each subpopulation, thus the system is at equilibrium with respect to the mobility dynamics. The social layer is constructed by dividing the population of each node into two groups according to the parameter α. The contact parameters ε and η define then the contact matrix ruling the transmission dynamics.
where the transmission rate β corresponding to the chosen value for R_{0} is computed from the largest eigenvalue of the next generation matrix – see Eq. (4). Recovery from the disease is also a binomial process, with every infectious individual having at each time step a probability μ to enter in the recovered compartment. We set R_{0} = 1.2 and μ = 0.5. The diffusion of individuals is implemented as a multinomial process by accounting the heterogeneities in individual travel frequency given by Eq. (7). Throughout this numerical exploration we always assumed that only individuals of group 2 travel, i.e. r = 0.
The epidemic is initialised by placing 5 infected individuals per each group within a randomly chosen subpopulation and it is simulated until the extinction of the virus is reached. The fraction of subpopulations reached by the disease D_{ ∞ }/V provides a clear quantification of the invasion potential of the disease. We consider the two scenarios introduced in the analytical treatment, the proportionate mixing case and the assortative one, and we provide a comparison between the outcome of the numerical simulations and the corresponding analytical results.
Panel B of Figure6 focuses on the assortative mixing case. Here we show the average fraction of infected subpopulations, D_{ ∞ }/V, as a function of the assortative parameter ε, for three different values of α and for η = 0.5. All the curves present a transition between local outbreak and global invasion in correspondence of a critical value of ε, above which the fraction of infected subpopulation becomes an increasing function of ε. The increase in α reduces the invasion potential of the disease. The threshold behaviour is in agreement with the theoretical analysis (Eq. (15)), whose threshold results are reported in the plot for comparison (coloured arrows).
Application to the 2009 H1N1 pandemic influenza
The values presented in the table describe an assortative system, where social activity is heterogeneous among the two groups, with children having on average more contacts than adults. Air-transportation statistics available for several airports yield an average of 7% of children occupancy[46], thus r = 7%. Finally we parametrize the mobility network and the distribution of traveling fluxes by setting γ = 2.3 and w_{0} = 1[2].
Epidemiological parameters were chosen among the estimates provided for the A(H1N1) pandemic. Throughout the analysis we consider an infectious period of 2.5 days[7] and three different estimates for R_{0}: R_{0} = 1.05 (corresponding to the estimate in[7] for the reproductive number in Europe during summer 2009), R_{0} = 1.20 (as estimated from the outbreak data in Japan[60]), and R_{0} = 1.40 (as estimated from the early outbreak data in Mexico[59]). We also consider a scenario in which a certain fraction of the adult population has a pre-existing immunity to the virus accounting in this way for the serological evidence indicating that about 30 to 37% of the individuals aged ≥60 years had an initial degree of immunity prior to exposure[61]. We assume that 33% of individuals aged ≥60 years are immune and completely protected against H1N1 pandemic virus[46], and for each country we compute the corresponding fraction of the adult group with pre-exposure immunity.
The comparison between the case r = 0 and r = 7% for Germany allows us to quantify the role of children as seeders of the epidemic in new locations in a data-driven situation. They contribute to the increase of the invasion potential of the epidemic, thus lowering the minimum value of the across-groups mixing for which the epidemic spatial spread is possible. The effect is small but appreciable.
Conclusions
This study presented a general theoretical framework to account for two different layers of heterogeneity relevant for the propagation of epidemics in a spatially structured environment, namely contact structure and heterogenous travel behaviour. The model presents a structure with two distinct scales – a social scale and a spatial one. Employing a subdivision into two host classes, we provide a mathematical formulation of the model and derive a semi-analytical solution of the invasion equation, encoding the conditions for the global invasion of the epidemic. The system is characterized by a very rich space of possible solutions, depending on the demographic profile of the population, the pattern of contacts across groups and their relative social activity, the travel attitude of each class, and the topological and traffic features of the mobility network. Two qualitatively different scenarios are found. The increase of the across-group mixing and of the social activity of the less active group (relative to the more active group) enhance the pandemic potential of the infectious disease, if seeders are mostly found in the less active group. Reductions of the number of contacts of individuals of the less active group is predicted to be the most efficient strategy for reducing the pandemic potential. If instead traveling is dominated by the most active class, the role of the contacts ratio between the two groups is negligible for a given population partition, whereas there exist an optimal across-groups mixing that maximizes the pandemic potential of the disease. Reductions or increases of this quantity with respect to the optimal value would decrease the probability that the epidemic, once seeded in a given region, would reach a global dimension. Such findings call for the need to develop further studies to identify appropriate intervention measures that can act on these socio-demographic aspects, depending on the type of partition and of population considered. Empirical data of contact patterns, demography and travel from eight European countries and from Mexico, and of the 2009 H1N1 influenza pandemic were used to parametrize our model in terms of two age classes of individuals – children and adults – and explain the spatial spread of the disease following emergence (in Mexico) and international seeding (in Europe). Despite the need to address some limitations of the model in future work (e.g. partition in more than two classes, and geographic dependence of population features), our approach offers a flexible theoretical framework – validated on historical epidemics – that can promptly assess the pandemic potential of an emerging infectious disease epidemic where a specific socio-demographic stratification is relevant in the disease transmission among individuals.
Declarations
Acknowledgements
This work has been partially funded by the ERC Ideas contract no. ERC-2007 -Stg204863 (EPIFOR) and the EC-Health contract no. 278433 (PREDEMICS) to VC and CP; the ANR contract no. ANR-12-MONU-0018 (HARMSFLU) to VC; the Ramón y Cajal program and the project MODASS of the Spanish Ministry of Economy (MINECO), and the EC projects EUNOIA and LASAGNE to JJR.
Authors’ Affiliations
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