# Graph, Spectra, Control and Epidemics: An Example with a SEIR Model

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## Abstract

**:**

## 1. Introduction

`(S)`and infected

`(I)`[2,3]. The subpopulation

`S`represents individuals who are healthy but susceptible to becoming infected, while

`I`represents individuals who became infected but are able to recover. If the model contains only these two compartments, a given population is initially divided into them. From this basic compartmentalization, there are numerous ways for introducing different interactions within the population. Most of these models for the disease evolution make two basic assumptions. The first assumption states that the population is well-mixed. In such a population, each individual has the same probability of encountering other infected individuals, and thus the resulting force of infection is equal for all. The second assumption states that there are a priori constraints upon the biological process, whilst gradual but random mutation of disease traits (such as transmission rate and infectious period) could occur. More refined epidemic models are required; the entire population cannot just be divided into two or more compartments/groups which are defined by a single quantity. In this paper, we consider the effect of the heterogeneity in the weighting of contacts between two individuals. Moreover, we focus on a meta-population model where the population is previously subdivided into subpopulations that can consist in spatially distinct groups of individuals (neighborhoods, towns, cities, etc.) or groups of individuals with different features. The resulting model is described by a dynamic system defined on a network (graph). We have also added the possibility of varying the weight of the connections between groups in order to formalize the problem of controlling the spread of the epidemic on the network. This generality could also allow the changing of disease features.

## 2. A Meta-Population Model on a Network

#### 2.1. Spatial Heterogeneity in Epidemiological Models

#### 2.2. A Prototype: SEIR Model on a Direct Graph

**Remark**

**1.**

- B has a positive real eigenvalue equal to its spectral radius $\rho \left(B\right)$;
- There exists an eigenvector $\mathit{v}\gg 0$ corresponding to $\rho \left(B\right)$;
- $\rho \left(B\right)$ increases when any entry of B increases;
- $\rho \left(B\right)$ is a simple eigenvalue of B;
- Collatz–Wielandt formula: $\rho \left(B\right)={min}_{\mathit{x}>\mathbf{0}}{max}_{i:{x}_{i}>0}\frac{{\left[{\mathit{x}}^{\top}B\right]}_{i}}{{x}_{i}}={max}_{\mathit{x}>\mathbf{0}}{min}_{i:{x}_{i}>0}\frac{{\left[{\mathit{x}}^{\top}B\right]}_{i}}{{x}_{i}}$ that are reached identically on every component of the eigenvector: $\rho \left(B\right)=\frac{{\left[{\mathit{v}}^{\top}B\right]}_{i}}{{v}_{i}}$, for any $i=1,\dots ,n$;
- There is no other, unless rescaled, non-negative eigenvector of B, different from $v$.

- $\mathit{\u0131}\left(t\right)>0$, and $\mathit{s}\left(t\right)\gg 0$ for all $t>0$;
- $\forall {t}_{1}>{t}_{2}>0$, $\mathit{s}\left({t}_{2}\right)\gg \mathit{s}\left({t}_{1}\right)$;
- ${\lambda}_{\mathit{s}}\left(t\right)$ is monotone decreasing.

- The threshold phenomenon that states that, under some condition, an epidemic propagates, in the sense that the introduction of a percentage of infected in the population triggers the contamination of many other individuals, otherwise the epidemic fades off.
- The asymptotic profiles of the steady states in order to understand if an endemic level can be reach.

**Theorem**

**1**

**Proof.**

**Theorem**

**2**

- (1)
- If for a time $\tau \ge 0$, $\beta {\lambda}_{\mathit{s}}\left(\tau \right)<\gamma $ then ${q}_{\tau}^{\left(\u03f5\right)}\left(t\right)={\mathit{v}}_{\mathit{s}}{\left(\tau \right)}^{\top}(\mathit{e}\left(t\right)(1+\u03f5)+\mathit{\u0131}\left(t\right))$ decreases exponentially to zero for $t\ge \tau $ and any $\u03f5\in \left(0,\frac{\gamma}{\beta {\lambda}_{\mathit{s}}\left(\tau \right)}-1\right)$.
- (2)
- If $\beta {\lambda}_{\mathit{s}}\left(0\right)>\gamma $, $\mathit{\u0131}\left(0\right)>\mathbf{0}$ and $\mathit{s}\left(0\right)\gg \mathbf{0}$, then $\exists {t}^{*}>0$ such that ${q}_{0}\left(t\right)={\mathit{v}}_{\mathit{s}}{\left(0\right)}^{\top}\phantom{\rule{0.166667em}{0ex}}(\mathit{e}\left(t\right)+\mathit{\u0131}\left(t\right))$ increases for $t\in (0,{t}^{*})$. Moreover, $\mathit{e}\left(t\right)\to \mathbf{0}$ and $\mathit{\u0131}\left(t\right)\to \mathbf{0}$.

**Proof.**

#### 2.3. A Control Problem

**Remark**

**2.**

## 3. Numerical Tests

- (BP)
- biological parameters related to the different epidemiological features of the disease (parameters $\gamma ,\phantom{\rule{0.166667em}{0ex}}\beta ,\phantom{\rule{0.166667em}{0ex}}\mu $ in (4));
- (MP)
- mobility data for the probability outgoing matrix ${P}^{o}$ and the probability incoming matrix ${P}^{i}$.

- (A)
- The parameters of the disease are $\gamma =0.14,\phantom{\rule{0.166667em}{0ex}}\beta =0.74,\phantom{\rule{0.166667em}{0ex}}\mu =0.5$.
- (B)
- The parameters of the disease are $\gamma =0.22,\phantom{\rule{0.166667em}{0ex}}\beta =1.0,\phantom{\rule{0.166667em}{0ex}}\mu =0.03$.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**,

**b**): Lockdown strategies for Scenario A and B, respectively. We show the optimal strategy $\mathit{V}$ that minimizes (7) the closer is to 0, the more severe the restrictions are. On the other hand, values close to 1 denote very mild restrictions on mobility.

**Figure 2.**(

**a**–

**d**): Evolution of the strategy and infected people in Scenario B for timestamps $t=0,20,40,179$, respectively. The color of the nodes represents the number of infected people in that node, while the color of the edges represents the percentage of people blocked.

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**MDPI and ACS Style**

Aletti, G.; Benfenati, A.; Naldi, G.
Graph, Spectra, Control and Epidemics: An Example with a SEIR Model. *Mathematics* **2021**, *9*, 2987.
https://doi.org/10.3390/math9222987

**AMA Style**

Aletti G, Benfenati A, Naldi G.
Graph, Spectra, Control and Epidemics: An Example with a SEIR Model. *Mathematics*. 2021; 9(22):2987.
https://doi.org/10.3390/math9222987

**Chicago/Turabian Style**

Aletti, Giacomo, Alessandro Benfenati, and Giovanni Naldi.
2021. "Graph, Spectra, Control and Epidemics: An Example with a SEIR Model" *Mathematics* 9, no. 22: 2987.
https://doi.org/10.3390/math9222987