The structured-population model has been widely used to study the spatial transmission of epidemics in human society. Many seminal works have demonstrated the impact of human mobility on the epidemic threshold, assuming that the contact pattern of individuals is mixing homogeneously. Inspired by the recent evidence of location-related factors in reality, we introduce two categories of location-specific heterogeneous human contact patterns into a phenomenological model based on the commuting and contagion processes, which significantly decrease the epidemic threshold and thus favor the outbreak of diseases. In more detail, we find that a monotonic mode presents for the variance of disease prevalence in dependence on the contact rates under the destination-driven contact scenario; while under the origin-driven scenario, enhancing the contact rate counterintuitively weakens the disease prevalence in some parametric regimes. The inclusion of heterogeneity of human contacts is expected to provide valuable support to public health implications.
To study the epidemic spreading of viruses and diseases in human society, various mathematical models have been proposed during the past decades1,2,3. In particular, the network-based models have attracted a great deal of attention from diverse disciplines4,5,6,7,8,9. Plenty of works10,11,12,13,14,15,16 explored the transmission of etiological agent on networks, where nodes are individuals and links describe the contacts between individuals to transmit the infection.
Using the particle-network framework to mould the entire system into structured populations, we arrive at an important class of models in modern epidemiology, namely, metapopulation. In this model, individuals live in discrete subpopulations and may transfer between subpopulations via the mobility pathways (or links)17,18,19,20,21,22,23,24,25,26. The reaction-diffusion or reaction-commuting mechanism is harnessed to sketch human daily contact and mobility. More precisely, the disease prevails inside each subpopulation (with homogeneously mixed as assumed), and transmits between subpopulations through the travel of infected individuals. Applied in simulating the spatial transmission at large geographic scale, this model manifests its power in predicting the pandemic outbreak and evaluating the effectiveness of intervention strategies27,28,29,30,31,32,33,34.
Many works17,20,21,23,24 have focused on the impact of human mobility on the threshold of disease outbreak, which generalized the reaction-diffusion process to deal with heterogeneous networking populations, and assumed the mobility scheme as a Markovian memoryless migration. Recent empirical findings on human mobility35,36,37,38,39 report that the commuting behaviors, characterized by individual recurrent movements between connected locations, dominates the human daily transportation. Refs. 25, 26 extended the metapopulation framework by introducing the element of recurrent commuting, which assumes that individuals have the memory of their original resident subpopulation and displaced commuters who stay at the ‘destination’ subpopulation cannot diffuse to other places but return to their residence with a certain rate. In contrast to the random diffusion scenario, the commuting system exhibits a novel epidemic threshold as well as the phenomenon of the saturation of propagation velocity.
It is well known that the contact pattern of individuals dramatically impacts the spatiotemporal dynamics of epidemic spreading in a population1,2,3,40. The evolution of the epidemic process is characterized by the force of infection, which identifies the probability of acquiring infection for a susceptible individual due to his contacts with infectious ones. The force of infection is defined proportional to the density of infectious individuals, the infection probability via a contact between susceptible (S) and infectious (I) individuals, and the contact pattern between individuals1. Since human contact pattern is often reflected in the disease transmission rate, which is usually assumed constant1,41,42, in metapopulation models, the general assumption that the disease transmission rate in all subpopulations is identical indicates the fact that the individual contacts inside each subpopulation are uniform.
Within each subpopulation, the assumption of homogeneous contact (homogeneously mixing) is supported with recent empirical findings. For instance, digital equipments such as Bluetooth43, active Radio Frequency Identification (RFID) devices44,45,46, wireless sensors47, and WiFi48,49, have been applied to collect the data of human close proximity contacts in various social environments, which unveils that the distribution of the number of distinct persons that each individual is in contact with per day has a small squared coefficient of variation (CV2)45,46,47,48,49. Therefore, human contacts inside each subpopulation may possess the characteristic contact rate reflecting the number of distinct persons each individual encounters per day.
At different subpopulations, human contact pattern may present evident discrepancies. The location-specific factor has been reported as a potential driver inducing the substantial variation of disease incidence between populations in reality50,51. Generally, because of the distinction in social conditions or lifestyles, the contact rate of individuals living in an urban area is largely different from what we can expect for small-town residents. Therefore, we consider the variety of contacts between connected subpopulations to study their impact, and in this paper, we theoretically analyze this issue with the phenomenological model proposed in Ref. 52, where the recurrent commuting of individuals couples two typical subpopulations. The standard susceptible-infectious-susceptible (SIS) model is considered to reflect individuals' compartment transitions.
We first consider a general case that the individual contact features depend on the location where an individual is. This scenario illustrates the influence of social environment of the located subpopulation, and may reflect human adaptive ability53,54,55. Intuitively, considering the commuting flows between the urban and the suburb areas, one can expect that the contact rate for an individual commuting from the suburb to the urban area will increase, and vice versa. Therefore, we assign each individual in the located subpopulation x with the same characteristic contact rate cx, which means that on average he contacts with cx other individuals in the same subpopulation per unit time. We define this destination-driven mode as the type-I contact scenario.
As the commuting mobility distinguishes individuals' original residences from their destinations, we further consider another case that the contacts of individuals correlate to their own residences. This scenario may derive from the anchoring effect of human56,57, and reflects that many social factors of one's home location, e.g., economy, cultural background, inevitably affect how gregarious a person is. For instance, the schools of related disciplines of a university are usually located in the same campus (here each campus is a subpopulation), while each day students from different campuses might commute among the campuses by the school buses. Students majored in humanities or social sciences usually have a tendency to participate in social activities, and they may have a high contact rate; whereas students majored in natural sciences or medicine (particularly, graduate students) are more prone to spend their time in the laboratories or libraries, and they may own a low contact rate. Despite students from different disciplines having distinct contact rates may reside at different campuses, the commuting movements still induce their intersectional couplings in each campus per day. For simplicity, we assume that each individual from subpopulation x has the characteristic contact rate . This means that each one from x will on average have contacts with other individuals per unit time, no matter where he locate at that time. We define this origin-driven mode as the type-II contact scenario.
In both two cases, the force of infection for a susceptible stems from all his contacts with infectious individuals per unit of time. By means of analytical arguments as well as extensive computer simulations, we demonstrate that these location-specific heterogeneous contact scenarios significantly decrease the epidemic threshold of the entire system, and thus favor the disease outbreaks in more broad parametric regimes. Under the destination-driven scenario, the variance of disease prevalence displays a monotonic mode as the contact rates change; while the results are more unexpected under the origin-driven scenario: Enhancing the contact rate will weaken the disease prevalence in some parametric regimes.
We first specify the mechanism of commuting mobility to transfer individuals between subpopulations. Consider two coupled subpopulations, x, y, each of which has a population size of the residents . They act as the reaction places where the contagion process occurs. The model proceeds at discrete time steps, with the unit interval as one hour. Individuals leave their original resident subpopulation x(y) to visit the neighboring ‘destination’ subpopulation y(x) with a per capita diffusion rate σxy(σyx) at each time step, and the displaced commuters will return to their residence x(y) with a per capita return rate τx(τy) per unit time. For simplicity, we assume the detailed balance as: σxy = σyx = σ, τx = τy = τ, and Figure 1 illustrates the commuting process.
According to Refs. 26, 52, the residents of each subpopulation can be partitioned into two subgroups based on the locations where they actually are at time t (see Figure 1). Define Nxx(t)(Nyy(t)) the subgroup size of the individuals staying in their original resident subpopulation x(y) at time t, and Nxy(t)(Nyx(t)) the subgroup size of the individuals from subpopulation x(y) presenting in the subpopulation y(x) at time t. Therefore, the population size of the residents of each subpopulation is , . We use the mean-field approximation to describe the commuting process with a set of rate equations governing the evolution of population dynamics26,52, which are discussed in Supplementary Information, and we have the following equilibria: with the condition σ−1 ≫ τ−1, i.e., averagely, individuals will remain at their residence most of the time, which mimics that in reality only a small fraction of residents leave to travel. This condition facilitates the analysis by considering that per unit time each subpopulation x(y) has an effective number of residents interacting with the individuals in subgroup Nyy(Nxx).
The contagion process takes place inside each subpopulation, which is composed of the infection dynamics and the contact dynamics here. For the infection dynamics, we consider the standard SIS compartment model. At time t, each individual falls in one of the disease compartments: susceptible or infectious. The population size, e.g., Nxx(t) (Nyx(t)), can be divided into Sxx(t)(Syx(t)), Ixx(t)(Iyx(t)), which is the number of susceptible and infectious individuals in each subgroup, respectively. At each time step, if a susceptible individual contacts an infectious one, he may acquire the infection with transmission rate β. An infectious individual recovers and becomes susceptible again with rate ν per unit time.
We now specify the contact dynamics in more detail. In the case of the type-I (destination-driven) contact scenario, we assume that per unit time each individual staying in subpopulation x(y) has the same characteristic contact rate cx(cy). Figure S1(a) in Supporting Information illustrates this case. In the unit interval, the infection probability, λ, for a susceptible individual is determined by all his contacts with the infectious ones. Therefore, the force of infection for any susceptible in subgroup Nxy or Nyy at time t is where λy(t) characterizes the probability that a random contact with any individual staying in subpopulation y will lead to the infection at time t. Similarly, the force of infection for any susceptible in subgroup Nyx or Nxx at time t is With the mean-field approximation, the contagion process in each subgroup under the type-I scenario can be delineated by a set of rate equations (see Eqs.(S5)–(S8) in Supplementary Information).
We next extend to the type-II (origin-driven) contact scenario. Per unit time, an individual from subpopulation x(y) has the characteristic rate of contacts, , which means that on average he encounters with persons, no matter where he is (see Figure S1(b) for the illustration). With the mean-field approximation, the contagion process in each subgroup under the type-II scenario can also be described by the formalism of rate equations as Eqs. (S5)–(S8), except the expressions of the force of infection. The force of infection for any susceptible staying in subgroup Nxy at time t obeys the following relation where denotes the probability that a random contact with any individual staying in subpopulation y will lead to the infection. It is worth emphasizing that to normalize the fraction of contacts of the infectious ones in each subpopulation, the denominator of Eq.(4) should be the total number of contacts of the population instead of the total number of individuals staying in the subpopulation. Similarly, the force of infection for a susceptible presenting in other subgroups at time t are as follows: Besides, we consider a baseline scenario that the contact rate in each subpopulation is equal to 〈c〉 = (cx + cy)/2, which corresponds to the traditional situation that the contact pattern in all subpopulations is uniform. In this case, the epidemic threshold is captured by the basic reproductive number, , which identifies the expected number of secondary infections produced by an infected individual during his infectious period within the entire susceptible population1,2,3,41,58. If , the whole system remains at the disease-free state.
With embedding the aforementioned contact features, the epidemic threshold can be characterized by an extended version of the basic reproductive number 2,41,58. We first build the next generation matrix (NGM) TΣ−1 from the rate equations (S5)–(S8) at the disease-free equilibrium (DFE). Under the type-I contact scenario, the transmission matrix T is defined as while under the type-II contact scenario, the transmission matrix T is In the above two matrices, each entry Tij(i, j ∈ (x, y)) represents the rate of generating new infections in subgroup Nij. In these two scenarios, the transition matrix Σ is where denotes the rate of transferring infectious individuals into subgroup Nij, while denotes the rate of transferring infectious people out of Nij. The recovery of infective individuals also decreases the number of infectious ones, and thus is included in . The matrices T and Σ are obtained at the disease-free equilibrium. The global version of the basic reproductive number is calculated by the spectral radius of the NGM41,58: Since the explicit expression of the threshold is unavailable, we present the numerical results in the following.
Figure 2 presents the phase diagrams of the global . As the empirical evidence tells that the commuting takes place at the time scale τ−1 ~ 1/3 day36, we choose the return rate τ = 0.125/hour. For the diffusion rate, we first set a large value σ = 0.042/hour, which implies that the time scale of diffusion is 1 day. We set other parameters of the disease as: ν = 0.042/hour (a short duration for the mean infectious period), and β = 0.021/hour. Note that fixing the disease parameters as other values does not change the main results of this paper. The white dashed line in each panel indicates the parametric values corresponding to the threshold . The evident difference of the thresholds between the two heterogeneous contact scenarios shows that the type-II contact manner may induce the disease outbreak at much lower contact rates. This is due to the fact that the commuting process facilitates each subpopulation x(y) to share an effective number of individuals into the neighboring subpopulation y(x). As the displaced commuters will closely interact with local residents, individuals with distinct contact rates have the opportunity to directly encounter in each subpopulation under the type-II scenario.
As shown in Figure 2(a), one can find a monotonic mode for the variance of global under the type-I scenario, which positively correlates to the increase of contact rates cx and cy. The results are more complex under the type-II scenario (see Figure 2(b)), because the values of global at the upper-left and lower-right corners of the phase diagram are also very large. The variance of does not show a monotonic mode when or is large (> 2.34) in the case of type-II scenario. For instance, when , gradually decreases until reaches the bottom, then monotonically grows with the increase of .
We further compare the parametric regions of the endemic phases among the baseline, type-I, and type-II scenarios with the aforementioned parameters, as shown in Figure 3(a). The upper triangular light orange area corresponds to the baseline with . The parametric space of the type-I or type-II scenario with the global is much broader than that of the baseline case. Therefore, the inclusion of both two location-specific heterogeneous contact patterns favors the epidemic outbreak. It is also of interest to inspect the effect of a smaller diffusion rate σ on the threshold, since reducing σ weakens the coupling between subpopulations. Figure 3(b) compares the parametric regions pertaining to the endemic phases among the baseline, type-I, and type-II scenarios with σ = 0.021/hour, which yields the same results.
To verify the above analysis, we use the dynamic Monte Carlo method to simulate the epidemic evolution. We focus on inspecting the parametric space of contact rates. With each set of parameters, we monitor the dynamical evolution of epidemic spreading in each subpopulation, and record the disease prevalence (the fraction of infectious individuals) in subpopulation x(y), Ix(t)/Nx(Iy(t)/Ny), where Nx = Nxx(t) + Nyx(t) (Ny = Nyy(t) + Nxy(t)). Due to the presence of stochasticity, each independent Monte Carlo simulation generates a random realization of the dynamic process. We perform 103 random Monte Carlo simulations for each set of parameters.
Initially, each subpopulation has 105 individuals, i.e., Nxx(0) = Nyy(0) = 105, Nxy(0) = Nyx(0) = 0. We fix the return rate τ = 0.125/hour and the diffusion rate σ = 0.042/hour to stress the coupling effect. Figure S2 presents the time series of population evolution in each subgroup. As analyzed in Supplementary Information, the population size of each subgroup quickly reaches to the equilibrium: , . Then we simulate the contagion process with a variety of typical sets of contact rates, fixing the transmission parameters β = 0.021, ν = 0.042/hour. The status of each individual is updated in parallel per unit time. The mobility and disease parameters are directly converted into the probabilities with the unitary time scale of one hour. The algorithm details of the simulation see the section of Methods.
We start the contagion process by introducing the seeds of infectious individuals into a given subgroup. For simplicity, the case of multiple introductions is not considered here, i.e., the seeds are only introduced into one subgroup. The contagion process begins with Ixx(0) = 50, , Iyx(0) = Iyy(0) = Ixy(0) = 0. By fixing the transmission parameters β = 0.021, ν = 0.042, we mainly study the stationary prevalence (SP) , with different parameters (SP reflects the average fraction of infectious individuals in each subpopulation at the equilibrium). To obtain each data point, we perform 103 Monte Carlo random experiments, each of which is simulated with 104 time steps. We define an outbreak run as the realization that the number of infectious individuals at the equilibrium is larger than 50. The stationary prevalence is calculated by averaging over all the outbreak realizations. We first record the mean value of the number of infectious individuals in each subpopulation over the last 10% time steps of each outbreak simulation, then average them to obtain the stationary prevalence. If there does not exist an outbreak run, .
In Figs. 4–5, we compare the phase diagrams of the stationary prevalence , among the baseline, type-I, and type-II contact scenarios with several typical . Figures 4(a)–(b) show the variance of , as gradually increases, fixing . The thresholds of contact rate , which separate the transition from the disease-free phase to the endemic phase, of both the type-I and type-II scenarios are much smaller than that of the baseline case, while the threshold of type-II scenario is the smallest. We also observe the well agreement between the simulations and analysis, as illustrated by the colored dashed lines in each panel. In the endemic phase, we find a counterintuitive phenomenon that, with the same parameters, the stationary prevalence , of the type-II scenario are larger than those of the type-I scenario. In subpopulation x, the larger of the type-II scenario might relate to the introduction of individuals with a higher contact rate. However, in subpopulation y, it seems a little unexpected that the type-II scenario leads to a larger , because under the type-II scenario the displaced commuters from subpopulation x to y have a smaller contact rate when the global . In Figure 6(a), we compare the phase diagrams of global between the type-I and type-II contact scenarios, fixing . When , with the same contact rate , the global of the type-II scenario is evidently larger than that of the type-I scenario, which means that with the same parameters, the prevalence is more serious in the former one. Since we assume that the individual contact rate keeps unchanged in the case of type-II scenario, this outcome may be due to the fact that individuals staying in subgroup Nyy increase their “within self-subgroup” contacts compared with those of the type-I scenario.
Figures 4(c)–(d) present the variance of stationary prevalence , as gradually increases, fixing . In this case, the thresholds of contact rate for the baseline, type-I, and type-II scenarios overlap at , which is also supported by Eq.(11) as indicated by the red dashed line in each panel. In the endemic phase of , the difference of stationary prevalence between the type-I and type-II processes is largely shrunk. Figure 6(b) compares the phase diagrams of global with fixing . In the case of a large , the difference of becomes smaller than that of the former situation .
When we further increase , e.g., or , the outcomes are more complex. Figures 5(a)–(b) show the variance of , as gradually increases, fixing . In the baseline case, both the simulations and analysis agree that the threshold of contact rate decreases to cy = 1. The heterogeneous contact scenarios yield that the threshold of vanishes. Figures 5(c)–(d) present the variance of , in dependence on , fixing . Since the threshold of contact rate of the baseline also vanishes, the baseline results are not included in Figures 5(c)(d). The analytical results of global (see Figures 6(c)(d)) indicate that the disease-free phase is eliminated, no matter how small is.
Moreover, under the type-I process, both and monotonically increase with the augment of cy, as shown in Figure 5. Under the type-II process, it is evident that also monotonically increases with the growth of , while first gradually decreases, after reaching the bottom it begins to monotonically grow with the increase of . The variance of global under the type-II scenario also experiences a valley as gradually increases (see Figure 6). To explain this untrivial phenomenon, we observe the stationary prevalence of each subgroup in detail. As an example, we present the results of in Figure 7. It is clear that and monotonically increase with the increment of , whereas the variances of and are nonmonotonic. In subpopulation x, the probability of an individual of subgroup Nxx to involve in a contact is , while the probability of an individual of subgroup Nyx to involve in a contact is . With the increase of , the individuals in subgroup Nxx will reduce their ‘within self-subgroup’ contacts. As initially grows from 0.1, the net loss of infectious individuals in subgroup Nxx suppresses the gain obtained from subgroup Nyx, because . In this stage, the presence of commuters from subpopulation y mitigates the epidemic situation in subpopulation x, and their influence will be enhanced with the increment of . Thus one can observe the counterintuitive drop of . In subpopulation y, with the increase of , the net increase of infectious individuals in subgroup Nyy can compensate the loss of subgroup Nxy, because . Therefore, keeps on increasing as shown in Figure 5(b). As subpopulations x, y are coupled together, increasing will eventually cease the decline of . After hitting the bottom, positively correlates with the increase of .
As a brief summary, we present the phase diagrams of the stationary prevalence and under the type-I and type-II scenarios, and give a holistic view about the impact of different contact patterns on the spatial transmission between populations. Figures 8(a)–(b) plot the cx−cy phase diagrams of and under the type-I scenario, respectively. One can clearly observe the monotonic increase of and with the increment of cx and cy. The white dashed line in each panel illustrates the threshold of contact rates leading to the global (calculated by the NGM method), which successfully separates the endemic phase from the disease-free phase. Similarly, Figures 8(c)–(d) show the phase diagrams of and under the type-II scenario, respectively. The parametric space corresponding to the endemic phase under the type-II scenario is broader than that of the type-I scenario. When or , the stationary prevalence and does not always monotonically increase with the augment of , . In this case, the increment of contact rates will reduce the disease prevalence or in some parametric regimes.
With the evidence of location-related factors in reality, we have introduced two categories of location-specific contact patterns in a phenomenological structured populations model based on the commuting and contagion processes. Through theoretical analysis and extensive computational simulations, we have shown that these heterogeneous contact scenarios favor the disease outbreaks and evidently decrease the epidemic threshold of the entire system. This finding is robust against the changing of system parameters. More specifically, under the destination-driven scenario, the variance of disease prevalence exhibits the monotonic dependence of contact rates, while under the origin-driven scenario the increment of contact rates unexpectedly reduces the disease prevalence in some parametric regimes. Compared to the traditional framework, where the contact pattern in all subpopulations is uniform, the models presented in this paper demonstrate that the inclusion of these extra factors raises the risk of infection, mainly due to the presence of couplings between populations. Therefore, the interventions to limit the mobility flows, e.g., entry screening, and travel restriction, may benefit to control the disease invasion. To better understand the impact of human contact behaviors on the spatial transmission of infectious disease, more efforts are deserved to deal with more heterogeneous networking populations in further studies.
The contagion and commuting processes at each time step proceed as follows:
(i) Contagion process. For the type-I scenario, the probability of a susceptible staying in subpopulation x to acquire the infection by each of his contacts at time t is λx(t) = β(Ixx(t) + Iyx(t))/(Nxx(t) + Nyx(t)). Therefore, the force of infection for each susceptible staying in subgroup Nxx or Nyx is λxx(t) = λyx(t) = cxλx(t). Similarly, the force of infection for each susceptible staying in subgroup Nyy or Nxy at time t is λyy(t) = λxy(t) = cyλy(t). At this time step, the number of new infections in subgroup Nxy is generated from a binomial distribution with probability λxy(t) and the number of trials Sxy(t). The number of new infections in other subgroups is generated in the same way. At the same time, any infectious individual recovers and becomes susceptible again with rate ν. The number of recovered individuals in each subgroup is generated from a binomial distribution with probability ν and the number of trials defined by the number of infectious individuals in that subgroup.
For the type-II scenario, each resident of subpopulation x(y) contacts with other individuals in his current location per unit time. At time t, the probability of a susceptible presenting in subpopulation y to acquire the infection by each of his contacts is . Thus the force of infection of each susceptible in subgroup Nxy is , while the force of infection of each susceptible in subgroup Nyy is . The forces of infection of susceptible individuals in other subgroups are obtained similarly. At this time step, the number of new infections in subgroup Nxy is generated from a binomial distribution with probability and the number of trials Sxy(t). The number of new infections in other subgroups is generated in the same way. At each time step, the number of recovered individuals in each subgroup is generated from a binomial distribution with probability ν and the number of trials defined by the number of infectious individuals in that subgroup.
(ii) Commuting process. At each time step, after all individuals update the contagion dynamics, the simulation moves to their commuting process. The number of susceptible travelers departing from their resident subpopulation x(y) per unit time is generated from a binomial distribution with probability σ and the number of trials Sxx(t)(Syy(t)). The number of susceptible commuters returning to their residence x(y) per unit time is also generated from a binomial distribution with probability τ and the number of trails Sxy(t)(Syx(t)). The number of infectious individuals leaving from or returning to their residence is acquired in the same way.
This work was partly supported by National Key Basic Research and Development Program (No.2010CB731403), the National Natural Science Foundation(No. 61273223), and the Research Fund for the Doctoral Program of Higher Education (No. 20120071110029) of China. LW also acknowledges the partial support by Fudan University Excellent Doctoral Research Program (985 Project).
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