Identification and Control of Game-Based Epidemic Models
<p>Schematic representation of the proposed Infection Game model. Two feedback mechanisms are at work, coupling the infection model and the RE: (1) The number of known infected people <span class="html-italic">I</span> and of deaths <span class="html-italic">D</span> influences the cooperation dynamics by changing the game through the term <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>I</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo> </mrow> </semantics></math>; (2) the cooperation <span class="html-italic">x</span> influences the infection rate <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 2
<p>Graphical representation of the statements of Propositions 1–3. The insets display the dynamics of <span class="html-italic">D</span> in each of the three regions. Inset (<b>a</b>): <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.4</mn> <mo>·</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> </mrow> </semantics></math>. Inset (<b>b</b>): <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math>. Inset (<b>c</b>): <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>0.24</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math>. For all the simulations, the other parameters have been set as follows: <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.07</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.002</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>3</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>σ</mi> <mi>C</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>σ</mi> <mi>N</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mo>−</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>60</mn> <mo>·</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>U</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Time course of <math display="inline"><semantics> <mover accent="true"> <mi>I</mi> <mo>^</mo> </mover> </semantics></math>, <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo>^</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mover accent="true"> <mi>D</mi> <mo>^</mo> </mover> </semantics></math> data (yellow dots), and of the corresponding numerical solutions <span class="html-italic">I</span>, <span class="html-italic">R</span> and <span class="html-italic">D</span>, obtained with IM (red lines) and IGM (blue lines).</p> "> Figure 4
<p>IGM simulation of <span class="html-italic">U</span> (blue line, left ordinate axis) and its percentage with respect to the total number of infected individuals <math display="inline"><semantics> <mrow> <mi>U</mi> <mo>+</mo> <mi>I</mi> </mrow> </semantics></math> (violet line, right ordinate axis).</p> "> Figure 5
<p>Estimation results of the time-varying parameters <math display="inline"><semantics> <mi>α</mi> </semantics></math> (subplot (<b>a</b>)), <math display="inline"><semantics> <mi>ω</mi> </semantics></math> (subplot (<b>b</b>)) and <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> (subplot (<b>c</b>)). The piecewise shape of the curves is due to the fact that these parameters are constant over each time window of <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>w</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> days.</p> "> Figure 6
<p>Subplot (<b>a</b>): time course of the estimated threshold parameter <span class="html-italic">a</span> (blue line, left ordinate axis) and of the “game-switching term” <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>I</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo> </mrow> </semantics></math> (violet line, right ordinate axis). The dashed black lines represents the 0 of the right coordinate axis. For values of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>I</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo> </mrow> </semantics></math> bigger than 0, the population plays a Harmony game, while for values lower that 0, a Prisoner’s Dilemma game is used. Subplot (<b>b</b>): time curse of the estimated efficacy parameter <span class="html-italic">e</span> (blue line), the simulated cooperation <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (violet line) and of the product the product <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> </mrow> </semantics></math> (green line). Subplot (<b>c</b>): time course of <math display="inline"><semantics> <mi>β</mi> </semantics></math> for the two models (red line for IM, blue line for IGM).</p> "> Figure 7
<p>Evolution of the estimated time varying reproduction number <math display="inline"><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, from real data (yellow circles) and IGM <math display="inline"><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (blue line). The <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> and <math display="inline"><semantics> <munder> <mi mathvariant="script">R</mi> <mo>̲</mo> </munder> </semantics></math> for IGM are reported in violet and green, respectively.</p> "> Figure 8
<p>Percentage variation between two successive months of the maximum values of infected (yellow) and dead (violet) individuals versus <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>. The labels indicate the reference months with respect to which the percentage variation is calculated, for example, 1 refers to the percentage variation between March and April 2020 and 17 to the percentage variation between August and September 2021. Regarding the variable <span class="html-italic">D</span>, since the model considers the total amount of dead people in each time instant, we have differentiated the variable in order to have the effective daily number of deaths. The clusters highlight crucial situations in the overall period of the pandemic indicated in the corresponding text.</p> "> Figure 9
<p>Two different scenarios obtained by varying the control parameter <span class="html-italic">e</span> at the beginning of the second wave (subplots (<b>a</b>,<b>b</b>)), reporting infected and dead people, respectively) and at the beginning of the fourth (subplots (<b>c</b>,<b>d</b>)), depicting infected and dead people, respectively). Yellow dots indicate real data, red curve is the simulation of the IGM using the estimated parameters, and the light blue area corresponds to the range of variations of the dynamics for different values of <span class="html-italic">e</span>.</p> "> Figure 10
<p>Scatter plots of the number of daily swabs (subplots (<b>a</b>–<b>g</b>)), 1st vaccine (subplots (<b>h</b>–<b>n</b>)) and 2nd vaccine (subplots (<b>o</b>–<b>u</b>)) dose administrations (<span class="html-italic">y</span>-axis) versus <math display="inline"><semantics> <mi>α</mi> </semantics></math>, <math display="inline"><semantics> <mi>ω</mi> </semantics></math>, <math display="inline"><semantics> <mi>ζ</mi> </semantics></math>, <math display="inline"><semantics> <mi>β</mi> </semantics></math>, <span class="html-italic">x</span>, <span class="html-italic">e</span> and <span class="html-italic">a</span> (<span class="html-italic">x</span>-axis). The colors indicate the time flow (blue for the first days, yellow for the last). The corresponding regression lines are reported (dashed black), as well as the correlation coefficients <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. Most significant correlations (<math display="inline"><semantics> <mrow> <mo>|</mo> <mi>ρ</mi> <mo>|</mo> <mo>></mo> <mn>0.5</mn> </mrow> </semantics></math>) are highlighted in red.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
- S: Susceptible individuals;
- U: Undetected infected individuals;
- I: Detected infected individuals (hospitalized or quarantined);
- R: Recovered individuals;
- D: Dead individuals,
2.1. How Epidemics Influence People’s Behavior
2.2. How People Behavior Influences the Epidemic
2.3. The Infection Game Model
2.4. Available Data
2.5. Parameters Estimation
3. Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Range | Param. i.c. | IM est. | IGM est. | Meaning and Definition | |
---|---|---|---|---|---|
t.v. | Natural infection rate. Ref. to Equations (9) and (18) for the relationship between and in IGM and IM, respectively. | ||||
n.a. | Net game payoff for cooperation. Ref. to Equation (5) for in IGM. | ||||
n.a. | Net game payoff for defection. Ref. to Equation (6) for in IGM. | ||||
Susceptibles initial condition. | |||||
n.a. | Cooperation initial condition. | ||||
e | n.a. | t.v. | Weight of cooperation in . | ||
a | n.a. | t.v. | Sensitivity to disease strength. | ||
t.v. | t.v. | Reinfection rate. | |||
t.v. | t.v. | Recovery rate for detected. | |||
t.v. | t.v. | Death rate. | |||
Recovery rate for undetected. | |||||
Detection rate. |
IM | IGM | |
---|---|---|
2.4% | 2.3% | |
7.0% | 7.1% | |
0.2% | 0.2% |
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Madeo, D.; Mocenni, C. Identification and Control of Game-Based Epidemic Models. Games 2022, 13, 10. https://doi.org/10.3390/g13010010
Madeo D, Mocenni C. Identification and Control of Game-Based Epidemic Models. Games. 2022; 13(1):10. https://doi.org/10.3390/g13010010
Chicago/Turabian StyleMadeo, Dario, and Chiara Mocenni. 2022. "Identification and Control of Game-Based Epidemic Models" Games 13, no. 1: 10. https://doi.org/10.3390/g13010010
APA StyleMadeo, D., & Mocenni, C. (2022). Identification and Control of Game-Based Epidemic Models. Games, 13(1), 10. https://doi.org/10.3390/g13010010