# The Importance of Being Hybrid for Spatial Epidemic Models:A Multi-Scale Approach

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## Abstract

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## 1. Introduction

## 2. Modeling Epidemic Spread in a Spatial Network

#### 2.1. The SIR Model

**Figure 1.**Example of the evolution of the number of susceptible, infected and recovered inhabitants in time for one node. To obtain this figure, a simulation was performed with the infection rate $\beta /N=0.5$ and the recovery rate $\alpha =0.2$, leading to the basic reproduction number ${R}_{0}=\beta /\alpha =2.5$. Initial numbers of susceptible, infected and recovered are ${S}_{init}=1000$, ${I}_{init}=10$ and ${R}_{init}=1$. The numerical integration method is Runge–Kutta 4, with a ${10}^{-3}$ step.

#### 2.2. Coupling SIR Models on a Network

#### 2.3. Metapopulation Approach

**Figure 2.**Interface of the metapopulation model, through which the user selects the network topology and mobility rates, as well as the initial population distribution (

**(left)**side of the interface). Typical outputs are computed and displayed during the simulation, such as the numbers of susceptible (S), infected (I) and recovered (R) at each time step (called here a “tick”) or at the end of the simulation, such as “epidemic duration” (

**(right)**side of the interface).

**(Central)**map displays the network of cities and their evolution in terms of number of Infected, during the simulation.

#### 2.4. Hybrid Micro-Macro Approach

**Figure 3.**Interface of the micro-macro hybrid model, through which the user selects the network topology, the mobility rates and the initial population distributions (

**(left)**side of the interface). At each time step, the difference between the total population in the network and the original population is calculated and plotted (“error plot”,

**(right)**side of the interface).

**(Central)**map displays the network of cities, their evolution in terms of number of Infected as well as flights occurring during the simulation (red airplanes contain at least one infected).

## 3. Mean Field Approximation as a Baseline

- the maximum value of infected people ($MaxI$);
- the time at which this value is reached ($TimeofMaxI$);
- the duration of the epidemic ($Duration$), that is the time to reach the condition that the number of infected people is below a small epsilon.

**Figure 4.**Identification of the three output indicators defined to describe the epidemic dynamics: $MaxI$ (maximum number of infected persons), $TimeofMaxI$ (the time at which this value is reached) and epidemic $Duration$.

- $MaxI$ of MetaPop > $MaxI$ of MicMac
- $TimeofMaxI$ of MetaPop < $TimeofMaxI$ of MicMac
- $Duration$ of MetaPop < $Duration$ of MicMac

## 4. Calibrating Time and Space

#### 4.1. Experimental Conditions

Parameter | Value | Parameter | Value |
---|---|---|---|

$N{B}_{Nodes}$ | 30 nodes | N | 40,000 people /nodes |

α | 0.2 | β | 0.5 |

Mobility rate g | 0.12 | Territory size | 300 km^{2} |

Aircraft capacity | 80 passengers | Initial number of infected people | 1 randomly localized |

Network topology | Complete |

#### 4.2. Impact of Traveling Distance in MicMac

**Figure 5.**Impact of traveling distances on epidemic dynamics defined by the three output indicators retained ($MaxI$ in

**(top)**, $TimeofMaxI$ in

**(middle)**and $Duration$ in

**(below)**figures) in MicMac model(number of replications: 50). “Territory size” (in $km$) displays no significant impact on the values of the three output indicators retained.

#### 4.2.1. Gillespie vs. ODE for Contagion during Flights

- We initialize stocks S, I and R.
- We setup the list of possible events. In the SIR model, a susceptible person can become infected, and infected people can recover. Therefore, there are two possible reactions, infection and recovery, given by the following expressions: $\frac{\beta}{N}SI$ and $\alpha I$. Both reactions can be broken down as follows:$$\begin{array}{cccccc}{\displaystyle \frac{\beta SI}{N}:}\hfill & S\hfill & \to S--\hfill & \phantom{\rule{56.9055pt}{0ex}}\alpha I:\hfill & I\hfill & \to I--\hfill \\ & I\hfill & \to I++\hfill & & R\hfill & \to R++\hfill \end{array}$$
- We define ${R}_{1}$ as a random number between zero and one. We calculate the time before the next event: $\tau =-ln\left({R}_{1}\right)/\left({\displaystyle \frac{\beta}{N}IS+\alpha I}\right)$.
- We define ${R}_{2}$ as a random number between zero and one. The event occurring after time τ is contagion if ${R}_{2}<\left({\displaystyle \frac{\beta}{N}SI}\right)/\left({\displaystyle \frac{\beta}{N}SI+\alpha I}\right)$ and healing otherwise.
- We update the values of S, I and R and repeat the previous steps.

**Figure 6.**Comparison of ODE (red dots) and Gillespie (200 replications) estimates of $MaxI$

**(left)**and $TimeofMaxI$

**(right)**for varying population sizes (between five and 1000 persons). The proximity of ODE red dots and boxplot median values highlights the remarkable matching between the two procedures, for ${S}_{init}$ parametervalues inferior to 500 individuals. Then, variance increases for Gillespie estimates, but impacting solely the $MaxI$ indicator.

#### 4.2.2. Impact of Contagion During Flights in MicMac

**Figure 7.**$MaxI$

**(top)**and $TimeofMaxI$

**(below)**with and without contagion during flights, MicMac. Central lines are LoWeSS (locally weighted scatterplot smoothing) estimates, displaying the underlying central tendencies of the two populations being compared.

#### 4.3. Impact of Network Topology in MetaPop and MicMac

**Figure 8.**Small-world (

**top**) and regular networks (

**below**). Number of nodes = 100. Small-world networks are random networks (in the sense that edges are defined randomly, based on the “rewiring probability”) characterized by short average path lengths and high clustering coefficients, varying with the “rewiring probability”, while the number of edges is held constant. Regular networks are characterized by an increasing number of edges here.

**Figure 9.**Impact of regular network density and small-world network rewiring probability on $MaxI$ on

**(top)**, $Duration$ in the

**(middle)**and $TimeofMaxI$

**(below)**. Red dots correspond to the MetaPopmodel, while the error bars are drawn from 50 replications of the MicMac model. The discrepancy displayed between red dots (MetaPop) and error bars (MicMac) is related to the continuous vs. discrete anchorage of the two models.

## 5. Containment Strategy and Adaptive Behavior

- In MetaPop, the infected mobility rate in the city i is multiplied by a coefficient ${\theta}_{3}$, varying between $[0,1]$. If ${\theta}_{3}=0$, city i will not emit infected people, and if ${\theta}_{3}=1$, the mobility rate is unchanged.
- In MicMac, we define ${\theta}_{3}$ directly as a threshold, to which we compare the individual probability to travel for infected agents.

#### 5.1. Quarantine

**Figure 10.**Impact of various thresholds of quarantine setting-up on the epidemic, for the three indicators retained ($MaxI$ on

**(top)**, $TimeOfMaxI$ in the

**(middle)**and $Duration$

**(below)**). Red dots correspond to MetaPop estimates and boxplots to MicMac estimates, for 50 replications. Threshold values correspond to the proportion of infected people in a node beyond which quarantine is setup (0.005 corresponds to five per-mille).

**Figure 11.**Time to reach all nodes from one infected node ($I=1$) for various thresholds of quarantine setting-up. Red dots correspond to MetaPop estimates and boxplots to MicMac estimates, for 50 replications. Threshold values correspond to the proportion of infected people in a node beyond which quarantine is setup (0.005 corresponds to five per-mille).

#### 5.2. Avoidance

**Figure 12.**Time to reach all nodes from one infected node ($I=1$) for various thresholds of avoidance setting-up. Red dots correspond to MetaPop estimates and boxplots to MicMac estimates, for 50 replications. Threshold values correspond to the proportion of infected people in a node beyond which avoidance is setup (0.005 corresponds to five per-mille).

**Figure 13.**Impact of various thresholds of avoidance setting-up on the epidemic, for the three indicators retained ($MaxI$ on

**(top)**, $TimeOfMaxI$ in the

**(middle)**and $Duration$

**(below)**). Red dots correspond to MetaPop estimates and boxplots to MicMac estimates, for 50 replications. Threshold values correspond to the proportion of infected people in a node beyond which avoidance is setup (0.005 corresponds to five per-mille).

#### 5.3. Risk Culture

**Figure 14.**Impact of risk culture on the epidemic, for the three indicators retained ($MaxI$ on

**(top)**, $TimeOfMaxI$ and $Duration$ in the

**(middle)**) plus a fourth indicator of the time needed to reach all nodes from a single infected one ($Time\phantom{\rule{4pt}{0ex}}to\phantom{\rule{4pt}{0ex}}reach\phantom{\rule{4pt}{0ex}}all\phantom{\rule{4pt}{0ex}}nodes$,

**(below)**). Red dots correspond to MetaPop estimates and boxplots to MicMac estimates, for 50 replications. Threshold values correspond to reduction coefficients for the mobility rate in MetaPop and probabilities to travel when infected for MicMac. In both cases, a zero value of the threshold implies the absence of mobility for infected people.

## 6. Conclusion and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Banos, A.; Corson, N.; Gaudou, B.; Laperrière, V.; Coyrehourcq, S.R. The Importance of Being Hybrid for Spatial Epidemic Models:A Multi-Scale Approach. *Systems* **2015**, *3*, 309-329.
https://doi.org/10.3390/systems3040309

**AMA Style**

Banos A, Corson N, Gaudou B, Laperrière V, Coyrehourcq SR. The Importance of Being Hybrid for Spatial Epidemic Models:A Multi-Scale Approach. *Systems*. 2015; 3(4):309-329.
https://doi.org/10.3390/systems3040309

**Chicago/Turabian Style**

Banos, Arnaud, Nathalie Corson, Benoit Gaudou, Vincent Laperrière, and Sébastien Rey Coyrehourcq. 2015. "The Importance of Being Hybrid for Spatial Epidemic Models:A Multi-Scale Approach" *Systems* 3, no. 4: 309-329.
https://doi.org/10.3390/systems3040309