## Hao Gao , Wuchen Li , Miao Pan , Zhu Han and H. Vincent Poor## |

bias-I | bias-R | [TeX:] $$R^{2} \text {-I }$$ | [TeX:] $$R^{2}-\mathrm{R}$$ | |
---|---|---|---|---|

MFEDs | 0.0953 | 0.0669 | 0.9969 | 0.9939 |

RDs | 0.9532 | 0.1120 | 0.9957 | 0.9893 |

We analyze the effect of one-time social distancing (SD) in the absence of seasonality in this subsection. Social distancing measures have been taken at the beginning of the transmission process of the COVID-19. From Figs. 6(a) to 6(e), the duration of social distancing varies from one month to an indefinite period. The length of the period is denoted in the blue shaded region. The reduction of infection rate varies from 0 to 80% representing the distinct effectiveness of SD. The total population size is assumed to be N = 50; 000. The basic infection rate is [TeX:] $$\eta=0.00004$$ without considering any seasonality, i.e., [TeX:] $$\eta$$ is held as a constant. With [TeX:] $$\eta=0.00004,$$ the effective reproduction number is [TeX:] $$R_{0}=N * \eta=2.$$

As shown in Fig. 6, depending on the strength of the impact of the infection rate, the peak number of infections varies from 10.000 to 27.500. To restrain the spreading of COVID-19, we need the peak number of infections to be less than the total capacity of patients in the hospital. These significant reductions in the peak number, due to the SD, are thus of vital importance. For a slight degree of SD, which

leads to a 40% reduction in the infection rate, there will be no significant increase in the number of infections when the quarantine period ends. For a severe degree of SD, which leads to a more than 60% reduction in the infection rates, there will be a significant increase in the number of infections when the quarantine period ends. The critical reason behind this phenomenon is that only when the number of immune individuals reaches a certain level can the spreading of the COVID-19 be stopped. Therefore, after a severe degree of SD, returning to the normal SD step by step and developing a vaccine should be of vital importance.

We analyze the seasonality behavior of COVID-19 in the post-pandemic period in this subsection. Specifically, we consider the effect of the duration of COVID-19 immunity [TeX:] $$d_{0},$$ the proportion of immunity loss [TeX:] $$L_{0},$$ and the seasonal variation [TeX:] $$v_{0}.$$ The duration of COVID-19 immunity [TeX:] $$d_{0}$$ refers to the length of the period when the recovered individuals can keep their immunity to COVID-19. After the immunity duration [TeX:] $$d_{0},$$ a certain proportion [TeX:] $$L_{0}$$ of the recovered population will lose their immunity. The seasonal variation [TeX:] $$v_{0}$$ refers to the variation of the infection rate of COVID-19 from wintertime to summertime.

In Fig. 7(a), periodic outbreaks of COVID-19 are depicted for [TeX:] $$d_{0}=6$$ months and [TeX:] $$d_{0}=12$$ months. The immunity loss is [TeX:] $$L_{0}=0.8, \text { i. e., } 80 \%$$ of the recovered population will lose their immunity after the duration [TeX:] $$d_{0}.$$ There is no seasonal variation [TeX:] $$\left(v_{0}=1\right).$$ The peak number of infections will gradually decrease through the outbreaks. The period of the outbreaks of COVID-19 is closely related to the immunity duration.

In Fig. 7(b), we show the impact of the proportion of immunity loss on the seasonality of COVID-19. The immunity duration is set as [TeX:] $$d_{0}=6$$ months and there is no seasonal variation [TeX:] $$\left(v_{0}=1\right).$$ The immunity loss [TeX:] $$L_{0}$$ varies from 0.4 to 0.8. Higher immunity loss will yield more severe outbreaks of COVID-19 every six months. However, the peak number of infections will decrease gradually.

In Fig. 7(c), the effect of seasonal variation is shown. The immunity loss is [TeX:] $$L_{0}=0.8$$ and the duration is [TeX:] $$d_{0}=6$$ months. Without loss of generality, we assume that the infection rate is higher in the winter. (One can also set a higher infection rate in the winter. In that case, the roles of summer and winter will switch but the results will be similar.)

In particular, the infection rate is [TeX:] $$\eta$$ during the wintertime and [TeX:] $$v_{0} \eta$$ during the summertime, where [TeX:] $$v_{0}$$ could be 0:2 or 0.8. The winter of 2019 is regarded as the starting point. A higher seasonal variation would significantly reduce the peak number of infections in the summertime but yield a more severe outbreak of COVID-19 in the following winter. The overall trend of the peak number of infections is also decreasing.

Mean field evolutionary dynamics, inspired by optimal transport theory and mean field games on graphs, have been proposed to model the evolution of COVID-19. This approach has been compared with the commonly used replicator dynamics with numerical simulation results. Applied into the new SIDR model, the mean field evolutionary dynamics have been seen to outperform the replicator dynamics in fitting the COVID-19 statistics of Wuhan, China. In our simulations, we have observed that one-time social distancing can reduce

the peak number of infections significantly while a second outbreak of COVID-19 can arise after a high level of social distancing. Finally, as per our model, people’s limited length of immunity will yield a periodic outbreak of COVID-19, and higher seasonal variation of infection rates will reduce the peak number of infections in summer but lead to a more severe outbreak in the following winter.

In this appendix, we derive the MFEDs in Theorem 1. First, we construct the Riemannian manifold on which the MFEDs are defined. Then we prove that the dynamics shown in Theorem 1 are well-defined evolutionary dynamics on the Riemannian manifold.

To measure distance in the state space [TeX:] $$\mathcal{P}(\mathcal{S}),$$ we need to define the following Wasserstein metric [29].

**Definition 1.** Given two discrete probability functions [TeX:] $$\rho^{0}, \rho^{1} \in \mathcal{P}_{o}(\mathcal{S}),$$ the Wasserstein metric W is defined by

[TeX:] $$\begin{aligned} W\left(\rho^{0}, \rho^{1}\right)^{2}= \inf \left\{\int_{0}^{1}(\nabla \Phi(t), \nabla \Phi(t))_{\rho(t)} d t:\right.\\ \left.\frac{d \rho}{d t}+\operatorname{div}(\rho \nabla \Phi)=0, \rho(0)=\rho^{0}, \rho(1)=\rho^{1}\right\} \end{aligned}.$$

where [TeX:] $$\nabla \Phi: \mathcal{S} \times \mathcal{S} \rightarrow \mathbb{R}$$ is given by

[TeX:] $$\nabla \Phi=\left\{\begin{array}{lr} \sqrt{\omega_{i j}}\left(\Phi_{i}-\Phi_{j}\right), \text { if }(i, j) \in \mathcal{E}, \\ 0, \text { otherwise }, \end{array}\right.$$

where [TeX:] $$\Phi$$ is a function and [TeX:] $$\Phi: \mathcal{S} \rightarrow \mathbb{R} \text { and } \omega_{i j}$$ is the weight on edge (i,j).

Besides the Wasserstein metric, we also need the following inner product [TeX:] $$g^{W}$$ to construct the Remannian manifold [TeX:] $$\left(\mathcal{P}_{o}(\mathcal{S}), g^{\tilde{W}}\right).$$

**Definition 2.** For any two tangent vectors [TeX:] $$\sigma^{1}, \sigma^{2} \in T_{\rho} \mathcal{P}_{o}(\mathcal{S}),$$ define the inner product [TeX:] $$g^{W}: T_{\rho} \mathcal{P}_{o}(\mathcal{S}) \times T_{\rho} \mathcal{P}_{o}(\mathcal{S}) \rightarrow \mathbb{R}$$ by

where [TeX:] $$\sigma^{i}=-\operatorname{div}\left(\rho \nabla \Phi^{i}\right) \text { for } i=1,2 . T_{\rho} \mathcal{P}_{o}(\mathcal{S})=\left\{\left(\sigma_{i}\right)_{i=1}^{n} \in\right. \left.\mathbb{R}^{n}: \sum_{i=1}^{n} \sigma_{i}=0\right\}$$ is the tangent space at a point [TeX:] $$\rho \in \mathcal{P}_{o}(\mathcal{S}).$$ [TeX:] $$\theta_{i j}$$ is the discrete probability on edge (i,j) which is defined by

where [TeX:] $$F_{i}: \mathcal{P}(\mathcal{S}) \rightarrow \mathbb{R}$$ is the payoff function and [TeX:] $$d_{i}$$ is the degree of node i (i.e. the total number of nodes in [TeX:] $$N(i)).$$

With the state space in (2) and the inner product in Definition 2, we can construct the Riemannian manifold [TeX:] $$\left(\mathcal{P}_{o}(\mathcal{S}), g^{W}\right)$$ [30], [27]. In this regard, we can give the proof of Theorem 1 as follows:

Proof. Given the tangent space [TeX:] $$T_{\rho} \mathcal{P}_{o}(\mathcal{S})=\left\{\left(\sigma_{i}\right)_{i=1}^{n} \in \mathbb{R}^{n}:\right. \left.\sum_{i=1}^{n} \sigma_{i}=0\right\},$$ there exists [TeX:] $$\Phi$$ such that [TeX:] $$\sigma=-\operatorname{div}(\rho \nabla \Phi)$$ for any [TeX:] $$\sigma \in T_{\rho} \mathcal{P}_{o}(\mathcal{S}) . \mathrm{As} \frac{d \rho}{d t}=\left(\frac{d \rho_{i}}{d t}\right)_{i}^{n}$$ is in [TeX:] $$T_{\rho} \mathcal{P}_{o}(\mathcal{S}),$$ we have

The noisy potential is given by

which is the summation of the potential and the Shannon- Bolztman entropy. Then we have

With (17) and (19), and the definition of gradient flow of [TeX:] $$-\bar{F}(\rho)$$ on the Remannian manifold [TeX:] $$\left(\mathcal{P}_{o}(\mathcal{S}), g^{W}\right),$$ we derive

[TeX:] $$\begin{aligned} 0 =g^{W}\left(\frac{d \rho}{d t}, \sigma\right)-d \bar{F}(\rho) \cdot \sigma \\ =\sum_{i=1}^{n} \frac{d \rho_{i}}{d t}+\operatorname{div}(\rho \nabla F(\rho))_{i} \Phi_{i}. \end{aligned}$$

As the above is true for all [TeX:] $$\left(\Phi_{i}\right)_{i=1}^{n} \in \mathbb{R}^{n},$$ we finally obtain

[TeX:] $$\frac{d \rho_{i}}{d t}+\sum_{j \in N(i)} \omega_{i j} \theta_{i j}(\rho)\left(\bar{F}_{j}(\rho)-\bar{F}_{i}(\rho)\right)=0.$$

Replacing [TeX:] $$\theta_{i j}$$ with (16), the mean field evolutionary dynamics in Theorem 1 are proved.

Hao Gao (S’19) received his B.E. degree in Electrical and Information Engineering from Huazhong University of Science and Technology, Wuhan, China, in 2018. He started pursuing his Ph.D. degree in Electrical Engineering in University of Houston, USA, since 2018. His current research interests include mean field game and related applications in wireless communication.

Wuchen Li received his BSc in Mathematics from Shandong university in 2009. He obtained M.S. degree in Statistics, and Ph.D. degree in Mathematics from Georgia institute of Technology in 2016. He was a CAM Assistant Adjunct Professor in the Department of Mathematics at University of California, Los Angeles from 2016 to 2020. Now, he is an Assistant Professor at University of South Carolina. His research interests include optimal transport, information geometry, mean field games with applications in data science.

Miao Pan (S’07-M’12-SM’18) received his BSc degree in Electrical Engineering from Dalian University of Technology, China, in 2004, MASc degree in Electrical and Computer Engineering from Beijing University of Posts and Telecommunications, China, in 2007 and Ph.D. degree in Electrical and Computer Engineering from the University of Florida in 2012, respectively. He is now an Associate Professor in the Department of Electrical and Computer Engineering at University of Houston. He was a recipient of NSF CAREER Award in 2014. His research interests include Wireless/AI for AI/Wireless, deep learning privacy, cybersecurity, underwater communications and networking, and cyber-physical systems. His work won IEEE TCGCC (Technical Committee on Green Communications and Computing) Best Conference Paper Awards 2019, and Best Paper Awards in ICC 2019, VTC 2018, Globecom 2017 and Globecom 2015, respectively. Dr. Pan is an Editor for IEEE Open Journal of Vehicular Technology and an Associate Editor for IEEE Internet of Things (IoT) Journal (Area 5: Artificial Intelligence for IoT), and used to be an Associate Editor for IEEE Internet of Things (IoT) Journal (Area 4: Services, Applications, and Other Topics for IoT) from 2015 to 2018. He has also been serving as a Technical Organizing Committee for several conferences such as TPC Co-Chair for Mobiquitous 2019, ACM WUWNet 2019. He is a member of AAAI, a member of ACM, and a senior member of IEEE.

Zhu Han (S’01-M’04-SM’09-F’14) received the B.S. degree in Electronic Engineering from Tsinghua University, in 1997, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of Maryland, College Park, in 1999 and 2003, respectively. From 2000 to 2002, he was an RD Engineer of JDSU, Germantown, Maryland. From 2003 to 2006, he was a Research Associate at the University of Maryland. From 2006 to 2008, he was an Assistant Professor at Boise State University, Idaho. Currently, he is a John and Rebecca Moores Professor in the Electrical and Computer Engineering Department as well as in the Computer Science Department at the University of Houston, Texas. His research interests include wireless resource allocation and management, wireless communications and networking, game theory, big data analysis, security, and smart grid. Dr. Han received an NSF Career Award in 2010, the Fred W. Ellersick Prize of the IEEE Communication Society in 2011, the EURASIP Best Paper Award for the Journal on Advances in Signal Processing in 2015, IEEE Leonard G. Abraham Prize in the field of Communications Systems (best paper award in IEEE JSAC) in 2016, and several best paper awards in IEEE conferences. Dr. Han was an IEEE Communications Society Distinguished Lecturer from 2015-2018, AAAS fellow since 2019 and ACM distinguished Member since 2019. Dr. Han is 1% highly cited researcher since 2017 according to Web of Science. Dr. Han is also the winner of 2021 IEEE Kiyo Tomiyasu Award, for outstanding early to mid-career contributions to technologies holding the promise of innovative applications, with the following citation: "For contributions to game theory and distributed management of autonomous communication networks."

H. Vincent Poor received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is currently the Michael Henry Strater University Professor. During 2006 to 2016, he served as the dean of Princeton’s School of Engineering and Applied Science. He has also held visiting appointments at several other universities, including most recently at Berkeley and Cambridge. His research interests are in the areas of information theory, machine learning and network science, and their applications in wireless networks, energy systems and related fields. Among his publications in these areas is the forthcoming book Machine Learning and Wireless Communications (Cambridge University Press). Dr. Poor is a Member of the U.S. National Academy of Engineering and the U.S. National Academy of Sciences, an Honorary Member of the National Academy of Sciences, Republic of Korea, and a Foreign Member of the National Academy of Engineering of Korea. He received the IEEE Alexander Graham Bell Medal in 2017.

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