Figure 2
Epidemic spread due to a variation of disease transmission in one patch. Modifying the transmission rate at one patch changes the epidemic threshold Eq. (2) to the more complicated expression Eq. (9). Transmission changes from \(\beta\) to \(\beta ^{*}\), ranging from no transmission at the perturbed patch (yellow) to an increased transmission (purple). Disease transmission occurs at a rate \(\beta =0.15\) and the network has \(N=50\) patches. (a) Sum of infected individuals across all patches over time for different values of \(\beta ^{\star }\). A change of regime is represented by the dashed line defined by the threshold value of \(\beta ^{\star }\). (b) Infected individuals over time for a system with equal transmission rates (left) and a system with an increased transmission rate at one node (right, perturbed node in purple). Even though the perturbation in transmission only affects one patch the system switches from stable to unstable. (c) Eigenvalue distribution for the system with different transmission rate in one patch. As the perturbation increases the epidemic threshold given by the rightmost eigenvalue increases. The stars correspond to the examples presented in (b), yellow for the left plot and purple for the right one. One more outlier arises for the perturbed case due to the specific form of the structure matrix (see “Methods” section). (d) Maximum number of infected individuals at one patch (left) and time to reach the epidemic peak (right) in terms of \(\beta ^*\). The maximum increases nonlinearly as the perturbation in transmission increases, while the time to reach this maximum decreases since the initial growth is steeper for higher \(\beta ^*\).
