Epidemic spreading in random rectangular networks

The use of network theory to model disease propagation on populations introduces important elements of reality to the classical epidemiological models. The use of random geometric graphs (RGGs) is one of such network models that allows for the consideration of spatial properties on disease propagation. In certain real-world scenarios—like in the analysis of a disease propagating through plants—the shape of the plots and fields where the host of the disease is located may play a fundamental role in the propagation dynamics. Here we consider a generalization of the RGG to account for the variation of the shape of the plots or fields where the hosts of a disease are allocated. We consider a disease propagation taking place on the nodes of a random rectangular graph and we consider a lower bound for the epidemic threshold of a susceptible-infected-susceptible model or a susceptible-infected-recovered model on these networks. Using extensive numerical simulations and based on our analytical results we conclude that (ceteris paribus) the elongation of the plot or field in which the nodes are distributed makes the network more resilient to the propagation of a disease due to the fact that the epidemic threshold increases with the elongation of the rectangle. These results agree with accumulated empirical evidence and simulation results about the propagation of diseases on plants in plots or fields of the same area and different shapes.

Figure 1

Illustration of an RRG created with 250 nodes embedded into a unit square, with $a=1$ (a), and a unit rectangle, with $a=2$ (b). In both cases the nodes are connected if they are at a Euclidean distance smaller than or equal to $r=0.15$.

Figure 2

(a) Change of the connectivity of RRGs with the change of the connection radius for different values of the rectangle elongation. (b) Illustration of the way in which the critical radius for an RRG is obtained. Also the upper bound (5) (red dotted line) is illustrated. (c) Plot of the connection radius versus the rectangle elongation for the RRGs. The line dividing the two regions represents the critical values of the radius and elongation. All RRGs studied here have $n=1000$ nodes and all the calculations are the result of averaging 20 random realizations of the RRG with the given parameters.

Figure 3

Scatter plots of the spectral radius versus the average degree for RRGs with $a=1$ (a), $a=30$ (b) and $a=1,2.5,5,7.7,15,20,25,30$ (c) for different values of the connection radius.

Figure 4

Values of the spectral radius ${\lambda }_1$ of the adjacency matrix of RRGs with $n=1000$ nodes as a function of the rectangle size length $a$ and the connection radius $r$. The bottom-right part of the plot corresponds to networks which are created with a radius below the critical radius, $r<r_c$ [see plot (c) in Fig. 2], and consequently are disconnected. All the calculations are the result of averaging 20 random generations of the RRG with the given parameters.

Figure 5

(a) Fraction of infected nodes at the stationary state $\rho$ as a function of the infection rate $\beta$ for different values of $a=1$, 10, 20, and 30. $a=1$ represents the first case ($0\le r\le a^{-1}$) of Eq. (3) while $a=10$, 20, and 30 fall in the second case ($a^{-1}\le r\le a$). Other parameters are $n={10}^3$ nodes, $r=0.35$, and $\mu =1.0$. Each point is an average over 250 independent runs. The values shown by arrows are the analytical ones obtained using Eq. (14).

Figure 6

(a) Comparison between the theoretical bound and the epidemic threshold obtained via numerical simulations. Lines represent the theoretical prediction of Eq. (13) while points represent the numerical threshold. The inset shows a zoom for the first case of Eq. (3), $0\le r\le a^{-1}$, (solid line) and the second case, $a^{-1}\le r\le a$ (dashed line). Other parameters are $n={10}^3$ nodes, $r=0.35$, and $\mu =1.0$. Each point is average over 250 independent runs. (b) Fraction of infected nodes at the steady state $\rho$ as a function of the infection rate $\beta$ for $a\le r\le \sqrt{a^2+a^{-2}}$. In the simulations $a=3$ and $r=3.01$. Other parameters are $n={10}^3$ nodes and $\mu =1.0$. Each point is average over 250 independent runs.