Letter

# Critical regimes driven by recurrent mobility patterns of reaction–diffusion processes in networks

Accepted:
Published online:

## Abstract

Reaction–diffusion processes1 have been widely used to study dynamical processes in epidemics2,3,4 and ecology5 in networked metapopulations. In the context of epidemics6, reaction processes are understood as contagions within each subpopulation (patch), while diffusion represents the mobility of individuals between patches. Recently, the characteristics of human mobility7, such as its recurrent nature, have been proven crucial to understand the phase transition to endemic epidemic states8,9. Here, by developing a framework able to cope with the elementary epidemic processes, the spatial distribution of populations and the commuting mobility patterns, we discover three different critical regimes of the epidemic incidence as a function of these parameters. Interestingly, we reveal a regime of the reaction–diffussion process in which, counter-intuitively, mobility is detrimental to the spread of disease. We analytically determine the precise conditions for the emergence of any of the three possible critical regimes in real and synthetic networks.

## Main

Epidemic processes in complex networks have attracted the attention of physicists during the last two decades6. Several outstanding results have been the consequence of a mathematical analysis that borrows ideas from other physical processes. In particular, epidemic spread in networks can be thought of as reaction–diffusion processes, referring to the change of the concentration of two or more types of element: local reactions in which the elements are transformed into each other, and diffusion that causes the substances to spread out over the available space. In epidemiology, the elements in play are the subjects (humans or animals), characterized by their states in the evolution of the sickness (for example, susceptible, infected, recovered and so on). In complex networks, the reaction phase corresponds to the infections produced by the local interaction of subjects within a subpopulation (node), and the diffusion phase corresponds to their mobility through the network according to the connections (links) between nodes.

This approach to epidemic spread using reaction–diffusion processes, usually referred to as metapopulation models, has been largely studied in network science10,11,12,13,14; however, several challenges remain open15,16. The most representative of these challenges, from a physicist’s perspective, is to complement large-scale agent-based simulations17,18,19, by deriving models amenable to mathematical analysis20 that capture the influence of human behaviour21 and the existence of complex social structures.

Our proposal to fill this gap is to formulate a general ‘microscopic’ Markovian model describing the metapopulation reaction–diffusion dynamics. We start by analysing, at the individual level, the probabilities of infection in the scope of the susceptible–infected–susceptible (SIS) epidemic model12. We denote as λ and μ the infection and recovery probabilities respectively. This way, a susceptible (S) individual is infected with probability λ when interacting with an infected (I) subject. In turn, infected (I) individuals become susceptible (S) again with probability μ. Note that if there is no recovery (that is, μ = 0), the phenomenology we will report does not hold.

Motivated by the recurrent (commuting) nature of most urban and regional movements reported and modelled8,9, let us explain two key assumptions of our model. First, we assume that each individual is associated with a certain subpopulation (her residence). Second, to incorporate the recurrence of human mobility patterns, we force all of the agents who have decided to move from their residence to return to them after each time step. This way, given N subpopulations (nodes), the variable ρ i (t) (i = 1,..., N) denotes the fraction of infected individuals associated with node i at time t. The time evolution of ρ i (t) is as follows:

$ρ i ( t + 1 ) = ( 1 - μ ) ρ i ( t ) + 1 - ρ i ( t ) Π i ( t )$
(1)

where the first term in the right-hand side denotes the fraction of infected individuals associated with i that do not recover and the second term accounts for the fraction of healthy individuals associated with i that pass to infected at time t + 1. In this second term, Π i (t) is the probability that a healthy individual associated with node i becomes infected at time t. This probability reads:

$Π i ( t ) = ( 1 - p d ) P i ( t ) + p d ∑ j = 1 N W i j ∑ l = 1 N W i l P j ( t )$
(2)

where pd denotes the probability of moving and W ij denotes the weight of the connection between nodes i and j. The first term in the right-hand side denotes the probability that a susceptible individual associated with node i becomes infected when remaining at node i, and the second one accounts for the probability that this individual catches the disease when moving to any neighbour of i. In addition, P i (t) denotes the probability that a healthy individual in (but not necessarily associated with) node i at time t becomes infected by any of the infected individuals placed in the same node i at the same time t. This probability, under the well-mixing approximation for the dynamics within the subpopulations, reads:

$P i ( t ) =1- ∏ j = 1 N 1 - λ ρ j ( t ) n j → i$
(3)

where n ji denotes the number of individuals that moved from node j to node i:

$n j → i = δ i j 1 - p d n i + p d W j i ∑ l = 1 N W j l n j$
(4)

where δ ij  = 1 when i = j and δ ij  = 0 otherwise. The effective population of node i is then computed as $n i eff = ∑ j n j → i$.

Equations (1)–(4) form a closed set of equations covering the evolution of an SIS disease in a metapopulation of commuting agents following a microscopic Markovian description. Here, we are interested in the critical behaviour of the system as a function of the reaction and diffusion parameters. To this end, we analyse the steady state of the dynamics, so that equation (1) reads:

$μ ρ i * = 1 - ρ i * Π i$
(5)

where $ρ i *$ is the stationary density of infected individuals associated with node i. Close to the critical point, the disease becomes endemic displaying a small incidence: $ρ i * = ϵ i * ≪1$ i. This allows a linerarization of equation (5) (see Supplementary Information) whose solution is given by:

$μ λ ϵ i * = ( M ϵ ⃗ * ) i$
(6)

where the entries of matrix M read:

$M i j = 1 - p d 2 δ i j n j + p d 1 - p d n j R + R T i j + p d 2 n j R ⋅ R T i j$
(7)

where $R i j = W i j ∑ l = 1 N W i l$. Each entry M ij accounts for the average number of contacts between one individual associated with node i and all of the individuals associated with node j. In particular, the first term accounts for the contacts with individuals associated with the same patch (i = j) that remain in it, the second term accounts for the number of contacts that the agent from i makes with agents from j, in either of the two associated patches i and j and, finally, the third term informs us about the number of contacts of an agent from i with others from j in a patch different from the ones they are associated with.

Equation (6) defines an eigenvalue problem where the epidemic threshold λc (the minimum λ value so that the above expression holds) is related to the maximum eigenvalue Λmax of matrix M as:

$λ c = μ Λ max ( M )$
(8)

To understand the role of the mobility parameter pd in the determination of the critical point λc, it is mandatory to analyse the maximum eigenvalue Λmax. Unfortunately, this is not possible for any configuration of the matrix M. Nevertheless, it is possible to gain some insight into the behaviour of λc for any metapopulation network, from the perturbation analysis of the eigenvalues22 at low mobility rates pd → 0 (see Supplementary Information). This analysis gives the following expression for the eigenvalues of M:

$λ i ≈ n i +2 p d n i R i i - 1 + p d 2 ∑ j ≠ i n j n i R i j + R j i 2 n i - n j$
(9)

including its maximum value Λmax.

Note the particular dependence of the eigenvalues of M on the volume of the different populations n i . This fact is crucial to understand the behaviour of the critical point of the epidemics. The asymptotic analysis of equation (9) shows that Λmax is governed by the largest population nmax when pd = 0. When mobility is pd = δ with δ 1, the second term of the right-hand side is negative because R ii  < 1 (eventually R ii  = 0) and the third term is positive (because nmax > n j , j ≠ max). Therefore, the character of the critical point will be governed by the balance between these two last terms.

Deriving equation (9) for Λmax with respect to the mobility parameter pd, we observe that the second-order algebraic equation presents a minimum at some value $p d *$ given by:

$p d * = 1 - R max,max ∑ j ≠ max n j R max , j + R j , max 2 n max - n j$
(10)

This result proves that Λmax (except for equally sized populations) will always decrease for low values of pd and hence the critical point, λc, will increase, pointing out the counter-intuitive effect that mobility is detrimental to the spread of the epidemic. However, the values of pd have to satisfy a certain constraint: 1/nmax ≤ pd ≤ 1. The mobility being larger than the inverse of the maximum population size of the subpopulations indicates that at least one individual moves; the other constraint is determined because pd is a probability. As shown in Fig. 1a, this defines three different regimes where the system can lie; namely, when Λmax(M) reaches its minimum value for: (I) $p d * <1∕ n m a x$; (II) in the interval $1∕ n m a x < p d * <1$; and (III) for $p d * >1$.

To highlight this effect, we study a simple star-like network configuration containing two relevant parameters, α (controlling the population difference between the hub and the leaves) and δ (controlling the flow from the leaves to the hub) (see Fig. 1b). Each pair of values (α, δ) together with pd yields a different matrix M, thus allowing a systematic study of the behaviour of the epidemic threshold λc. Moreover, for this particular configuration, it is possible to derive analytically the exact expression Λmax(M) (see Methods) and, for each pair (α, δ), analyse the behaviour of Λmax(M) as a function of pd to derive the value $p d *$ where Λmax(M) reaches its minimum.

In Fig. 1c we plot the value $p d *$ as a function of α and δ recovering the three regimes (I, II and III). To further validate these findings, in Fig. 1d–f we show the epidemic diagrams I(λ, pd) for three combinations (α, δ). The black solid line denotes the epidemic threshold as a function of pd, λc(pd), as predicted by equation (8) while the blue dashed line corresponds to the threshold as obtained from the perturbative analysis of Λmax(M) (equation (9)).

To conclude, we have tackled the analysis of a real urban system, the city of Cali (Colombia). To this end (see Methods for details), we have collected data about the demographic distribution of its 2.4 × 106 inhabitants, officially divided into 22 districts. Besides, we have collected the mobility patterns among these 22 subpopulations (see Fig. 2a) to construct matrix R. With these data, we have run Monte Carlo simulations of the reaction–diffusion dynamics and computed the incidence I as a function of λ and pd. In Fig. 2b we show these results (points) and make a comparison to the solution of the Markovian equations (1)–(4), revealing a perfect agreement between them. In the Supplementary Information we also show the agreement between the Monte Carlo simulations and the Markovian model equations (1)–(4) regarding the spatio-temporal contagion patterns. Finally, in Fig. 2c we compare the numerical results with the analytical expressions for λc in equations (8) (solid line) and (9) (dashed line). The agreement of equation (8) is almost perfect while, as expected, that of the approximated expression equation (9) is valid for small values of pd. Remarkably, the behaviour for the city of Cali lies in scenario II, thus displaying a rise and fall for λc that reveals the detrimental effect on the spread in the low-mobility regime. We have also studied synthetic spatially embedded networks (random geometric graphs23) and found the same qualitative behaviour regarding the detrimental effect on the epidemic (see Supplementary Information).

It is worth mentioning that the detrimental effect on the spread of epidemics has been theoretically predicted in contact networks24,25,26 by eliminating contacts between susceptible and infected nodes, thus pruning the contagion pathways and making the network more resistant to the spread of pathogens. Here, instead, the mobility rate of individuals between the patches of the metapopulations is what causes the detrimental effect. The mobility is dictated by the data; that is, we neither impose that healthy individuals avoid contact with infected ones or movement to infected patches, nor restrict the mobility of infected individuals. This difference is essential because the dynamics analysed in our manuscript is not designed to affect the epidemics but to capture the real mobility patterns of individuals between patches. Note that the main dependence of the epidemic threshold is linear on the largest population size of the metapopulation, for low values of the mobility, and we can exploit this fact to understand the roots of the phenomenon (see Supplementary Information). Indeed, at first order (pd → 0), it can be shown that the scenario that will always be detrimental to the epidemics corresponds to metapopulation systems in which mobility decreases the largest effective population size, homogenizing, as a by-product, the size of the rest of the populations. Out of this scenario (that is, when effective populations sizes are not homogenizing), the physics of the detrimental effect on the epidemic is rooted on the homogenization of the number of infected individuals across all subpopulations. A heuristic argument is as follows: mobility of individuals in large-size populations towards smaller ones acts as an epidemic prevention mechanism, since the distribution of infected individuals from large to small patches makes the relative number of infected individuals per population small enough for the epidemics not to survive. It is worth mentioning that similar observations of anomalous fixation probabilities for alleles in structured populations, akin to the detrimental effect we fine here, were first reported in population genetics27,28.

From a physics point of view, this work paves the way to a new paradigm of criticality in networked discrete reaction–diffusion processes, with an interplay of parameters defining different critical regions. In general terms, the detrimental effect of mobility on epidemics should be taken into account to ameliorate the policies towards epidemic containment4,29,30, since whether or not it may happen depends on the particular demographic distribution and mobility patterns of the region under study.

## Methods

### Analytical estimation of epidemic threshold in a star-like metapopulation

Here we derive the expression of the epidemic threshold of the star-like metapopulation. This toy network is composed of a central hub connected to k identical leaves. In turn, each leaf is connected to the hub and to one of the k − 1 leaves following the schematic plot in Fig. 1b. The k connections from the hub are identical in the sense that the probability that an agent associated with the hub moves to a particular leaf is: Rmax,j  = 1/k. On the other hand, agents associated with the leaves can move either to the hub with a probability R j,max = δ or to the next leaf with a probability R j,j+1 = 1 − δ. Finally, the hub has a population of nmax agents while each leaf contains αnmax, being α (0, 1]. To characterize the behaviour of the epidemic threshold, let us recall that the probability that a healthy agent associated with a node i becomes infected is given by:

$Π i =λ ∑ j = 1 N 1 - p d 2 δ i j n j + p d 1 - p d n j R i j + R j i + p d 2 n j ∑ l = 1 N R i l R j l$
(11)

As all of the k leaves in the star-like metapopulation are statistically equivalent, it is enough to compute two infection probabilities: that for the agents of the hub Πmax, and that corresponding to an agent associated with any of the leaves Πl. In particular, close to the epidemic threshold, where the disease incidence is small, we can write the probability that a healthy agent associated with the hub becomes infected as:

$Π max =λ 1 - p d 2 n max ϵ max + k p d 1 - p d 1 k + δ α n max ϵ l + p d 2 k 1 k 2 n max ϵ max + 1 - δ α n max ϵ l$
(12)

while the contagion probability for the agents associated with a leaf reads:

$Π l = λ 1 - p d 2 α n max ϵ l + p d 1 - p d 1 k + δ n max ϵ max + 2 1 - δ α n max ϵ l + p d 2 k δ 2 α n max ϵ l + 1 k 1 - δ n max ϵ max$
(13)

To obtain the expression of the epidemic threshold, we now make use of the expression governing the stationary state:

$μ λ ϵ i = Π i = ∑ j = 1 N M i j ϵ j$
(14)

For the star-like network the problem reduces from N = k + 1 equations to two: one for the hub and another for a leaf. Therefore, by using expressions of equations (12) and (13), we obtain the following effective matrix M:

$M= n max 1 - p d 2 + p d 2 k n max α p d 1 - p d 1 + k δ + p d 2 1 - δ n max p d 1 - p d 1 k + δ + p d 2 1 k 1 - δ n max α 1 - p d δ 2 + k p d 2 δ 2$
(15)

This matrix M allows us to define the following eigenvalue problem:

$μ λ ϵ max ϵ l = M ϵ max ϵ l$
(16)

from which the epidemic threshold λc can be calculated:

$λ c = μ Λ max ( M )$
(17)

The maximum eigenvalue of the 2 × 2 matrix M can be easily obtained as:

$Λ max ( M ) = Tr M + Tr M 2 - 4 det M 2$
(18)

where

$Tr M = n max 1 - p d 2 + n max p d 2 k +α n max 1 - p d δ 2 + k p d 2 δ 2$
(19)
$det M = n max 2 α 1 - p d 1 + δ 2$
(20)

The behaviour of the epidemic threshold λ c as a function of the mobility rate pd is determined by the set of parameters {α, δ, k}. In particular, we are interested in characterizing analytically the different epidemic regimes by evaluating the value of $p d *$ for which λ c (pd) reaches its maximum. To this end, by setting k = 10, we have computed $p d *$ as a function of (α, δ). The results are shown in Fig. 1c, where for clarity, we have bounded the colour code in the range $p d * ∈ [ 0 , 1 ]$. This way the areas in yellow correspond to $p d * ≥1$ and thus correspond to a critical behaviour of type III.

### Monte Carlo simulations

To validate the accuracy of the microscopic Markovian model, equations (1)–(4), we have compared the predictions of the epidemic incidence and its spatio-temporal evolution with numerical results from agent-based numerical Monte Carlo simulations. To this end, we start populating each node i with a set of inhabitants n i . To track the evolution of the epidemics, each agent has a label indicating her epidemic state (healthy or infected). Besides, it is necessary to impose an initial seed to allow the spread of the epidemic. In our case, we have considered that 1% of the agents of each node (subpopulation) are initially infected.

For each time step, reaction and diffusion processes take place. First, each agent has a probability pd of moving to any other patch different from her residence. To determine whether the agent moves or not, we generate an independent and identically distributed random variable (i.i.d.) r between [0, 1] in such a way that if r < pd the agent will move to a node other than its associated node. Otherwise, she will stay in her node. If the movement takes place, another i.i.d. r′ between [0,$∑ j W i j$] determines the target node of each movement so that the agent moves to the first node k that satisfies $∑ k W i k ≥ r ′$. We perform the same process for all of the agents so that a new spatial distribution of the whole population across the metapopulation for this time step is obtained.

Once all diffusion processes have taken place, contagion and recovery occur. Therefore, each healthy agent contacts with all infected people inside her node at this time step, becoming infected with a probability λ for each contact. On the other hand, infected agents recover with a probability μ. Finally, due to the commuting nature of displacements, all agents return to their associated node and another time step begins. Simulations finish when the epidemic reaches the stationary state, which is characterized by a constant value of the order parameter. In our case, we have identified this state when fluctuations of the total fraction of infected people I are lower than 10−5 during at least 100 time steps, so that the condition to stop simulations is |I(t + 100)–I(t)|<10–5.

### Cali

Santiago de Cali is the third most populated city in Colombia. It has about 2.4 × 106 inhabitants distributed across 22 neighbourhoods. To create the metapopulation associated with this city, we have extracted the population of each node, identified with each neighbourhood, from official reports31. Moreover, to determine the links of the metapopulation, it is necessary to know the mobility flows of people across the city. For this purpose, a representative sample of people from Cali were asked to complete a survey in which they indicated the neighbourhood where they reside i as well as the places to which they usually commute j during a work day. This way, the weight between patches i and j, W ij , is increased by 1 each time the survey reports a trip between these two patches. As a result, we obtained a network that encodes the information about the daily commutes among the 22 neighbourhoods of Cali (see Fig. 2a).

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

We acknowledge S. Meloni for the very many discussions and useful suggestions. J.G.-G. and A.A. acknowledge financial support from MINECO (projects FIS2015-71582-C2 and FIS2014-55867-P) and from the Departamento de Industria e Innovación del Gobierno de Aragón y Fondo Social Europeo (FENOL group E-19). A.A. acknowledges also financial support from the ICREA Academia, the James S. McDonnell Foundation.

## Author information

### Affiliations

1. #### GOTHAM Lab, Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain

• J. Gómez-Gardeñes
2. #### Department of Condensed Matter Physics, University of Zaragoza, Zaragoza, Spain

• J. Gómez-Gardeñes
•  & D. Soriano-Paños

• A. Arenas

### Contributions

All of the authors wrote the paper and contributed equally to the production of the manuscript.

### Competing interests

The authors declare no competing financial interests.

### Corresponding authors

Correspondence to J. Gómez-Gardeñes or A. Arenas.

## Supplementary information

1. ### Supplementary Information

Supplementary notes 1–8, Supplementary figures 1–7, Supplementary Notes