Epidemic Spreading on Metapopulation Networks with Finite Carrying Capacity

Abstract

In this study, we formulate and analyze a susceptible–infected–susceptible (SIS) dynamic on metapopulation networks, where each node has a finite carrying capacity and the motion of individuals is modulated by vacant space at the destination. We obtain that the vacancy-dependent mobility pattern results in various asymptotic population distributions on heterogeneous metapopulation networks. The resulting population distributions have remarkable impact on the behavior of SIS dynamics. We show that, for the given total number of individuals, higher heterogeneity in population distributions facilitates epidemic spreading in terms of both a smaller epidemic threshold and larger macroscopic incidence. Moreover, we analytically obtain a sufficient condition that the disease-free equilibrium becomes unstable and an endemic state arises. Contrary to the absence of an epidemic threshold in the standard diffusion case without excluded-volume effects, the finite carrying capacity induces a nonzero epidemic threshold under certain conditions in the limit of infinite network sizes with an unbounded maximum degree. Our analytical results agree well with numerical simulations.

1. Introduction

Metapopulation networks have been widely used to study the epidemic processes in a spatially structured population with mobile individuals [1]. This includes, for example, forecasting global outbreak trajectories of severe acute respiratory syndrome (SARS) [2] and assessing intervention strategies for COVID-19 [3]. In metapopulation networks, nodes represent subpopulations where individuals interact and edges allow for the movement of individuals [4,5,6]. Previous studies have shown that mobility patterns fundamentally influence the epidemic processes on metapopulation networks. Various mobility patterns have been considered, such as recurrent host mobility [7,8,9,10,11], prevalence-base mobility restrictions [12], mobility processes based on realistic transportation data [2], and density-dependent migration [13].

In many real systems, due to the finite carrying capacity of each node, diffusion behavior is usually affected by competition for the available space or resource. The actual migratory flows between nodes depend on vacant conditions [14]. This vacancy-dependent mobility has resulted in a nonlinear relationship between the stationary distribution of diffusing individuals and the node degree [14,15,16] at variance with the linear relationship in standard diffusion with independent movements among individuals [4,17]. There is growing concern about the effect of finite carrying capacity on epidemic spreading on metapopulation networks. Recently, Siebert et al. [18] proposed an epidemic model of optimal social distancing in the framework of metapopulation networks with finite carrying capacity.

In this paper, we investigate susceptible–infected–susceptible (SIS) dynamics on heterogeneous metapopulation networks with finite carrying capacity. The diffusion of individuals is dependent on the vacant space at the nearest neighbors. This vacancy-dependent mobility generates various steady population distributions on heterogeneous metapopulation networks. We find that, for the given total number of individuals, greater heterogeneity in population distribution favors epidemic spreading with a smaller epidemic threshold and a larger macroscopic incidence. Furthermore, we analytically obtain a sufficient condition for the instability of the disease-free state. We find that when the number of individuals at the maximum-degree nodes is bounded, the epidemic threshold is nonzero in the limit of infinite network sizes with a diverging maximum degree, at variance with the absence of an epidemic threshold in the standard diffusion case (which corresponds to the situation where each node has unlimited carrying capacity in the context of our studied model) [17,19,20]. Our analytical results are confirmed by numerical simulations.

2. The Model

We consider a generic undirected network composed of N nodes and characterized by its $N\times N$ adjacency matrix A, with elements $A_{ij}=1$ when nodes i and j are connected and zero otherwise. The degree $k_i$ of node i is the number of connected neighbors, i.e., $k_i={\sum }_j{A}_{ij}$. Let $c_i$ represents the carrying capacity of node i, which indicates that node i is able to host a maximum number $c_i$ of individuals. The number of individuals in node i is denoted as ${\rho }_i$. Each individual attempts to perform diffusion with rate d, while the actual jump between nodes depends on vacant conditions of the destination. The jump can only occur if the destination node j has vacant space, i.e., with probability $(c_j-{\rho }_j)/c_j$ [14]. So, when moving, individuals prefer destinations with a higher vacant level. Thus, the hopping probability $T_{ij}$ from node i to j is

$$T_{ij}={\displaystyle \frac{A_{ij}}{k_i}}{\displaystyle \frac{c_j-{\rho }_j}{c_j}},$$

(1)

where $1/k_i$ is the probability that the individuals sitting on node i randomly select one of the $k_i$ neighboring nodes as its destination. In Appendix A, Figure A1a shows a schematic of the movement process between two nodes of the network.

In addition to the movement processes, local SIS reaction dynamics take place simultaneously. Each individual can be either susceptible (S, who can catch the disease) or infected (I, who has the disease and transmits it). Let ${\rho }_{S,i}$ and ${\rho }_{I,i}$ denote the number of susceptible and infected individuals in node i, respectively. Hence, ${\rho }_{S,i}+{\rho }_{I,i}={\rho }_i$ is the total number of individuals in node i. We assume the two different classes are uniformly mixed and there is all-to-all interaction within each node. The transmission of the infection involves each $S-I$ pair having an infection rate $\beta$ at which a susceptible individual becomes infected,

$$\begin{array}{ccc}\hfill S+I& \stackrel{\beta }{\to }& 2I.\hfill \end{array}$$

(2)

Under well-mixed assumption, within node i, there are a total of ${\rho }_{S,i}{\rho }_{I,i}$ distinct $S-I$ pairs (i.e., susceptible individuals interact with all of the infected individuals in the same node). Thus, the infection event rate within node i is $\beta {\rho }_{S,i}{\rho }_{I,i}$. One infection event means that the number of susceptible individuals is reduced by one. The gain in the infected individuals is equal to the loss of susceptible individuals. The recovery process is that each infected individual spontaneously becomes susceptible with recovery rate $\mu$,

$$\begin{array}{ccc}\hfill I& \stackrel{\mu }{\to }& S.\hfill \end{array}$$

(3)

Then, within node i, the recovery event rate is $\mu {\rho }_{I,i}$. An infection event reduces the number of infected individuals by one. In Appendix A, Figure A1b shows a schematic of the local SIS reaction dynamics.

To capture the heterogeneity in the carrying capacity [21], we assume the following nonlinear form for carrying capacity $c_i$:

$$c_i={\displaystyle \frac{k_i^{\alpha }}{\langle k^{\alpha }\rangle }}\overline{c},$$

(4)

where $\overline{c}={\sum }_i{c}_i/N$ is the average carrying capacity per node and $\langle k^{\alpha }\rangle ={\sum }_i{k}_i^{\alpha }/N$. The tunable exponent $\alpha$ controls the correlation between the carrying capacity and the node’s degree. For $\alpha >0$, we have a scenario in which larger degree nodes have a higher carrying capacity. On the other hand, an exponent $\alpha <0$ implies a scenario in which smaller degree nodes have a higher carrying capacity. Finally, for $\alpha =0$, each node has the same carrying capacity.

3. Simulation Procedures

We use Gillespie’s algorithm [22] to simulate our studied model for confirming analytical predictions. For a given set of parameters $\overline{c}$ and $\alpha$, we calculate from Equation (4) each node’s carrying capacity $c_i$. In the numerical simulation, initially, a total population $N\overline{\rho }$ ($\overline{\rho }$ is the average number of individuals per node, and $\overline{\rho }\le \overline{c}$) are randomly assigned on the network nodes, with the constraint ${\rho }_i\le c_i$. At first, we perform the movement process only, until the population distribution arrives at a steady state (see Appendix A, Figure A2 for a schematic of this simulation procedure). Then a fraction $i_0(=0.01)$ of individuals are randomly selected as the initial infected individuals, while the remaining ones are susceptible. At each time step, we perform the following procedures: (i) a node i is selected with probability $V_i/V$, where $V_i=\beta {\rho }_{S,i}{\rho }_{I,i}+\mu {\rho }_{I,i}+d{\rho }_i$, i.e., the summation of the rates of the infection, recovery, and movement event within node i, and $V={\sum }_i{V}_i$. (ii) The next event is chosen from three possibilities according to the following rules: (a) recovery event (${\rho }_{S,i}\leftarrow {\rho }_{S,i}+1$, ${\rho }_{I,i}\leftarrow {\rho }_{I,i}-1$) with probability $\mu {\rho }_{I,i}/V_i$; (b) infection event (${\rho }_{S,i}\leftarrow {\rho }_{S,i}-1$, ${\rho }_{I,i}\leftarrow {\rho }_{I,i}+1$) with probability $\beta {\rho }_{S,i}{\rho }_{I,i}/V_i$; and (c) movement event with probability $d{\rho }_i/V_i$. If the movement event is chosen, one neighboring node is selected randomly, say j, as the destination. With probability $(c_j-{\rho }_j)/c_j$, one randomly selected individual at node i hops into node j; otherwise, nothing happens.

Time is updated as $t\leftarrow t+1/V$. Finally, steps (i) and (ii) are iterated until the dynamics arrive at a steady state (see Appendix A, Figure A3 for a schematic of this simulation procedure). Considering that many realistic networks have a heterogeneous degree distribution, we build the random uncorrelated scale-free networks, generated according to Ref. [23], as the substrate network of the metapopulation. For any system of finite population, the system will eventually enter the absorbing state where all individuals are susceptible due to finite-size fluctuations. Based on this fact, we allow the system to evolve up to $t_{max}=500$ and average the related quantities over the last 100 times as the results in a steady state in one realization. The results are averaged over 50 runs.

4. Results

According to the model’s assumptions, we use the continuous-time formulation developed in [17] to describe the dynamics at the single-node level,

$$\left\{\begin{array}{c}{\partial }_t{\rho }_{S,i}\left(t\right)=\left(\mu -\beta {\rho }_{S,i}\right){\rho }_{I,i}-d{\rho }_{S,i}\sum _j{T}_{ij}+d\sum _j{T}_{ji}{\rho }_{S,j},\hfill \\ {\partial }_t{\rho }_{I,i}\left(t\right)=\left(\beta {\rho }_{S,i}-\mu \right){\rho }_{I,i}-d{\rho }_{I,i}\sum _j{T}_{ij}+d\sum _j{T}_{ji}{\rho }_{I,j}.\hfill \end{array}\right.$$

(5)

The first term in each sub-equation of Equation (5) corresponds to the infection and recovery processes. The last two terms represent the movement process. The negative term counts the number of individuals leaving from node i, while the positive term is the sum of the flow of individuals arriving at node i from its neighbors.

Adding the two sub-equations in Equation (5), we obtain the rate equation for the evolution of the number ${\rho }_i\left(t\right)$ of individuals at node i,

$${\partial }_t{\rho }_i\left(t\right)=-d\sum _j{\displaystyle \frac{A_{ij}}{k_i}}{\displaystyle \frac{c_j-{\rho }_j}{c_j}}{\rho }_i+d\sum _j{\displaystyle \frac{A_{ij}}{k_j}}{\displaystyle \frac{c_i-{\rho }_i}{c_i}}{\rho }_j,$$

(6)

where we have used Equation (1). Summing the overall nodes in both sides of Equation (6), we obtain ${\partial }_t{\sum }_i{\rho }_i\left(t\right)=0$, which is consistent with the fact that the total number ${\sum }_i{\rho }_i$ of individuals in the system is conserved. Note that when ${\rho }_i/c_i\to 0$, for all i, Equation (6) is reduced to the standard diffusion equation [4,17]. The equilibrium solution ${\rho }_i^{\ast }$ of Equation (6) can be obtained by setting ${\displaystyle \frac{1}{k_i}}{\displaystyle \frac{c_j-{\rho }_j^{\ast }}{c_j}}{\rho }_i^{\ast }={\displaystyle \frac{1}{k_j}}{\displaystyle \frac{c_i-{\rho }_i^{\ast }}{c_i}}{\rho }_j^{\ast }$. Obviously, we can obtain

$${\rho }_i^{\ast }={\displaystyle \frac{c_i{k}_i}{k_i+a{c}_i}},$$

(7)

where the constant a is determined from the conservation condition, that is, ${\sum }_i{\rho }_i^{\ast }={\sum }_i{\displaystyle \frac{c_i{k}_i}{k_i+a{c}_i}}$ is equal to the total number of individuals in the system. Substituting Equation (4) into Equation (7), we can obtain the number ${\rho }_k^{\ast }$ of individuals in the node of degree k at steady state, i.e., the asymptotic population distribution

$${\rho }_k^{\ast }={\displaystyle \frac{k}{a+\frac{\langle k^{\alpha }\rangle }{\overline{c}{k}^{\alpha -1}}}},$$

(8)

where the constant a is also determined from the conservation condition ${\sum }_k{\rho }_k^{\ast }P\left(k\right)=\overline{\rho }$. From Equation (8), we can obtain ${\rho }_k^{\ast }=k\overline{\rho }/\langle k\rangle$ for the standard diffusion process (i.e., by setting $\overline{c}\to \infty$) [4,17]. When $\alpha =0$, we have ${\rho }_k^{\ast }=k\overline{c}/(a+k)$ for the case that the carrying capacity of each node is identical [14].

The theoretical results are compared to those obtained from numerical simulations, where the underlay network is generated by the uncorrelated configuration model [23] with degree distribution $P\left(k\right)\propto k^{-3}$, minimum degree $k_{min}=3$, maximum degree $k_{max}=N^{1/2}$, and network size $N=1000$. As shown in Figure 1, the numerical solutions for stationary population distribution ${\rho }_k^{\ast }$ (obtained from Equation (8)) are in good agreement with the results of numerical simulations. Note that the stationary population distribution ${\rho }_k^{\ast }$ is independent of the infection dynamics and the diffusion rate d. The simulation data were obtained from the simulation algorithm of the movement process without considering the infection status of each individual (see Appendix A, Figure A2 for a schematic of this simulation procedure). We denote the macroscopic incidence (fraction of infected population) at steady state as ${\rho }_I$, which is computed as the number of infected individuals after transient divided by the total number of individuals, i.e.,

$${\rho }_I\equiv {\displaystyle \frac{1}{N\overline{\rho }}}\sum _i{\rho }_{I,i}^{\ast },$$

(9)

where ${\rho }_{I,i}^{\ast }$ is the number of infected individuals within node i at steady state. Figure 2a and Figure 3a show that the numerical solutions for macroscopic incidence ${\rho }_I$, obtained from Equations (5) and (9), agree well with computer simulations.

For the fixed carrying capacity exponent $\alpha =0.0$ and average number of individuals per node $\overline{\rho }=250$, Figure 1a shows that ${\rho }_k^{\ast }$ is an increasing function of degree k for any value of $\overline{c}$. As $\overline{c}$ increases, ${\rho }_k^{\ast }$ increases for high k values, whereas it decreases for lower k values. Thus, the increasing $\overline{c}$ causes a larger range of values for ${\rho }_k^{\ast }$’s, that is, more heterogeneity in the distribution of individuals. Figure 2 shows that higher values of $\overline{c}$ enhance epidemic spreading in terms of both a smaller epidemic threshold and larger macroscopic incidence.

For fixed average population per node $\overline{\rho }=250$ and fixed $\overline{c}=10\overline{\rho }$, Figure 1b shows ${\rho }_k^{\ast }$ with different carrying capacity exponent $\alpha$. As $\alpha$ increases, i.e., carrying capacity correlates more strongly with node degree, the asymptotic population distribution becomes more heterogeneous (see Figure 1b), which enhances epidemic spreading in terms of both smaller epidemic threshold and larger macroscopic incidence (as shown in Figure 3).

Next, we will analyze the sufficient condition that the disease-free equilibrium becomes unstable and investigate how the carrying capacity exponent $\alpha$ affects the epidemic threshold. For the sake of analytical treatment described, we will assume statistical equivalence for nodes with the same degree. Under this heterogeneous mean-field approximation [1,24], we can introduce degree-block variables ${\rho }_{S,k}$ and ${\rho }_{I,k}$ represents the numbers of susceptible and infected individuals in the node of degree k, respectively, i.e.,

$${\rho }_{S,i}\left(t\right)\equiv {\rho }_{S,k}\left(t\right),{\rho }_{I,i}\left(t\right)\equiv {\rho }_{I,k}\left(t\right),\forall i\in \mathcal{V}\left(k\right),$$

(10)

where $\mathcal{V}\left(k\right)$ is the set of nodes of degree k. We can split the sum with index j into two sums over $k^{\prime }$ and $\mathcal{V}\left(k^{\prime }\right)$, i.e.,

$$\sum _j{A}_{ij}=\sum _{k^{\prime }}\sum _{j\in \mathcal{V}\left(k^{\prime }\right)}{A}_{ij}=\sum _{k^{\prime }}kP\left(k^{\prime }\right|k),\forall i\in \mathcal{V}\left(k\right).$$

(11)

The double sum over $A_{ij}$ is related to the conditional probability $P\left(k^{\prime }\right|k)$ that a node of given degree k has a neighbor which has degree $k^{\prime }$. Furthermore, we restrict ourselves to the case of uncorrelated networks in which the conditional probability takes the simple form $P\left(k^{\prime }\right|k)=k^{\prime }P\left(k^{\prime }\right)/\langle k\rangle$ [1,24]. Thus, after some reformulation, Equation (5) simplifies to

$$\left\{\begin{array}{c}{\partial }_t{\rho }_{S,k}\left(t\right)=\left(\mu -\beta {\rho }_{S,k}\right){\rho }_{I,k}-d{\rho }_{S,k}\sum _{k^{\prime }}{\displaystyle \frac{k^{\prime }P\left(k^{\prime }\right)}{\langle k\rangle }}(1-{\displaystyle \frac{{\rho }_{k^{\prime }}}{c_{k^{\prime }}}})+d(1-{\displaystyle \frac{{\rho }_k}{c_k}}){\displaystyle \frac{k}{\langle k\rangle }}\sum _{k^{\prime }}P\left(k^{\prime }\right){\rho }_{S,k^{\prime }},\hfill \\ {\partial }_t{\rho }_{I,k}\left(t\right)=\left(\beta {\rho }_{S,k}-\mu \right){\rho }_{I,k}-d{\rho }_{I,k}\sum _{k^{\prime }}{\displaystyle \frac{k^{\prime }P\left(k^{\prime }\right)}{\langle k\rangle }}(1-{\displaystyle \frac{{\rho }_{k^{\prime }}}{c_{k^{\prime }}}})+d(1-{\displaystyle \frac{{\rho }_k}{c_k}}){\displaystyle \frac{k}{\langle k\rangle }}\sum _{k^{\prime }}P\left(k^{\prime }\right){\rho }_{I,k^{\prime }}.\hfill \end{array}\right.$$

(12)

To obtain the sufficient condition that the disease-free equilibrium becomes unstable, we use the analytical argument presented in [17,19,20]. The Jacobian matrix of Equation (12) at the disease-free equilibrium (i.e., ${\rho }_{S,k}={\rho }_k^{\ast }$ and ${\rho }_{I,k}=0$ for all k) can be written in blocks as

$$\left(\begin{array}{cc}{d}_{eff}(\mathcal{F}-I)& -\mathrm{diag}\left(\beta {\rho }_k^{\ast }-\mu \right)\\ 0& d_{eff}\mathcal{F}+\mathrm{diag}\left(\beta {\rho }_k^{\ast }-\mu -d_{eff}\right)\end{array}\right),$$

(13)

where each block is an $n_k\times n_k$ matrix, with $n_k$ being the number of degrees in the network, 0 as the null matrix, I as the identical matrix, and $\mathrm{diag}\left(\beta {\rho }_k^{\ast }-\mu \right)$ representing the diagonal matrix with diagonal element $\left(\beta {\rho }_k^{\ast }-\mu \right)$. It is notable that $d_{eff}$ can be considered as an effective diffusion rate

$$d_{eff}\equiv d\sum _k{\displaystyle \frac{kP\left(k\right)}{\langle k\rangle }}(1-{\displaystyle \frac{{\rho }_k^{\ast }}{c_k}}).$$

(14)

$\mathcal{F}$ stands for the matrix with elements

$${\mathcal{F}}_{k{k}^{\prime }}={\displaystyle \frac{(1-\frac{{\rho }_k^{\ast }}{c_k})\frac{kP\left(k^{\prime }\right)}{\langle k\rangle }}{{\sum }_k\frac{kP\left(k\right)}{\langle k\rangle }(1-\frac{{\rho }_k^{\ast }}{c_k})}}.$$

(15)

Obviously, $\mathcal{F}$ is a rank-one matrix. The eigenvalues of $\mathcal{F}$ are $\lambda =0$, with algebraic multiplicity $n_k-1$ and a simple eigenvalue $\lambda =1$, owing to the fact that the sum of the diagonal elements of $\mathcal{F}$ is equal to 1. Thus, the characteristic polynomial of the upper diagonal block $d_{eff}(\mathcal{F}-I)$ of the Jacobian matrix is $\lambda {(\lambda +d_{eff})}^{n_k-1}$, and its largest eigenvalue is equal to 0. The lower diagonal block of the Jacobian matrix has the form $\mathrm{diag}\left(\beta {\rho }_k^{\ast }-\mu -d_{eff}\right)$, which is perturbed by the rank-one matrix $d_{eff}\mathcal{F}$. According to the general interlacing theorem of eigenvalues for this type of matrix [25], its largest eigenvalue ${\lambda }_1$ satisfies ${\lambda }_1>{\max }_k\{\beta {\rho }_k^{\ast }-(\mu +d_{eff})\}$. The triangular structure of the Jacobian matrix means its spectrum is the union of the spectra of these two diagonal blocks. Finally, the largest eigenvalue of the Jacobian matrix is $\max \{0,{\lambda }_1\}$. Therefore, a sufficient condition for the disease-free equilibrium to be unstable is given by

$${\displaystyle \frac{\beta }{\mu +d_{eff}}}{\rho }_{k_{max}}^{\ast }\ge 1.$$

(16)

For the standard diffusion scenario, the inequality reduces to ${\displaystyle \frac{\beta }{\mu +d}}{\displaystyle \frac{k_{max}}{\langle k\rangle }}\rho \ge 1$, as reported in Refs. [17,19]. The left-hand side of this inequality can be approximately treated as a basic reproductive number at maximum-degree nodes. Note that we only consider the case that ${\rho }_k^{\ast }$ is an increasing function of k. Rewriting Equation (16), we obtain the sufficient condition of $\beta$ as

$$\beta \ge {\beta }_c\equiv {\displaystyle \frac{\mu +d_{eff}}{{\rho }_{k_{max}}^{\ast }}},$$

(17)

which indicates that if $\beta \ge {\beta }_c$, the system certainly reaches an endemic state. As the arrows show in Figure 2a and Figure 3a, ${\beta }_c$ can be approximately considered as an epidemic threshold which separates the endemic state from the disease-free state. Note that it may be possible to have an endemic state if $\beta$ is less than ${\beta }_c$. Equation (17) shows that ${\beta }_c$ is an increasing function of $d_{eff}$, which implies that diffusion suppresses epidemic spreading in the sense of an increase in the epidemic threshold. This is consistent with the result obtained for the standard diffusion scenario in [20].

In the thermodynamics limit (i.e., $N\to \infty$ and $\overline{\rho }$ is a finite fixed value), for networks with a bounded average degree $\langle k\rangle$ and an unbounded largest degree $k_{max}\to \infty$, from Equation (8), we have

$$\left\{\begin{array}{c}{\rho }_{k_{max}}^{\ast }\to \infty \mathrm{for}\alpha >0,\hfill \\ {\rho }_{k_{max}}^{\ast }=\overline{c}\mathrm{for}\alpha =0,\hfill \\ {\rho }_{k_{max}}^{\ast }<\overline{c}\mathrm{for}\alpha <0.\hfill \end{array}\right.$$

(18)

Combining Equations (17) and (18), we obtain

$$\left\{\begin{array}{c}{\beta }_c=0\mathrm{for}\alpha >0,\hfill \\ {\beta }_c>0\mathrm{for}\alpha \le 0,\hfill \end{array}\right.$$

(19)

for a network with $k_{max}\to \infty$ in the thermodynamics limit. Note that the lack of an epidemic threshold, i.e., ${\beta }_c=0$, is also obtained for the standard diffusion scenario [17,19,20], while in the finite carrying capacity scenarios, the conditions for the lack of an epidemic threshold depend on the value of the carrying capacity exponent $\alpha$. When $\alpha \le 0$, the number of individuals at maximum-degree nodes ${\rho }_{k_{max}}^{\ast }$ is bounded (see Equation (18)), which results in the presence of a nonzero epidemic threshold. As shown in Figure 4, for $\alpha =0$, ${\beta }_c$ is almost unchanged as the network size N increases, which indicates the presence of a positive epidemic threshold in limit $N\to \infty$. On the other hand, when $\alpha >0$, $log\left({\beta }_c\right)$ is linear in $-log\left(N\right)$, which indicates the absence of an epidemic threshold in limit $N\to \infty$.

5. Conclusions

In summary, motivated by the fact that the mobility pattern of individuals in many real systems is usually affected by competition for available space or resources, we investigated an SIS metapopulation model with finite carrying capacity, where the mobility of individuals is modulated by the vacant space at the destination. Considering that many realistic networks have heterogeneous degree distribution, we have used random uncorrelated scale-free networks [23] as the substrate network of the metapopulation. To capture the heterogeneity in the carrying capacity [21], we have assumed that the carrying capacity scales as a power law of the node’s degree. The introduced exponent $\alpha$ allows us to tune the correlation between the carrying capacity and node degree. We obtained both a higher value of the average carrying capacity per node $\overline{c}$ and a higher value of the carrying capacity exponent $\alpha$, generating greater heterogeneity in the population distribution, which promotes the spread of a disease in terms of both a smaller epidemic threshold and larger stationary macroscopic incidence. We found that the value of $\alpha$ determines whether a nonzero epidemic threshold exists or not in the limit of infinite network sizes with an unbounded maximum degree. Our study enriches the understanding of how the mobility patterns induced by finite carrying capacity influence the epidemic processes on metapopulation networks. To reduce the complexity of our model, we have not considered the demographic evolution, i.e., there are no natural birth and death processes. It would be interesting, in the future, to consider the scenario in which the finite carrying capacity can drive mobility and shape demographics simultaneously.