Fig 1.
Description of the vaccination dynamics.
We apply the strategy to the same original contact pattern network to study the different vaccination dynamics. As a first differentiation, we consider i) Preventive scenario: in which the vaccination dynamic occurs before the appearance of the zero patient, leading the disease to propagate in a diluted original network; ii) Containment scenario: where the vaccination dynamic occurs simultaneously with the spreading process, every time new infected individuals appear a set of their neighbors are chosen to be vaccinated. Besides, we explore the cases of (un)limited resources: a) Block vaccination: in which all the individuals within the ring are vaccinated; and b) ring vaccination: where resources are limited hence, only individuals in the largest radio (border of the ring) are vaccinated. In the colored panels, we display a one-time step of the different vaccination dynamics considering a radio of vaccination of second neighbors (r = 2). For the propagation of the disease, we consider a modified version of the SIR model where individuals can be in one of four different states: Susceptible (green), Infected (red), Recovered (gray), or Vaccinated (Blue). Yellow panels correspond to the asymptotic state of the preventive scenario (after the disease propagates, without infected individuals). Islands of susceptible (ring) or vaccinated (block) individuals appear depending on the strategy, shaping the reach of the epidemic. Pink panels correspond to an intermediate step of the disease propagation. Conversely, in this case, for every newly infected individual, with a certain probability, a portion of their neighbors will get vaccinated with the intent of breaking the chain of infection. We develop a mathematical framework for this family of vaccination strategies and validate our results with stochastic simulations. Through a case study, we show how to build maps of epidemic risk in spatially embedded networks. Data for the creation of the maps taken from: https://www.istat.it/notizia/confini-delle-unita-amministrative-a-fini-statistici-al-1-gennaio-2018-2/.
Table 1.
Summary of parameters occurring in the proposed equations - Preventive scenario.
Fig 2.
Validation of the theoretical framework for the case of Preventive—static—scenarios.
For the cases of ring (left panel) and block (right panel) vaccinations. The radius of vaccination spans from 0 (baseline that corresponds to random vaccination) to 3. As the vaccination strategy occurs before the propagation of the disease, we validate our results by analyzing the evolution of the giant component as a function of the percentage of vaccinated nodes ϕ, i.e., the resulting diluted network after the percolation process. Each horizontal panel corresponds to the same network structure with N = 104 nodes, ranging from more homogeneous on the top, a random regular graph with , to more heterogeneous on the bottom, a scale-free network with degree exponent
and
and
. In the middle, results are shown for Erdős-Rényi nets with
. See the Methods Section for details further details on these network ensembles. Experimental results are averaged over 103 independent stochastic simulations.
Table 2.
Summary of parameters occurring in the proposed equations — Preventive scenario.
Fig 3.
Validation of the theoretical framework for the case of Containment—dynamic—scenario.
For the cases of ring (left panel) and block (right panel) vaccination. The radius of vaccination considered is r = 2. As the vaccination occurs along with the propagation of the disease, we validate our results by analyzing the evolution over time of the individuals in each compartment of the SIR-V model. The epidemiological parameters values are ,
and
, and the process occurs on top of a Scale-Free network with
,
and
, and size N = 104. Experimental results are averaged over 103 independent stochastic simulations.
Fig 4.
Asymptotic size of the epidemic (total fraction of recovered individuals R) as a function of the virulence of the diseases β for the preventive scenario.
Solid lines are theoretical predictions, and symbols correspond to stochastic simulations. Grey lines correspond to the case with vaccination strategies, while violet lines correspond to the standard SIR. The initial size is N = 104, and the results are the average computed over 103 realizations. We consider scale-free networks with exponent ,
and
. The recovery probability is set in
, and the vaccination radius is equal to r = 2. Different lines refer to different initial percentages of vaccinated nodes ϕ.
Fig 5.
Epidemic impact in the containment scenario.
We consider scale-free networks with an exponent ,
and
, and size N = 104. The recovery probability is set in
. 1st and 3rd rows display the fraction of vaccinated (top) and recovered (bottom) individuals at the asymptotic state of the epidemic. Each line corresponds to different values of vaccination radius,
, with a probability of vaccination
. Panel on top. Dotted lines are theoretical predictions, and symbols correspond to stochastic simulations. 2nd and 4th rows: Heatmaps show the dependence of the vaccinated (top) and recovered (bottom) individuals on both the probability of vaccination ω (y-axis) and the virulence of the disease β (x-axis) for the specific case of r = 2.
Fig 6.
Impact of block vaccination strategy for a containment scenario on the olive tree empirical network: evolution of the diseases.
Snapshots of the advance of the disease at different times, spanning from the early stages of the propagation (A) to the asymptotic state when there is no more presence of infected trees (C). Single nodes are colored according to their state during the process: Susceptible (green), Infected (red), Recovered (gray), and Vaccinated (blue). The epidemiological parameter values are: probability of infection , probability of recovery
, and probability of vaccination
. Data for the creation of the maps taken from: https://www.istat.it/notizia/confini-delle-unita-amministrative-a-fini-statistici-al-1-gennaio-2018-2/.
Fig 7.
Impact of the vaccination strategy in the containment scenario applied to the empirical olive tree network.
(A) displays the distance between the starting point of the epidemics and the farthest R-node, as a function of the probability of vaccination ω. Heatmaps (bottom) show the probability of each node to be in the recovered state at the end of the process, for, respectively, dynamic ring (B) and block (C) vaccination, for the case of . The epidemiological parameters are: probability of infection
, probability of recovery
. Data for the creation of the maps taken from: https://www.istat.it/notizia/confini-delle-unita-amministrative-a-fini-statistici-al-1-gennaio-2018-2/.