# Parrondo’s paradox in susceptible-infectious-susceptible dynamics over
periodic temporal networks

Maisha Islam Sejunti  Dane Taylor<EMAIL_ADDRESS>Naoki Masuda
<EMAIL_ADDRESS>Department of Mathematics, State University of New York
at Buffalo, NY, 14260-2900, USA  School of Computing and Department of
Mathematics and Statistics, University of Wyoming, Laramie, WY, 82071-3036,USA
Institute for Artificial Intelligence and Data Science, State University of
NewYork at Buffalo, NY, 14260-5030, USA  Center for Computational Social
Science, Kobe University, Kobe 657-8501, Japan

###### Abstract

Many social and biological networks periodically change over time with daily,
weekly, and other cycles. Thus motivated, we formulate and analyze
susceptible-infectious-susceptible (SIS) epidemic models over temporal
networks with periodic schedules. More specifically, we assume that the
temporal network consists of a cycle of alternately used static networks, each
with a given duration. We show Parrondo’s paradox behavior in this model, with
which the periodic alternation of two static networks that are supra-threshold
(i.e., above the epidemic threshold of the SIS model) yields a sub-threshold
dynamics (i.e., with the number of infectious nodes exponentially decaying
over time) in many cases. We find community structure to play an important
role in shaping this phenomenon, and we study its dependence on the
connectivity between and number of communities. Our findings associate such
paradoxical behavior with anti-phase oscillatory behavior of the number of
infectious nodes in different communities of the network.

###### keywords:

temporal network, SIS model, epidemic threshold, Floquet theory, anti-phase
oscillation

[table]capposition=top capbtabboxtable[][]

##  1 Introduction

The connectivity of social contact networks is a main determinant for how
contagions, i.e., spreading of communicative diseases, opinions, rumor, ideas,
and so forth, occur in a population. Epidemic modeling on networks has been a
useful tool that can help us understand, predict, and intervene into
contagious processes occurring in the physical world and online [1, 2, 3, 4,
5, 6]. In fact, the structure of social and transportation networks can also
dynamically change on the timescale of disease spreading. For example, human
and animal mobility is a main obvious reason why contact networks vary over
time. Airport transportation networks, which consist of airports as nodes and
direct commercial flights between two airports as edges, also alter over days,
weeks, and longer time [7, 8]. Time-varying networks are collectively called
temporal networks, and epidemic processes on temporal networks have been
extensively studied [9, 3, 10, 11, 12, 13, 14, 15, 16, 17]. Ignoring the time
change of a network may lead to wrong or inaccurate conclusions about dynamics
on networks. For example, the number of eventually infected nodes tends to be
systematically different between simulations on a given temporal network and
those on a counterpart static network that ignores temporal nature of the
original network, even if the infection and recovery rates are assumed to be
the same between the two sets of simulations (e.g., [12, 17]). Therefore,
simulations on the static network would over-or under-estimate epidemic spread
occurring on the given temporal network.

One strategy for mathematically and computationally modeling temporal networks
and dynamical processes on them is to use periodic temporal networks. Periodic
temporal networks are useful because they allow us to derive and analyze a
mapping of the given system’s state from one time point to one period after,
reminiscent of the Poincaré map. A useful subclass of periodic temporal
networks is periodic switching temporal networks, which are defined as time-
varying networks in which one static network is used for a certain duration
and then the network switches to another network, which is used for a certain
duration, and so forth, and we come back to the first static network after the
last one to complete one period, and then repeat the same cycle. It is common
to use each static network for the same duration for simplicity. Periodic
temporal networks in discrete time are by definition periodic switching
temporal networks because each one static network is present in each time
step, which generally changes over discrete times. Popularity of using
periodic switching temporal networks is presumably due to their analytical
tractability. Periodically switching networks have been applied to modeling,
e.g., contagion processes [18, 19, 20, 21, 22, 23, 24] evolutionary dynamics
[25, 26, 27, 28, 29, 30, 31, 32, 33], gene regulation [34, 35], network
control [36, 37, 38, 39], and random walks [40, 41, 42].

A phenomenon that can occur in periodically switching dynamical systems or
stochastic processes is the Parrondo’s paradox. Originally, the Parrondo’s
paradox refers to the situation in which a combination of two losing
strategies generates a winning strategy [43, 44, 45]. When a person
alternately and repetitively plays two different losing gambling games (i.e.,
losing regardless of the player’s strategy), the player may still win the
game, even on average. The Parrondo’s paradox has also been found in
theoretically motivated dynamical systems [46, 47, 48, 49, 50, 51, 52], social
dynamics models [53, 54, 55, 56], evolutionary dynamics on networks [57],
financial investment models [58, 59], chemical engineering [60], molecular
transport [61], dynamics of allele frequency [62, 63, 64], cancer biology
[65], cellular biology [66], and so on. Models of epidemic dynamics are no
exception [67, 68, 69, 70, 71]. For example, Cheong et al. considered a model
of COVID-19 outbreak [69]. In their model, when the community is not under
lockdown, the infection rate is higher and the hospital cost is increased for
individual persons. In contrast, when a lockdown is implemented, the total
economic costs are increased despite the rate of epidemic spreading being
reduced. They argue that implementing either one of these two strategies is a
losing strategy. However, switching between the two strategies can reduce the
overall cost per day; Cheong et al. interpreted this to be a winning strategy,
thereby representing a Parrondo’s paradox [69]. As another example, in [71],
the authors implemented the Parrondo’s paradox idea on a periodic parameter
switching algorithm with the aim of controlling the COVID-19 outbreak in their
model.

In the present article, we study the Parrondo’s paradox occurring on the
susceptible-infectious-susceptible (SIS) dynamics over periodically switching
temporal networks. We first find that the Parrondo’x paradox often occurs for
small networks (i.e., those with a few nodes) modeling interaction between a
small number of subpopulations of individuals. We corroborate our results with
analysis of larger networks having the same structure of interaction between
subpopulations. We characterize the observed paradoxical dynamics by
quantifying anti-phase oscillation between the time courses of the fraction of
infectious individuals in different subpopulations. We finally develop a
perturbation theory, which also assists in explaining the reason for the
paradoxical dynamics. Code used in this paper is available on GitHub [72].

##  2 Preliminaries

We first review the SIS epidemic model over static networks, definition of
temporal networks, and the SIS model over periodic temporal networks.

###  2.1 SIS model over static networks

In this section, we present the SIS model over static networks and its
epidemic threshold. We consider a static weighted network on $N$ nodes,
$G(\mathcal{V},\mathcal{E})$, that may have self-loops;
$\mathcal{V}=\\{v_{1},\ldots,v_{N}\\}$ is the set of nodes, and
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is the set of edges. We
denote the weighted adjacency matrix by $\textbf{A}=(a_{ij})$. By assumption,
each node takes one of the two states, i.e., susceptible (denoted by S) or
infectious (denoted by I), at any given time $t\in\mathbb{R}$. If node $v_{i}$
is susceptible and its neighbor $v_{j}$ is infectious, then $v_{j}$ infects
$v_{i}$ at a constant infection rate, denoted by $\beta$ ($>0$). If $v_{i}$ is
adjacent to multiple infectious nodes, each infectious neighbor infects
$v_{i}$ independently of each other. In addition, any infectious node recovers
to the susceptible state at a constant recovery rate, denoted by $\mu$ ($>0$).
Therefore, the SIS model is a Markov process in continuous time with $2^{N}$
states.

The binary random variable, $X_{i}(t)$, encodes the state of $v_{i}$, i.e.,
$X_{i}(t)=0$ and $X_{i}(t)=1$ if $v_{i}$ is susceptible or infectious,
respectively. Let $x_{i}(t)$ be the probability that the node $v_{i}$ is
infectious at time $t$. Because there is no closed system of ODEs that
describes the evolution of $x_{i}(t)$ (with $i\in\\{1,\ldots,N\\}$), a system
of ODEs approximating the evolution of $x_{i}(t)$, called the individual-
based-approximation (IBA), has been studied [5]. The IBA assumes independence
of $\\{x_{1}(t),\ldots,x_{N}(t)\\}$. Under the IBA, the SIS dynamics is given
by [73, 74, 75]

$\frac{\text{d}x_{i}(t)}{\text{d}t}=\beta\left[1-x_{i}(t)\right]\sum_{j=1}^{N}a_{ij}x_{j}(t)-\mu
x_{i}(t),\quad i\in\\{1,\ldots,N\\}.$ (1)

Equation (1) can be expressed in the succinct form as

$\frac{\text{d}\textbf{x}(t)}{\text{d}t}=\left[\beta\textbf{A}-\beta\text{diag}(x_{i}(t))\textbf{A}-\mu\textbf{I}\right]\textbf{x}(t),$
(2)

where $\textbf{x}(t)=[x_{1}(t),\dots,x_{N}(t)]^{\top}$, the transposition is
denoted by ⊤, $\text{diag}(x_{i}(t))$ represents the diagonal matrix with
diagonal entries $x_{1}(t)$, $\ldots$, $x_{N}(t)$, and I is the $N\times N$
identity matrix.

The linearized ODE for Eq. (2) under the assumption that $x_{i}(t)\approx 0$,
$\forall$ $i\in\\{1,\ldots,N\\}$, or that the infection rate is posed such
that epidemic dynamics is near the disease-free equilibrium, is given by

$\frac{\text{d}\textbf{x}(t)}{\text{d}t}=(\beta\textbf{A}-\mu\textbf{I})\textbf{x}(t)\equiv\textbf{M}\textbf{x}(t).$
(3)

Equation (3) leads to

$\textbf{x}(t)=e^{\textbf{M}t}\textbf{x}(0)=e^{(\beta\textbf{A}-\mu\textbf{I})t}\textbf{x}(0).$
(4)

It is known that Eq. (4) provides an upper bound of the fraction of infectious
nodes in the original stochastic SIS dynamics. Specifically, the probability
that the infection persists in the network is upper-bounded by [76, 77, 73,
78]

$Pr[X(t)\neq
0]\leq\sqrt{N||X(0)||_{1}}e^{\lambda_{\max}(\textbf{M})t}=\sqrt{N||X(0)||_{1}}e^{(\beta\lambda_{\max}(\textbf{A})-\mu)t},$
(5)

where $X(t)=(X_{1}(t),\ldots,X_{N}(t))$,
$\left\|X(0)\right\|_{1}=\sum_{i=1}^{N}X_{i}(0)$ represents the number of
initially infectious nodes, and $\lambda_{\max}$ represents the largest
eigenvalue of a matrix in modulus. Note that the Perron-Frobenius theorem
guarantees that $\lambda_{\max}(\textbf{A})$ is real and positive, since the
entries in A are nonnegative and at least one entry is positive.

Under the IBA, the epidemic grows if and only if [73, 78]

$\lambda_{\max}(\textbf{M})>0.$ (6)

Since $\lambda_{\max}(\textbf{M})=\beta\lambda_{\max}(\textbf{A})-\mu$, Eq.
(6) is equivalent to $\beta\lambda_{\max}(\textbf{A})>\mu$, or
$\frac{\beta}{\mu}>\frac{1}{\lambda_{\max}(\textbf{A})}.$ Therefore, under the
IBA, the epidemic threshold, i.e., the value of the infection rate above which
the epidemic grows, is given by
$\beta_{\text{c}}=\mu/\lambda_{\max}(\textbf{A})$.

###  2.2 Periodic switching temporal networks

The time-varying $N\times N$ weighted adjacency matrix of the temporal network
is denoted by $\textbf{A}(t)=(a_{ij}(t))$, where $a_{ij}(t)$ represents the
weight of the edge from node $v_{i}$ to $v_{j}$ at time $t\in\mathbb{R}$. When
$\textbf{A}(t+T)=\textbf{A}(t)$ for any $t$, we say that the temporal network
is periodic and that $T$ is the period.

We consider a special case of periodic temporal networks in which
$\textbf{A}(t)$ switches between a finite number of static adjacency matrices
according to a periodic schedule. We call such a periodic temporal network a
periodic switching (temporal) network. The periodic switching network is
defined by a sequence of adjacency matrices
$\\{\textbf{A}^{(1)},\textbf{A}^{(2)},\ldots,\textbf{A}^{(\ell)}\\}$ and the
duration of each $\ell^{\prime}$th static network denoted by
$\tau_{\ell^{\prime}}$ (with $\ell^{\prime}\in\\{1,\ldots,\ell\\}$). The
periodic switching network is defined by

$\textbf{A}(t)=\begin{cases}\textbf{A}^{(1)}&(nT\leq t<nT+\tau_{1}),\\\
\textbf{A}^{(2)}&(nT+\tau_{1}\leq t<nT+\tau_{1}+\tau_{2}),\\\
\;\;\vdots&\quad\quad\quad\vdots\\\
\textbf{A}^{(\ell)}&(nT+\tau_{1}+\cdots+\tau_{\ell-1}\leq
t<(n+1)T),\end{cases}$ (7)

where $n\in\mathbb{Z}$. Note that
$T=\sum_{\ell^{\prime}=1}^{\ell}\tau_{\ell^{\prime}}$.

In Fig. 1, we visualize a periodic switching network with $\ell=2$ that obeys
a weekly cycle (i.e., period $T=7$ days) and contains $N=2$ nodes. In each
cycle, the first adjacency matrix, $\textbf{A}^{(1)}$, is used in the first
five days (i.e., $\tau_{1}=5$ days), corresponding to the weekdays, and then
the second adjacency matrix, $\textbf{A}^{(2)}$, is used in the next two days
(i.e., $\tau_{2}=2$ days), corresponding to the weekend. Each node of this
periodic switching network may represent a group of people, and there are
different disease transmission rates within and between members of two groups,
and at different times (i.e., weekday versus weekend).

Figure 1: A periodically switching network with $N=2$ nodes, a period of $T=7$
days, and two sub-intervals of lengths $\tau_{1}=5$, corresponding to
weekdays, and $\tau_{2}=2$, corresponding to weekends.

###  2.3 SIS model over periodic temporal networks

We now consider the SIS model in continuous time on periodic temporal
networks. First of all, the IBA for temporal networks in continuous time is
given by [24]

$\frac{\text{d}x_{i}(t)}{\text{d}t}=\beta\left[1-x_{i}(t)\right]\sum_{j=1}^{N}a_{ij}(t)x_{j}(t)-\mu
x_{i}(t).$ (8)

Around the disease-free equilibrium, the linearized SIS dynamics (which
assumes small $x_{i}(t)$ values) is given by

$\frac{\text{d}\textbf{x}(t)}{\text{d}t}=\left[\beta\textbf{A}(t)-\mu\textbf{I}\right]\textbf{x}(t)\equiv\textbf{M}(t)\textbf{x}(t).$
(9)

Equation (9) yields

$\textbf{x}(T)=e^{\int_{0}^{T}\textbf{M}(t)\text{d}t}\textbf{x}(0)\equiv\mathcal{T}\textbf{x}(0).$
(10)

To prove that $\lambda_{\max}(\mathcal{T})$ is real and positive, we show that
$\mathcal{T}$ is a positive matrix. To this end, we start with

$\mathcal{T}=e^{\int_{0}^{T}(\beta\textbf{A}(t)-\mu\textbf{I})\text{d}t}=e^{\beta\textbf{E}(T)-\mu
T\textbf{I}},$ (11)

where $\textbf{E}(T)=\int_{0}^{T}\textbf{A}(t)\text{d}t$. Because the identity
matrix commutes with any matrix, $\textbf{E}(T)$ commutes with the identity
matrix. Therefore, we obtain [79]

$\mathcal{T}=e^{\beta\textbf{E}(T)}\times e^{-\mu
T\textbf{I}}=e^{\beta\textbf{E}(T)}\times e^{-\mu
T}\textbf{I}=e^{\beta\textbf{E}(T)}\times e^{-\mu T}.$ (12)

Because $\beta>0$ and all the entries in $\textbf{A}(t)$ are nonnegative, all
the entries of $\textbf{E}(T)$ are nonnegative. Let us assume that
$\textbf{E}(T)$ is an irreducible matrix (i.e., $\left[e_{ij}(T)\right]^{k}>0$
$\forall i,j$ for some integer $k$). This condition is equivalent to assuming
that the time-averaged network is strongly connected. The irreducibility of
$\textbf{E}(T)$ implies that $e^{\beta\textbf{E}(T)}$ is a positive matrix
(i.e., $e^{\beta e_{ij}(T)}>0$ $\forall i,j$) [79, 80]. This observation
combined with Eq. (12) guarantees that $\mathcal{T}$ is a positive matrix.
Therefore, $\lambda_{\max}(\mathcal{T})$ is real and positive owing to the
Perron-Frobenius theorem.

The epidemic threshold is given by the value of $\beta$ at which
$\lambda_{\max}(\mathcal{T})=1$. Eigenvalue $\lambda_{\max}(\mathcal{T})$ is
equivalent to the largest Floquet multiplier in Floquet theory (see Appendix A
for a brief review of a Floquet theory for periodic linear dynamical systems).
We also define $\lambda_{\text{F}}\equiv
T^{-1}\ln\lambda_{\max}(\mathcal{T})$, which gives the average growth rate of
the infection. Under the IBA, the epidemic grows on a periodic temporal
network if and only if

$\lambda_{\text{F}}>0,$ (13)

thereby generalizing Eq. (6) for the temporal network case. Note that
$\lambda_{\text{F}}>0$ is equivalent to $\lambda_{\max}(\mathcal{T})>1$ and
vice versa. Because $\lambda_{\text{F}}$ is the largest of the Floquet
exponents in Floquet theory (see Appendix A), we refer to $\lambda_{\text{F}}$
as the largest Floquet exponent in the following text.

In the case of periodic switching networks, Eq. (8) is reduced to [20, 81]

$\frac{\text{d}x_{i}(t)}{\text{d}t}=\beta(1-x_{i}(t))\sum_{j=1}^{N}a^{(\ell^{\prime})}_{ij}x_{j}(t)-\mu
x_{i}(t),$ (14)

where $a^{(\ell^{\prime})}_{ij}$ is the $(i,j)$ entry of
$\textbf{A}^{(\ell^{\prime})}$ , and $\ell^{\prime}\in\\{1,\ldots,\ell\\}$ is
the unique value satisfying
$t\in[nT+\tau_{1}+\cdots+\tau_{\ell^{\prime}-1},nT+\tau_{1}+\cdots+\tau_{\ell^{\prime}-1}+\tau_{\ell^{\prime}})$,
$\exists n\in\mathbb{Z}$. Linearization of Eq. (14) around the disease-free
equilibrium yields

$\frac{\text{d}\textbf{x}(t)}{\text{d}t}=(\beta\textbf{A}^{(\ell^{\prime})}-\mu\textbf{I})\textbf{x}(t)$
(15)

with the same value of $\ell^{\prime}$ as that in Eq. (14). Equation (15)
gives Eq. (10) with

$\mathcal{T}=e^{(\beta\textbf{A}^{(\ell)}-\mu I)\tau_{\ell}}\cdots
e^{(\beta\textbf{A}^{(1)}-\mu I)\tau_{1}}.$ (16)

Note that $\mathcal{T}=e^{\beta\textbf{A}^{(\ell)}\tau_{\ell}}\cdots
e^{\beta\textbf{A}^{(1)}\tau_{1}}e^{-\mu T}$ because any $e^{-\mu
I\tau_{\ell^{\prime}}}$ $(=e^{-\mu\tau_{\ell^{\prime}}}I)$ commutes with any
matrix and that each
$e^{\beta\textbf{A}^{(\ell^{\prime})}\tau_{\ell^{\prime}}}$,
$\ell^{\prime}\in\\{1,\ldots,\ell\\}$ is a positive matrix. Therefore, the
Perron-Frobenius theorem guarantees that $\lambda_{\max}(\mathcal{T})$ is real
and positive.

##  3 Results

In this section, we investigate the Parrondo’s paradox for the SIS dynamics
over periodic switching networks. Specifically, epidemics on a temporal
network can be sub-critical (i.e., the number of infectious nodes
exponentially decays over time) over periods even if the SIS dynamics at any
point of time is super-critical (i.e., the number of infectious nodes
exponentially grows over time if the momentary static network is used
forever). In section 3.1, we showcase the Parrondo’s paradox on a two-node
periodic switching network. Section 3.2 shows the relationship between the
amount of interaction and epidemic spreading. Section 3.3 illustrates the
generality of the Parrondo’s paradox. We show relationships between the
Parrondo’s paradox and anti-phase oscillation in section 3.4. Section 3.5
presents the perturbation theory for the largest Floquet exponent.

###  3.1 Parrondo’s paradox for a periodic switching network with two nodes

We start by studying the SIS dynamics on a minimal periodic switching network
with two nodes and $\ell=2$ static networks to be switched between. We say
that the Parrondo’s paradox occurs if the largest eigenvalues of
${\textbf{M}}^{(1)}\equiv\beta{\textbf{A}}^{(1)}-\mu{\textbf{I}}$ and that of
${\textbf{M}}^{(2)}\equiv\beta{\textbf{A}}^{(2)}-\mu{\textbf{I}}$, i.e.,
$\lambda_{\max}(\textbf{M}^{(1)})$ and $\lambda_{\max}(\textbf{M}^{(2)})$, are
both positive and the largest Floquet exponent, $\lambda_{\text{F}}$, is
negative. In this case, the SIS dynamics at any point of time is above the
epidemic threshold such that the fraction of infectious nodes exponentially
grows over time if either ${\textbf{A}}^{(1)}$ or ${\textbf{A}}^{(2)}$ is
permanently used, whereas the epidemic exponentially decays over time on the
periodic switching network. Generalizing this definition to the case of
$\ell>2$ is straightforward. We only study the case of $\ell=2$ in this
article.

Figure 2: Parrondo’s paradox for the periodic switching network with two nodes
shown in Fig. 1. We set $\mu=0.5$ and vary $\beta$. (a) Largest Floquet
exponent, $\lambda_{\text{F}}$ (black line), largest eigenvalue of
$\textbf{M}^{(1)}$ (blue line), and that of $\textbf{M}^{(2)}$ (magenta line)
as a function of $\beta$. The shaded region shows the range of $\beta$ in
which the Parrondo’s paradox occurs. (b)–(e) Time courses of $x_{1}(t)$ and
$x_{2}(t)$, which represent the infection probabilities in two nodes, for the
four values of $\beta$ indicated in (a): (b) $\beta=0.01$, (c) $\beta=0.028$,
(d) $\beta=0.032$, and (e) $\beta=0.04$.

We show an example of the Parrondo paradox in Fig. 2, where we set $\mu=0.5$.
Figure 2(a) shows $\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$,
and $\lambda_{\max}(\textbf{M}^{(2)})$ as a function of $\beta$ for the
periodic switching network shown in Fig. 1, i.e., the one with $\ell=2$,
$T=7$, $\tau_{1}=5$, $\textbf{A}^{(1)}=\begin{pmatrix}17&5\\\
5&0\end{pmatrix}$, and $\textbf{A}^{(2)}=\begin{pmatrix}0&5\\\
5&15\end{pmatrix}$. As expected, $\lambda_{\text{F}}$,
$\lambda_{\max}(\textbf{M}^{(1)})$, and $\lambda_{\max}(\textbf{M}^{(2)})$
increase as the infection rate, $\beta$, increases. For static networks with
adjacency matrices $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$, the epidemic
threshold, which is the value of $\beta$ at which
$\lambda_{\max}(\textbf{M}^{(1)})$ or $\lambda_{\max}(\textbf{M}^{(2)})$ is
equal to $0$, denoted by $\beta_{1}^{*}$ and $\beta_{2}^{*}$, respectively, is
$\beta_{1}^{*}\approx 0.0272$ and $\beta_{2}^{*}\approx 0.0303$, where
$\approx$ represents “approximately equal to”. However, the epidemic threshold
for the periodic switching network, denoted by $\beta_{\text{F}}^{*}$, is
larger than both of these values, i.e., $\beta_{\text{F}}^{*}\approx 0.0335$.

The shaded region in Fig. 2(a) represents the range of $\beta$ for which we
have super-critical (i.e., above the epidemic threshold) dynamics on the
static networks (i.e.,
$\lambda_{\max}(\textbf{M}^{(1)}),\lambda_{\max}(\textbf{M}^{(2)})>0$) and
sub-critical dynamics on the periodic switching network (i.e.,
$\lambda_{\text{F}}<0$). In this situation, the Parrondo’s paradox is
occurring such that the disease-free equilibrium is the unique stable
equilibrium for Eq. (15), whereas, if the network were static then the
disease-free equilibrium would be unstable and the number of infectious nodes
would exponentially grow over time. Now we further explore the SIS dynamics at
the $\beta$ values indicated by the circles in Fig. 2(a). In Fig. 2(b), (c),
(d), and (e), we show $x_{1}(t)$ and $x_{2}(t)$ for $\beta=0.01$,
$\beta=0.028$, $\beta=0.032$, and $\beta=0.04$ respectively. We recall that
$x_{1}(t)$ and $x_{2}(t)$ are the probability that the $i$th node is
infectious. We selected these four values of $\beta$ because each of them
represents a qualitatively different situation regarding whether
$\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$, or
$\lambda_{\max}(\textbf{M}^{(2)})$ is positive or negative. In Fig. 2(b),
$x_{1}(t)$ and $x_{2}(t)$ exponentially decay at any instantaneous time and in
a long term. This result is expected because the growth rates of infection,
$\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$, and
$\lambda_{\max}(\textbf{M}^{(2)})$, are all negative at this value of $\beta$.
The time courses of $x_{1}(t)$ and $x_{2}(t)$ show anti-phase behavior when
$\beta$ is poised near (see Fig. 2(c)) or within (see Fig. 2(d)) the region in
which the Parrondo’s paradox is present. In Fig. 2(e), both $x_{1}(t)$ and
$x_{2}(t)$ exponentially grow over time in the long term. This last result is
expected because $\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$, and
$\lambda_{\max}(\textbf{M}^{(2)})$ are all positive at this value of $\beta$.

To show that the Parrondo’s paradox can also be present in agent-based SIS
dynamics, we simulated the stochastic, agent-based SIS model by the direct
method of the Gillespie algorithm (e.g., [82]). To this end, we generated a
network with $N=2000$ nodes and two equally sized communities using a
stochastic block model (SBM) [83, 84]. See Appendix B for the network
generation. We used the entries of $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$
to inform the edge probabilities for the periodic switching SBM and ran SIS
simulations for the four values of $\beta$ used in Fig. 2(b)–(e). We assumed
that there are initially 100 infectious nodes out of the 1000 nodes, which we
selected uniformly at random, in each block, or community. The results shown
in Appendix B indicate the presence of the Parrondo’s paradox in these
stochastic simulations although the paradox’s outcome is less eminent than in
the case of ODE simulations presented in this section. This discrepancy is
partly due to the nonlinearity of the SIS dynamics, which is also present in
the deterministic SIS dynamics, i.e., the IBA before the linearization (see
Eq. (14)).

###  3.2 Relationships between the amount of interaction and epidemic
spreading

We propose that an advantage of the Parrondo’s paradox, or using periodic
switching networks showing it, is to have comparably less epidemic spreading
for the given total amount of interaction between individuals. To examine this
point, we define the total amount of interaction between individuals in the
case of static networks by the sum of all the entries of the adjacency matrix.
In the case of undirected networks, this definition double counts the off-
diagonal entries but provides a correctly normalized total amount of
interaction. Imagine a static SBM with two blocks in which each block has
$N/2$ individuals and the edge exists between two individuals in the first
subpopulation with probability $a_{11}/N$, for example. Then, the expected
number of edges, which is equivalent to the amount of interaction in the case
of static networks, within the first subpopulation is
$\frac{a_{11}}{N}\times\left[\frac{1}{2}\times\frac{N}{2}\left(\frac{N}{2}-1\right)\right]\approx\frac{Na_{11}}{8}$.
The expected number of edges between the two subpopulations is
$\frac{a_{12}}{N}\times\left(\frac{N}{2}\right)^{2}=\frac{Na_{12}}{4}$,
justifying the double counting. For periodic switching networks, we further
take the time average over one cycle of the total amount of interaction. The
total amount of interaction for periodic switching networks is equal to
$\sum_{\ell^{\prime}=1}^{\ell}\tau_{\ell^{\prime}}\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}^{(\ell^{\prime})}/T$.

For the periodic switching network used in Fig. 2, we show in Fig. 3 the
leading eigenvalues (i.e., $\lambda_{\text{F}}$,
$\lambda_{\max}(\textbf{M}^{(1)})$, and $\lambda_{\max}(\textbf{M}^{(2)})$) as
a function of the total amount of interaction multiplied by the infection
rate, $\beta$; this product represents the total infection potential in the
population. Figure 3 indicates that the two static networks yield larger
leading eigenvalues than the periodic switching network with the same total
amount of interaction, supporting our proposal.

Figure 3: Leading eigenvalues as a function of the total amount of interaction
for a two-node periodic switching network and the constituent static networks.
The value of $\mu$ $(=0.5)$, $\textbf{A}^{(1)}$, $\textbf{A}^{(2)}$, and the
switching schedule (i.e., $\ell=2$, $T=7$, and $\tau_{1}=5$, and hence
$\tau_{2}=T-\tau_{1}=2$) are the same as those used in Fig. 2. The shaded
region represents the region of “$\beta\times(\text{amount of interaction})$”
in which the Parrondo’s paradox occurs.

###  3.3 Generality of the Parrondo’s paradox

Next we examine the generality of the Parrondo’s paradox. First, we analyze
some other two-node periodic switching networks. We use the same switching
schedule (i.e., $\ell=2$ alternating static networks, period $T=7$, the
duration of the first static network $\tau_{1}=5$, and hence
$\tau_{2}=T-\tau_{1}=2$) in the remainder of this paper including here. We
consider three one-parameter families of two-node periodic switching networks
and calculated the epidemic threshold for each periodic switching network. We
show in Fig. 4(a)–(c) the epidemic threshold for the periodic switching
network (i.e., $\beta_{\text{F}}^{*}$), first static network (i.e.,
$\beta_{1}^{*}$), and second static network (i.e., $\beta_{2}^{*}$). We set
$\textbf{A}^{(1)}=\begin{pmatrix}\gamma&5\\\ 5&0\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}0&5\\\ 5&15\end{pmatrix}$ in Fig. 4(a),
$\textbf{A}^{(1)}=\begin{pmatrix}17&5\\\ 5&\gamma\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}\gamma&5\\\ 5&15\end{pmatrix}$ in Fig. 4(b),
and $\textbf{A}^{(1)}\begin{pmatrix}17&\gamma\\\ \gamma&0\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}0&\gamma\\\ \gamma&15\end{pmatrix}$ in Fig.
4(c). The shaded regions in the figure are the ranges of $\gamma$ and $\beta$
for which the Parrondo’s paradox occurs, i.e., where $\lambda_{\text{F}}<0$,
$\lambda_{\max}(\textbf{M}^{(1)})>0$, and $\lambda_{\max}(\textbf{M}^{(2)})>0$
are simultaneously satisfied. We find that the paradox occurs for some ranges
of $\gamma$.

To verify that the Parrondo’s paradox can also be present in large networks,
we similarly calculated the leading eigenvalues for networks with $N=2000$
nodes generated by an SBM that creates large networks with community structure
as discussed in Appendix B. We define the SBMs to have two communities, each
with $N/2=1000$ nodes, and obtain their edge probabilities using the entries
in the adjacency matrices of the three families of $2\times 2$ periodic
switching networks used in Fig. 4(a)–(c). Summing the entries across SBM
blocks recovers the original $2\times 2$ matrices, in expectation. We show in
Fig. 4(d)–(f) the epidemic thresholds and the region of pairs of $\gamma$ and
$\beta$ in which the Parrondo’s paradox is present for the networks with
$N=2000$ nodes generated by the SBM. The results for the SBM are close to
those for the corresponding two-node networks shown in Fig. 4(a)–(c).

Figure 4: Parrondo’s paradox over other periodically switching networks with
two nodes. We show the epidemic threshold for the periodic switching network
(i.e., $\beta_{\text{F}}^{*}$; black lines), $\textbf{A}^{(1)}$ (i.e.,
$\beta_{1}^{*}$; blue lines), and $\textbf{A}^{(2)}$ (i.e., $\beta_{2}^{*}$;
magenta lines). The shaded region shows the values of $\gamma$ and $\beta$ for
which the paradox occurs. (a)–(c): Two-node networks. (d)–(f): Networks with
$N=2000$ nodes and two communities generated by the SBM. (a) and (d):
$\textbf{A}^{(1)}=\begin{pmatrix}\gamma&5\\\ 5&0\end{pmatrix}$,
$\textbf{A}^{(2)}=\begin{pmatrix}0&5\\\ 5&15\end{pmatrix}$. (b) and (e):
$\textbf{A}^{(1)}=\begin{pmatrix}17&5\\\ 5&\gamma\end{pmatrix}$,
$\textbf{A}^{(2)}=\begin{pmatrix}\gamma&5\\\ 5&15\end{pmatrix}$. (c) and (f):
$\textbf{A}^{(1)}=\begin{pmatrix}17&\gamma\\\ \gamma&0\end{pmatrix}$,
$\textbf{A}^{(2)}=\begin{pmatrix}0&\gamma\\\ \gamma&15\end{pmatrix}$.

Next we conduct an experiment with random networks to study the fraction of
periodic switching networks for which the paradox occurs. To this end, we
assume that the adjacency matrices $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$
are symmetric and parameterize them as
$\textbf{A}^{(1)}=\begin{pmatrix}\gamma_{1}&\gamma_{2}\\\
\gamma_{2}&\gamma_{3}\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}\gamma_{4}&\gamma_{5}\\\
\gamma_{5}&\gamma_{6}\end{pmatrix}$. Without loss of generality, we set
$\gamma_{1}=1$; multiplying $\gamma_{1}$, $\ldots$, $\gamma_{6}$ by a factor
of $c$ ($>0$) is equivalent to using the original values of $\gamma_{1}$,
$\ldots$, $\gamma_{6}$ and changing $\beta$ to $c\beta$. Then, we
independently draw $\gamma_{2}$, $\gamma_{3}$, $\gamma_{4}$, $\gamma_{5}$, and
$\gamma_{6}$ from the uniform density on $[0,5]$. For each pair of
$\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$, we computed $\beta_{\text{F}}^{*}$,
$\beta_{1}^{*}$, and $\beta_{2}^{*}$ as the values of $\beta$ at which
$\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$, or
$\lambda_{\max}(\textbf{M}^{(2)})$ is equal to $0$, respectively.

We defined the Parrondo’s paradox as a situation in which
$\lambda_{\text{F}}<0$, $\lambda_{\max}(\textbf{M}^{(1)})>0$, and
$\lambda_{\max}(\textbf{M}^{(2)})>0$ simultaneously hold true. Because these
leading eigenvalues monotonically increase as $\beta$ increases, a periodic
switching network shows the Parrondo’s paradox for some values of $\beta$ if
and only if $\beta_{\text{F}}^{*}>\max\\{\beta_{1}^{*},\beta_{2}^{*}\\}$, as
shown by the shaded regions in Figs. 2 and 4. We generated $10^{4}$
independent pairs of $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$ to find that
the paradox is present in 28.7% of the periodic switching networks. Therefore,
we conclude that the Parrondo’s paradox is not a rare phenomenon, at least for
$2\times 2$ adjacency matrices. We did not find any periodic switching network
that showed the opposite paradoxical behavior, i.e.,
$\beta_{\text{F}}^{*}<\min\\{\beta_{1}^{*},\beta_{2}^{*}\\}$.

As our final set of experiments to investigate the generality of the
Parrondo’s paradox, we study how it is affected by the number of node. We
first examine the possibility of the paradox with $3\times 3$ periodic
switching networks. Let us consider arbitrarily selected one-parameter
families of three-node periodic switching networks. The epidemic thresholds
for the periodic switching network (i.e., $\beta_{\text{F}}^{*}$) and those
for the static networks (i.e., $\beta_{1}^{*}$, $\beta_{2}^{*}$) are shown in
Fig. 5(a) for $\textbf{A}^{(1)}=\begin{pmatrix}\gamma&5&5\\\ 5&0&5\\\
5&5&15\end{pmatrix}$ and $\textbf{A}^{(2)}=\begin{pmatrix}14&5&5\\\ 5&0&5\\\
5&5&\gamma\end{pmatrix}$ and Fig. 5(b) for
$\textbf{A}^{(1)}=\begin{pmatrix}24&\gamma&5\\\ 5&0&\gamma\\\
\gamma&5&15\end{pmatrix}$ and $\textbf{A}^{(2)}=\begin{pmatrix}14&\gamma&5\\\
5&0&\gamma\\\ \gamma&5&24\end{pmatrix}$. The shaded regions represent the
region of $\gamma$ and $\beta$ values in which the Parrondo’s paradox occurs.
We observe that the paradox occurs for a range of $\gamma$ in both families of
periodic switching networks. For the periodic switching network shown in Fig.
5(a), the ranges of $\gamma$ for which the paradox occurs are disjoint; the
paradox occurs for small and large $\gamma$ values, but not for intermediate
$\gamma$ values.

Figure 5: Epidemic thresholds for periodic switching networks and static
networks with three nodes. (a): $\textbf{A}^{(1)}=\begin{pmatrix}\gamma&5&5\\\
5&0&5\\\ 5&5&15\end{pmatrix}$ and $\textbf{A}^{(2)}=\begin{pmatrix}14&5&5\\\
5&0&5\\\ 5&5&\gamma\end{pmatrix}$. (b):
$\textbf{A}^{(1)}=\begin{pmatrix}24&\gamma&5\\\ 5&0&\gamma\\\
\gamma&5&15\end{pmatrix}$ and $\textbf{A}^{(2)}=\begin{pmatrix}14&\gamma&5\\\
5&0&\gamma\\\ \gamma&5&24\end{pmatrix}$. See the caption of Fig. 4 for the
legends. The shaded regions represent the values of $\gamma$ and $\beta$ for
which the Parrondo’s paradox occurs. In (b), the blue line is not visible
because it is almost completely hidden behind the magenta line.

To further assess the generality of the Parrondo’s paradox for periodic
switching networks with three nodes, we retain the same switching schedule for
simplicity and generate symmetric adjacency matrices parameterized as
$\textbf{A}^{(1)}=\begin{pmatrix}\gamma_{1}&\gamma_{2}&\gamma_{3}\\\
\gamma_{2}&\gamma_{4}&\gamma_{5}\\\
\gamma_{3}&\gamma_{5}&\gamma_{6}\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}\gamma_{7}&\gamma_{8}&\gamma_{9}\\\
\gamma_{8}&\gamma_{10}&\gamma_{11}\\\
\gamma_{9}&\gamma_{11}&\gamma_{12}\end{pmatrix}$. We set $\gamma_{1}=1$
without loss of generality. Then, we independently draw $\gamma_{2}$,
$\ldots$, $\gamma_{12}$ from the uniform density on $[0,5]$. We find that,
among $10^{4}$ pairs of the generated $\textbf{A}^{(1)}$ and
$\textbf{A}^{(2)}$, the paradox is present, with
$\beta_{\text{F}}^{*}>\max\\{\beta_{1}^{*},\beta_{2}^{*}\\}$, in 12.0% of the
periodic switching networks, and that there is no case with
$\beta_{\text{F}}^{*}<\min\\{\beta_{1}^{*},\beta_{2}^{*}\\}$.

We similarly analyzed randomly generated periodic switching networks with four
and five nodes. We found that among $10^{4}$ pairs of randomly generated
$\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$, the paradox is present in $5.21\%$
and $1.2\%$ of the periodic switching networks with four and five nodes,
respectively. There was no periodic switching network showing
$\beta_{\text{F}}^{*}<\min\\{\beta_{1}^{*},\beta_{2}^{*}\\}$. These results
are qualitatively the same as those for periodic switching networks with fewer
(i.e., two or three) nodes.

###  3.4 Relationships between the Parrondo’s paradox and anti-phase
oscillations

To better understand mechanisms of the Parrondo’s paradox, we carried out
additional experiments motivated by the following observation. Figure 2
suggests that the Parrondo’s paradox may be characterized by an anti-phase
oscillation between $x_{1}(t)$ and $x_{2}(t)$. However, this association is
not straightforward already with Fig. 2 because the figure shows anti-phase
oscillations for a $\beta$ value at which the paradox occurs (see Fig. 2(d))
as well as for a $\beta$ value at which it does not (see Fig. 2(c)).
Nevertheless, the anti-phase oscillation is absent for $\beta$ values that are
far from where the paradox is present (see Fig. 2(b) and 2(e)).

We measure the extent of anti-phase oscillation by the fraction of time, $q$,
during one period in which $\text{d}x_{1}(t)/\text{d}t$ and
$\text{d}x_{2}(t)/\text{d}t$ have the opposite sign. We measure $q$ in the
fifth cycle, i.e., using $t\in[28,35]$, and use the initial condition
$(x_{1}(0),x_{2}(0))=(0.1,0.1)$. Consider the $10^{4}$ periodic switching
networks with two communities used in section 3.3. We recall that $28.7\%$ of
the periodic switching networks (i.e., 2,870 networks) show the Parrondo’s
paradox. For each of these 2,870 networks, we select the middle value of the
range of $\beta$ in which the paradox occurs and compute $q$ at the selected
$\beta$ value. At this value of $\beta$, perfect anti-phase synchronization,
i.e., $q=1$, occurs for 2867 ($99.9\%$) of the 2870 networks. For comparison,
we have also computed $q$ for the rest of the networks (i.e., 7,130 out of the
$10^{4}$ networks). We find that the maximum, minimum, mean and standard
deviation of $q$ for these 7,130 networks is $0.974$, $0$, $0.366$, and
$0.305$, respectively. Therefore, although some networks show strong anti-
phase oscillation despite the lack of the Parrondo’s paradox behavior, most
networks not showing the paradox have substantially smaller $q$ values than
the networks showing the paradox.

We show examples of anti-phase dynamics in Appendix C. It should be noted that
$q=1$ holds true for both inside and outside the range of $\beta$ in which the
paradox is present for some networks. In our examples, $q$ is much larger for
the periodic switching networks showing the paradox than those not showing the
paradox. We also observe some anti-phase behavior in the four periodic
switching networks that do not show the Parrondo’s paradox for any value of
$\beta$, but to a limited extent (i.e., $q\leq 0.7$). These results are
consistent with the population results with $10^{4}$ networks described above.
In sum, we conclude that anti-phase oscillation occurs more frequently in the
presence or approximate presence of the Parrondo’s paradox than otherwise.

###  3.5 Perturbation theory for the largest Floquet exponent

Regardless of the presence or absence of the Parrondo’s paradox as one varies
$\beta$, Fig. 2 suggests that the relationship between the largest Floquet
exponent (i.e., $\lambda_{\text{F}}$) and $\beta$ is apparently close to
linear for an extended range of $\beta$. Therefore, we derive the first-order
approximation of $\lambda_{\text{F}}$ in terms of $\beta$ and assess its
accuracy in this section.

######  Theorem 1.

(Perturbation of simple eigenvalues [85]). Let $\hat{\textbf{M}}$ be a
symmetric $N\times N$ matrix with the left eigenvector
$\\{\textbf{u}^{(i)}\\}$ and right eigenvector $\\{\textbf{v}^{(i)}\\}$
associated with eigenvalue $\lambda_{i}$, which is assumed to be simple (with
$i\in\\{1,\ldots,N\\}$). Without loss of generality, the Euclidean norm of
each eigenvector is equal to $1$. Consider a perturbation of
$\hat{\textbf{M}}$ given by
$\hat{\textbf{M}}(\beta)=\hat{\textbf{M}}+\beta\Delta\hat{\textbf{M}}$. The
eigenvalues of $\hat{\textbf{M}}(\beta)$ when $\left|\beta\right|\ll 1$
satisfy

$\lambda_{i}(\beta)=\lambda_{i}+\beta\lambda_{i}^{\prime}(0)+\mathcal{O}(\beta^{2}),$
(17)

where

$\lambda_{i}^{\prime}(0)=\frac{[\textbf{u}^{(i)}]^{\top}\Delta\hat{\textbf{M}}\textbf{v}^{(i)}}{[\textbf{u}^{(i)}]^{\top}\textbf{v}^{(i)}}.$
(18)

######  Lemma 1 (First-order approximation of matrix $\mathcal{T}$).

The first-order perturbation of matrix $\mathcal{T}$ (defined in Eq. (16)) is
given by

$\begin{split}\mathcal{T}(\beta)&=\mathcal{T}(0)+\beta\Delta\mathcal{T}(0)+\mathcal{O}(\beta^{2})\\\
&=\left(\textbf{I}+\beta
T\overline{\textbf{A}}\right)e^{-T\mu\textbf{I}}+\mathcal{O}(\beta^{2}).\end{split}$
(19)

where

$\overline{\textbf{A}}\equiv\frac{1}{T}\sum_{\ell^{\prime}=1}^{\ell}\tau_{\ell^{\prime}}\textbf{A}^{(\ell^{\prime})}$
(20)

is the time-averaged adjacency matrix.

###### Proof.

Using Eq. (16), we obtain

$\begin{split}\mathcal{T}(\beta)&=e^{\beta\tau_{\ell}\textbf{A}^{(\ell)}}e^{\beta\tau_{\ell-1}\textbf{A}^{(\ell-1)}}\cdots
e^{\beta\tau_{1}\textbf{A}^{(1)}}e^{-\mu T\textbf{I}},\end{split}$ (21)

which leads to

$\mathcal{T}(0)=e^{-\mu T\textbf{I}}$ (22)

and

$\begin{split}\Delta\mathcal{T}(0)&=\left.\frac{\text{d}\mathcal{T}(\beta)}{\text{d}\beta}\right|_{\beta=0}\\\
&=\tau_{\ell}\textbf{A}^{(\ell)}e^{\beta\tau_{\ell}\textbf{A}^{(\ell)}}e^{\beta\tau_{\ell-1}\textbf{A}^{(\ell-1)}}\cdots
e^{\beta\tau_{1}\textbf{A}^{(1)}}e^{-\mu T\textbf{I}}\\\
&+e^{\beta\tau_{\ell}\textbf{A}^{(\ell)}}\tau_{\ell-1}\textbf{A}^{(\ell-1)}e^{\beta\tau_{\ell-1}\textbf{A}^{(\ell-1)}}\cdots
e^{\beta\tau_{1}\textbf{A}^{(1)}}e^{-\mu T\textbf{I}}+\cdots\\\
&+e^{\beta\tau_{\ell}\textbf{A}^{(\ell)}}e^{\beta\tau_{\ell-1}\textbf{A}^{(\ell-1)}}\cdots\tau_{1}\textbf{A}^{(1)}e^{\beta\tau_{1}\textbf{A}^{(1)}}e^{-\mu
T\textbf{I}}\Bigg{.}\Bigg{|}_{\beta=0}\\\
&=\left(\tau_{\ell}\textbf{A}^{(\ell)}+\tau_{\ell-1}\textbf{A}^{(\ell-1)}+\cdots+\tau_{1}\textbf{A}^{(1)}\right)e^{-\mu
T\textbf{I}}\\\ &=Te^{-\mu T\textbf{I}}\overline{\textbf{A}}.\end{split}$ (23)

Equations (22) and (23) imply (19). ∎

######  Lemma 2 (First-order approximation of the largest eigenvalue of
$\mathcal{T}$).

For $\left|\beta\right|\ll 1$, we obtain

$\lambda_{\max}(\mathcal{T})(\beta)=e^{-T\mu}+\beta\lambda_{\max}(\overline{\textbf{A}})Te^{-T\mu}+\mathcal{O}(\beta^{2}).$
(24)

###### Proof.

Equation (22) implies that

$\lambda_{\max}(\mathcal{T})(0)=e^{-T\mu}.$ (25)

Let u and v be the dominant left and right eigenvectors of
$\overline{\textbf{A}}$, respectively. By combining Eqs. (18) and (23), we
obtain

$\begin{split}\lambda^{\prime}_{\max}(\mathcal{T})(0)&=\frac{\textbf{u}^{\top}(Te^{-T\mu}\overline{\textbf{A}})\textbf{v}}{\textbf{u}^{\top}\textbf{v}}\\\
&=\lambda_{\max}(\overline{\textbf{A}})Te^{-T\mu}.\end{split}$ (26)

By substituting Eqs. (25) and (26) in Eq. (17), we obtain Eq. (24). ∎

######  Lemma 3 (First-order approximation of $\lambda_{\text{F}}$).

For $\left|\beta\right|\ll 1$, we obtain

$\lambda_{\text{F}}(\beta)=\lambda_{\text{F}}(0)+\beta\lambda^{\prime}_{\text{F}}(0)+\mathcal{O}(\beta^{2})=-\mu+\beta\lambda_{\max}(\overline{\textbf{A}})+\mathcal{O}(\beta^{2}).$
(27)

###### Proof.

By definition, we obtain
$\lambda_{\text{F}}(\beta)=\frac{\ln\lambda_{\max}(\mathcal{T})(\beta)}{T}$.
Using this, Eq. (25), and Eq. (26), we obtain

$\lambda_{\text{F}}(0)=\frac{\ln\lambda_{\max}(\mathcal{T})(0)}{T}=-\mu$ (28)

and

$\lambda^{\prime}_{\text{F}}(0)=\left.\frac{d\lambda_{\text{F}}}{d\beta}\right|_{\beta=0}=\left.\frac{\partial\lambda_{\text{F
}}}{\partial\lambda_{\max}(\mathcal{T})}\frac{\partial\lambda_{\max}(\mathcal{T})}{\partial\beta}\right|_{\beta=0}=\frac{1}{T\lambda_{\max}(\mathcal{T})(0)}\cdot\lambda_{\max}(\overline{\textbf{A}})Te^{-T\mu}=\lambda_{\max}(\overline{\textbf{A}}).$
(29)

Equations (28) and (29) imply Eq. (27). ∎

Equation (27) implies the following corollary.

######  Corollary 1.

The epidemic threshold for the periodic switching network under the first-
order approximation is given by

$\beta_{\text{F}}^{*}=\frac{\mu}{\lambda_{\max}(\overline{\textbf{A}})}.$ (30)

Now we validate our first-order approximation for $2\times 2$ periodic
switching networks with $\ell=2$. We compare $\lambda_{\text{F}}$ between the
exact value and linear approximation in Fig. 6 for eight arbitrarily selected
two-node periodic switching networks. Figures 6(a)–(d) are for four networks
showing the Parrondo’s paradox. We find that the first-order approximation is
sufficiently accurate for a range of $\beta$ including the range in which the
paradox occurs. The first-order approximation is also accurate for four
periodic switching networks in which the paradox is absent in the entire range
of $\beta$ (see Fig. 6(e)–(h)). Figure 6 suggests that the first-order
approximation tends to be more accurate when the paradox is absent (see Fig.
6(e)–(h)) than present (see Fig. 6(a)–(d)).

Figure 6:  First-order approximation to the largest Floquet exponent for eight
two-node periodic switching networks with $k=2$, $T=7$, and $\tau_{1}=5$. We
set $\mu=0.5$. (a) $\textbf{A}^{(1)}=\begin{pmatrix}17&5\\\ 5&0\end{pmatrix}$
and $\textbf{A}^{(2)}=\begin{pmatrix}0&5\\\ 5&15\end{pmatrix}$. (b)
$\textbf{A}^{(1)}=\begin{pmatrix}12&5\\\ 5&1\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}1&5\\\ 5&15\end{pmatrix}$. (c)
$\textbf{A}^{(1)}=\begin{pmatrix}17&0\\\ 0&0\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}0&0\\\ 0&15\end{pmatrix}$. (d)
$\textbf{A}^{(1)}=\begin{pmatrix}25&2\\\ 3&4\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}6&1\\\ 7&30\end{pmatrix}$. (e)
$\textbf{A}^{(1)}=\begin{pmatrix}3&2\\\ 2&3\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}4&8\\\ 6&26\end{pmatrix}$. (f)
$\textbf{A}^{(1)}=\begin{pmatrix}10&5\\\ 2&15\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}4&3\\\ 6&8\end{pmatrix}$. (g)
$\textbf{A}^{(1)}=\begin{pmatrix}2&4\\\ 4&2\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}3&8\\\ 8&3\end{pmatrix}$. (h)
$\textbf{A}^{(1)}=\begin{pmatrix}30&5\\\ 10&7\end{pmatrix}$ and
$\textbf{A}^{(2)}=\begin{pmatrix}3&6\\\ 6&3\end{pmatrix}$.

As an application of the first-order approximation of $\lambda_{\text{F}}$, we
examine the fraction of randomly generated periodic switching networks for
which the Parrondo’s paradox occurs, as we did in section 3.3. Note that the
epidemic threshold is given in the form of $\mu/\lambda_{\max}$ for the
periodic switching network under the first-order approximation as well as for
the static networks. Therefore, if the first-order approximation is accurate,
the Parrondo’s paradox happens if $\lambda_{\max}(\overline{\textbf{A}})$ is
larger than both $\lambda_{\max}(\textbf{A}^{(1)})$ and
$\lambda_{\max}(\textbf{A}^{(2)})$ or smaller than both of them. In the same
manner as that in section 3.3, we generated $10^{4}$ pairs of
$\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$ and computed
$\lambda_{\max}(\overline{\textbf{A}})$, $\lambda_{\max}(\textbf{A}^{(1)})$,
and $\lambda_{\max}(\textbf{A}^{(2)})$. We found that
$\lambda_{\max}(\overline{\textbf{A}})$ is smaller than both
$\lambda_{\max}(\textbf{A}^{(1)})$ and $\lambda_{\max}(\textbf{A}^{(2)})$ for
$31.7\%$ pairs of $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$. There was no case
in which $\lambda_{\max}(\overline{\textbf{A}})$ is larger than
$\lambda_{\max}(\textbf{A}^{(1)})$ and $\lambda_{\max}(\textbf{A}^{(2)})$.
These results are similar to those obtained for $\lambda_{\text{F}}$ (see
section 3.3).

##  4 Discussion

We showed a Parrondo’s paradox in the SIS dynamics on periodically switching
temporal networks in which an epidemic decays over a long time while the
dynamics is super-critical (i.e., above the epidemic threshold) for the
momentary static network at any point of time. We primarily showed the paradox
in simple cases involving a periodic alternation of two $2\times 2$ adjacency
matrices. Larger networks (with $N=2000$ nodes) with two subpopulations
generated by the SBM also showed the paradox, even for a single run of
stochastic SIS dynamics. We further verified that the paradox occurred with
adjacency matrices of larger sizes, up to five nodes. However, we found that
the fraction of random networks that show the Parrondo’s paradox decreased as
the number of nodes increases. This is an important limitation of the present
study. Further investigating the generality of the Parrondo’s paradox in terms
of the number of nodes or subpopulations, the number of static networks
constituting a periodic switching network, different epidemic process models,
and different network models such as the metapopulation model and hypergraphs,
warrants future work.

We pointed out the relationship between the growth rate of the epidemics
(including the epidemic threshold, at which the growth rate is equal to $0$)
and the Floquet exponent. Leveraging Floquet theory to characterize and
intervene into epidemic processes on periodic temporal networks may be
fruitful. For example, the periodic matrix obtained by the Floquet
decomposition (i.e., matrix $\textbf{P}(t)$ in Appendix A) may help us to
quantify the amplitude of antiphase oscillation between $x_{i}(t)$ and
$x_{j}(t)$ (with $j\neq i$) or change it through manipulation of
$\textbf{P}(t)$.

In the context of COVID-19, effects on mitigating spread of infection were
numerically simulated on physical proximity and face-to-face contact data and
compared between “rotating” and “on-off” strategies with two-day or biweekly
periodic schedules [86]. The rotating strategy by definition imposes that half
the population is allowed to go out in odd days or odd weeks, and the other
half is allowed to go out in even days or even weeks. In contrast, the on-off
strategy allows the entire population to go for work in one day (or one week)
and not on the next day (or week), and such a two-day or biweekly pattern is
repeated. As expected, any of these four strategies helped towards decreasing
the basic reproduction number below $1$. Other studies also investigated
effects of different on-off strategies on suppressing epidemics [87, 88].
Additionally, it was shown in Ref. [86] that among the four strategies, the
best and worst strategies were the rotating strategy with the biweekly cycle
and the on-off strategy with two-day cycle, respectively. Superiority of a
rotating strategy to an on-off strategy was also shown in a different study
[89]. These comparative results suggest that different quarantine or curfew
strategies based on periodic switching networks are feasible and that their
efficiency depends on the periodic switching network. Investigating the
possibility of the Parrondo’s paradox and its possible roles in suppressing
population-wide infection in these and other practical disease spreading
models warrants future study. In particular, both these previous studies and
the present work suggest that eliciting anti-phase oscillations between
different communities might be a key to suppressing infection while keeping
the overall activity level of individuals relatively high.

##  Appendix Appendix A Floquet theory

In this section, we give a brief review of Floquet theory for periodic linear
dynamical systems in continuous time.

######  Definition A.1 (Periodic system).

Consider a non-autonomous continuous-time dynamical system given by

$x^{\prime}(t)=M(t,x(t)).$ (31)

Dynamical system (31) is called a $T$-periodic system with period $T$ $(>0)$
if $M(t+T,x)=M(t,x)$ $\forall t$.

We present Floquet theory for linear $T$-periodic dynamical systems in
continuous time given by

$\frac{\text{d}\textbf{x}(t)}{\text{d}t}=\textbf{M}(t)\textbf{x}(t),$ (32)

where $\textbf{M}(t)\in\mathcal{R}^{N\times N}$ is a $T$-periodic matrix, and
$\textbf{x}(t)\in\mathcal{R}^{N}$ represents the system’s state. Let
$\textbf{X}(t)\in\mathcal{R}^{N\times N}$ be the fundamental matrix of system
(32). By definition of the fundamental matrix, the columns of the fundamental
matrix are linearly independent solutions of Eq. (32).

######  Theorem A.2.

[Floquet’s theorem for continuous-time linear periodic systems [90]] For
system (32) that has a fundamental matrix, there exists a nonsingular
$T$-periodic matrix, $\textbf{P}(t)\in\mathbb{R}^{N\times N}$, and a constant
matrix, $\textbf{B}\in\mathbb{C}^{N\times N}$, such that

$\textbf{X}(t)=\textbf{P}(t)e^{\textbf{B}t}.$ (33)

Equation (33) is known as the Floquet normal form for $\textbf{X}(t)$.

######  Remark A.1.

Floquet theory guarantees that there exists a nonsingular constant matrix
$\tilde{\textbf{M}}\in\mathcal{R}^{N\times N}$ such that
$\textbf{X}(t+T)=\textbf{X}(t)\tilde{\textbf{M}}$ for all $t\in\mathcal{R}$
and $\tilde{\textbf{M}}=\textbf{X}^{-1}(0)\textbf{X}(T)$ [90, 91]. Matrix
$\tilde{\textbf{M}}$ is called the monodromy matrix.

######  Remark A.2.

If $\textbf{X}(0)=I$, then $\tilde{\textbf{M}}=\textbf{X}(T)$.

######  Remark A.3.

Because $\tilde{\textbf{M}}$ is non-singular, we obtain
$\tilde{\textbf{M}}=e^{\textbf{B}T}$, or equivalently,
$\textbf{B}=\ln(\tilde{\textbf{M}})/T$.

Theorem A.2 implies that the solution of (32) is represented as

$\textbf{x}(t)=\textbf{X}(t)\textbf{x}(0)=\textbf{P}(t)e^{\textbf{B}t}\textbf{x}(0),$
(34)

where $\textbf{x}(0)$ is the initial state.

Each eigenvalue $\lambda_{i}$ of the monodromy matrix is known as the Floquet
or characteristic multiplier. The eigenvalues of B are known as Floquet or
characteristic exponents, and they are equal to $\ln(\lambda_{i})/T$. We
define the largest Floquet exponent by

$\lambda_{\text{F}}=\frac{\ln\left|\lambda_{\max}\right|}{T},$ (35)

where $\lambda_{\max}$ is the Floquet multiplier with the largest modulus. The
largest Floquet exponent gives the average growth rate along the dominant
eigenvector direction over one period. The equilibrium $\textbf{x}(t)=0$ of
the dynamical system given by Eq. (32), corresponding to the disease-free
equilibrium in our context, is asymptotically stable if and only if
$\lambda_{\text{F}}<0$.

We are interested in $T$-periodic and switching ${\bf M}(t)$, that is, ${\bf
M}(t)$ is constant for some duration, then it discontinuously switches to a
different matrix that is used for another duration of time, and so on. In this
case, the Floquet theory is simplified as follows.

######  Theorem A.3.

[Floquet’s theorem for linear dynamical systems with switching matrices [91]]
Let $\textbf{M}(t)$ be a switching $T$-periodic matrix given by

$\textbf{M}(t)=\begin{cases}\textbf{M}^{(1)}&0\leq t<\tau_{1},\\\
\textbf{M}^{(2)}&\tau_{1}\leq t<\tau_{1}+\tau_{2},\\\
\;\;\vdots&\quad\quad\;\vdots\\\
\textbf{M}^{(\ell)}&\tau_{1}+\cdots+\tau_{\ell-1}\leq t<T.\end{cases}$ (36)

Then, the monodromy matrix is represented as

$\tilde{\textbf{M}}=e^{\tau_{\ell}\textbf{M}^{(\ell)}}e^{\tau_{\ell-1}\textbf{M}^{(\ell-1)}}\cdots
e^{\tau_{1}\textbf{M}^{(1)}}.$ (37)

######  Remark A.4.

Matrix $\tilde{\textbf{M}}$ for the deterministic SIS model on periodic
switching networks is the same as $\mathcal{T}$ given by Eq. (16).

##  Appendix Appendix B Simulations of the stochastic SIS model and nonlinear
IBA

We used SBMs to generate the networks with two blocks, each of which
represents a community and nodes within the same block have statistically the
same connectivity patterns [83, 84]. Each community has $N/2=1000$ nodes. Each
edge $(i,j)$ exists with probability $[\textbf{A}^{(1)}]_{c_{i}c_{j}}/N$ for
the first duration of one cycle and $[\textbf{A}^{(2)}]_{c_{i}c_{j}}/N$ for
the second duration, where $c_{i}\in\\{1,2\\}$ is the block to which the $i$th
node belongs. The $2\times 2$ matrix informing the SBM is the one used in Fig.
2. In this manner, we obtain $2000\times 2000$ adjacency matrices
$\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$, and hence $\hat{\textbf{M}}$.

Figure 7:  Agent-based simulations of the SIS model on periodic switching
networks. (a)–(d) Results for a single run. (e)–(h) Comparison between the
results of the agent-based simulations, linear IBA, and nonlinear IBA. The
value of $\mu$ $(=0.5)$, $\textbf{A}^{(1)}$, $\textbf{A}^{(2)}$, and the
switching schedule (i.e., $\ell=2$, $T=7$, and $\tau_{1}=5$) are the same as
those used in Fig. 2. (a) and (e): $\beta=0.01$. (b) and (f): $\beta=0.028$.
(c) and (g): $\beta=0.032$. (d) and (h): $\beta=0.04$. In (e)–(h), the lines
and error bars for the agent-based simulations represent the mean and standard
deviation on the basis of 100 simulations with the same initial condition. In
all panels, we initially infected 10% of nodes in each of the two communities,
i.e., $(x_{1}(0),x_{2}(0))=(0.1,0.1)$.

We show in Fig. 7(a)–(d) the time course of $x_{1}(t)$ and $x_{2}(t)$ in a
single run of stochastic SIS dynamics simulated using the Gillespie algorithm.
We employed the four values of $\beta$ used in Fig. 2(b)–(e). The fluctuations
in $x_{1}(t)$ and $x_{2}(t)$ are large because the results shown are from a
single run of stochastic SIS dynamics. However, there is some indication of
the Parrondo’s paradox and anti-phase oscillation in Fig. 7(b) and (c). For a
value of $\beta$ at which the paradox occurs according to our theoretical
analysis (i.e., $\beta=0.032$; shown in Fig. 7(c)), we further carried out a
single run of stochastic simulation on each of the two static networks with
$N=2000$ nodes constituting the periodic switching network used in Fig. 7.
Figure 8(a) and (b) represents an example time course of $x_{1}(t)$ and
$x_{2}(t)$ on the first and second static network, respectively. We find that
$x_{1}(t)$ and $x_{2}(t)$ do not decay exponentially over time in each static
network in this run. Therefore, although not all the runs show this behavior,
the Parrondo’s paradox can occur in a single run of stochastic simulation.

Figure 8: A single run of stochastic SIS simulation over the two static
networks with $N=2000$ nodes generated by the SBM. The block adjacency
matrices $\textbf{A}^{(1)}$ and $\textbf{A}^{(2)}$ are the same as those used
in Fig. 7. We set $\beta=0.032$, corresponding to Fig. 7(c) and (g), and
$\mu=0.5$. (a) $\textbf{A}^{(1)}$. (b) $\textbf{A}^{(2)}$.

We then simulate the same stochastic SIS dynamics 100 times for each value of
$\beta$ and average $x_{1}(t)$ and $x_{2}(t)$ over the 100 runs. We show the
results, together with the standard deviation of $x_{1}(t)$ and $x_{2}(t)$, in
Fig. 7(e)–(h). We find a consistent tendency of the Parrondo’s paradox and
anti-phase oscillation in the averaged time course of $x_{1}(t)$ and
$x_{2}(t)$ (see Fig. 7(f) and (g)).

For comparison, we also show numerically simulated solutions of the nonlinear
IBA, Eq. (2), and its linearized variant, Eq. (3), for the $2\times 2$
periodic switching network in the same figure. The time courses of $x_{1}(t)$
and $x_{2}(t)$ for the linearized IBA shown in Fig. 7(e)–(h) are the same as
those shown in Fig. 2(b)–(e). We observe that the nonlinear IBA shows less
pronounced Parrondo’s paradox and that the average time courses of the
stochastic simulations when the paradox occurs are close to those for the
nonlinear IBA (see Fig. 7(f) and (g)). We conclude that stochastic agent-based
simulations also show the Parrondo’s paradox, albeit to a lesser extent than
the linearized IBA with a small number of communities shown in the main text.

##  Appendix Appendix C Anti-phase oscillatory behavior with examples

We measure $q$, i.e., the extent of the anti-phase oscillation defined in
section 3.4, for the eight two-node periodic switching networks that we used
in Fig. 6. The upper subpanel in each of Fig. 9(a)–(h) shows
$\lambda_{\text{F}}$, $\lambda_{\max}(\textbf{M}^{(1)})$, and
$\lambda_{\max}(\textbf{M}^{(2)})$, and the range of $\beta$ in which the
paradox occurs, similar to Fig. 2. We find that the first four networks
showing the Parrondo’s paradox (i.e., Fig. 9(a)–(d)) produce large $q$ values
(i.e., above approximately $0.8$) when $\beta$ is close to the epidemic
threshold. The largest $q$ value attained in these four networks is $1$,
indicating perfect anti-phase dynamics between $x_{1}(t)$ and $x_{2}(t)$. The
range of $\beta$ in which $q=1$ roughly coincides with where the Parrondo’s
paradox occurs. However, in one of the four periodic switching networks, shown
in Fig. 9(c), the range of $\beta$ in which $q=1$ extends to larger values of
$\beta$ far beyond where the paradox occurs. In contrast, for the other four
networks that do not show the paradox (Fig. 9(e)–(h)), the largest $q$ value
is substantially smaller than for the first four networks, while $q$ values
are peaked near the epidemic thresholds for each momentarily static network,
except in Fig. 9(g). Note that $q=0.5$ implies that the anti-phase behavior
exists only half of the time so one cannot really regard this as anti-phase
behavior. In Fig. 9(g), we find $q=0$, which is the complete absence of anti-
phase dynamics, for all values of $\beta$.

Figure 9: Fraction of the time showing anti-phase behavior, $q$, in eight two-
node periodic switching networks with $\ell=2$, $T=7$, and $\tau_{1}=5$. The
eight networks used are the same as those used in Fig. 6. We recall that the
Parrondo’s paradox behavior occurs for the first four networks in a range of
$\beta$ values, i.e., shaded regions, and it does not occur for the last four
networks. We compare $q$ with $\lambda_{\text{F}}$,
$\lambda_{\max}(\textbf{M}^{(1)})$, and $\lambda_{\max}(\textbf{M}^{(2)})$ as
a function of the infection rate, $\beta$, for each periodic switching
network. We set $\mu=0.5$. The lines for $\lambda_{\text{F}}$,
$\lambda_{\max}(\textbf{M}^{(1)})$, and $\lambda_{\max}(\textbf{M}^{(2)})$ are
the same as those shown in Fig. 6. Observe that there is a strong association
between the Parrondo’s paradox behavior and the appearance of large $q$
values.

## Acknowledgments

D.T. and N.M. acknowledge support from National Science Foundation (under
Grant No. 2052720). D.T. also acknowledges NSF award DMS-2401276 and Simon’s
Foundation Grant 587333. N.M. also acknowledges support from the Japan Science
and Technology Agency (JST) Moonshot R&D (under Grant No. JPMJMS2021), the
National Science Foundation (under Grant No. 2204936), and JSPS KAKENHI (under
Grant No. JP 21H04595, 23H03414, 24K14840).

## References

  * [1] M. J. Keeling, K. T. Eames, Networks and epidemic models, Journal of the Royal Society Interface 2 (2005) 295–307.
  * [2] D. Easley, J. Kleinberg, Networks, Crowds, and Markets: Reasoning about a Highly Connected World, Cambridge University Press, Cambridge, UK, 2010.
  * [3] L. Danon, A. P. Ford, T. House, C. P. Jewell, M. J. Keeling, G. O. Roberts, J. V. Ross, M. C. Vernon, Networks and the epidemiology of infectious disease, Interdisciplinary Perspectives on Infectious Diseases 2011 (2011) 284909\.
  * [4] J. C. Miller, I. Z. Kiss, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena 9 (2014) 4–42.
  * [5] R. Pastor-Satorras, C. Castellano, P. V. Mieghem, A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics 87 (2015) 925.
  * [6] I. Z. Kiss, J. C. Miller, P. L. Simon, Mathematics of Epidemics on Networks, Springer, Cham, Switzerland, 2017.
  * [7] L. E. Rocha, Dynamics of air transport networks: A review from a complex systems perspective, Chinese Journal of Aeronautics 30 (2017) 469–478.
  * [8] K. Sugishita, N. Masuda, Recurrence in the evolution of air transport networks, Scientific Reports 11 (2021) 5514.
  * [9] S. Bansal, J. Read, B. Pourbohloul, L. A. Meyers, The dynamic nature of contact networks in infectious disease epidemiology, Journal of Biological Dynamics 4 (2010) 478–489.
  * [10] P. Holme, J. Saramäki, Temporal networks, Physics Reports 519 (2012) 97–125.
  * [11] B. Blonder, T. W. Wey, A. Dornhaus, R. James, A. Sih, Temporal dynamics and network analysis, Methods in Ecology and Evolution 3 (2012) 958–972.
  * [12] N. Masuda, P. Holme, Predicting and controlling infectious disease epidemics using temporal networks, F1000Prime Reports 5 (2013) 6.
  * [13] P. Holme, Modern temporal network theory: a colloquium, European Physical Journal B 88 (2015) 234.
  * [14] N. Masuda, P. Holme, Temporal Network Epidemiology, Springer, Berlin, Germany, 2017\.
  * [15] P. Holme, J. Saramäki, Temporal Network Theory, Springer, Cham, Switzerland, 2019.
  * [16] N. Masuda, R. Lambiotte, A Guide to Temporal Networks, World Scientific, Singapore, 2020.
  * [17] N. Masuda, J. C. Miller, P. Holme, Concurrency measures in the era of temporal network epidemiology: A review, Journal of the Royal Society Interface 18 (2021) 20210019.
  * [18] S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Discrete-time Markov chain approach to contact-based disease spreading in complex networks, EPL 89 (2010) 38009.
  * [19] E. Valdano, L. Ferreri, C. Poletto, V. Colizza, Analytical computation of the epidemic threshold on temporal networks, Physical Review X 5 (2015) 021005.
  * [20] L. Speidel, K. Klemm, V. M. Eguíluz, N. Masuda, Temporal interactions facilitate endemicity in the susceptible-infected-susceptible epidemic model, New Journal of Physics 18 (2016) 073013.
  * [21] T. Onaga, J. P. Gleeson, N. Masuda, Concurrency-induced transitions in epidemic dynamics on temporal networks, Physical Review Letters 119 (2017) 108301.
  * [22] L. Zino, M. Cao, Analysis, Prediction, and Control of Epidemics: A survey from scalar to dynamic network models, IEEE Circuits and Systems Magazine 21 (2021) 4–23.
  * [23] V. L. Somers, I. R. Manchester, Sparse resource allocation for spreading processes on temporal-switching networks, IFAC-PapersOnLine 56 (2023) 7387–7393.
  * [24] A. J. Allen, C. Moore, L. Hébert-Dufresne, Compressing the chronology of a temporal network with graph commutators, Physical Review Letters 132 (2024) 077402\.
  * [25] R. Olfati-Saber, Evolutionary dynamics of behavior in social networks, 46th IEEE Conference on Decision and Control (2007) 4051–4056.
  * [26] Á. Kun, I. Scheuring, Evolution of cooperation on dynamical graphs, Biosystems 96 (2009) 65–68.
  * [27] A. Li, L. Zhou, Q. Su, S. P. Cornelius, Y.-Y. Liu, L. Wang, S. A. Levin, Evolution of cooperation on temporal networks, Nature Communications 11 (2020) 2259.
  * [28] Y. Guan, A. Li, L. Wang, Structural controllability of directed signed networks, IEEE Transactions on Control of Network Systems 8 (2021) 1189–1200.
  * [29] A. Sheng, A. Li, L. Wang, Evolutionary dynamics on sequential temporal networks, PLOS Computational Biology 19 (2023) 1011333.
  * [30] Q. Su, A. McAvoy, J. B. Plotkin, Strategy evolution on dynamic networks, Nature Computational Science 3 (2023) 763–776.
  * [31] J. Bhaumik, N. Masuda, Fixation probability in evolutionary dynamics on switching temporal networks, Journal of Mathematical Biology 87 (2023) 64.
  * [32] I. Belykh, M. Di Bernardo, J. Kurths, M. Porfiri, Evolving dynamical networks, Physica D 267 (2014) 1–6.
  * [33] A. Sheng, Q. Su, L. Wang, J. B. Plotkin, Strategy evolution on higher-order networks, Nature Computational Science 4 (2024) 274–284.
  * [34] R. Edwards, Analysis of continuous-time switching networks, Physica D 146 (2000) 165–199.
  * [35] D. J. Jenkins, B. Finkenstädt, D. A. Rand, A temporal switch model for estimating transcriptional activity in gene expression, Bioinformatics 29 (2013) 1158–1165.
  * [36] B. Hou, X. Li, G. Chen, Structural controllability of temporally switching networks, IEEE Transactions on Circuits and Systems I 63 (2016) 1771–1781.
  * [37] A. Li, S. P. Cornelius, Y.-Y. Liu, L. Wang, A.-L. Barabási, The fundamental advantages of temporal networks, Science 358 (2017) 1042–1046.
  * [38] Y. Guan, L. Tian, L. Wang, Controllability of switching signed networks, IEEE Transactions on Circuits and Systems II 67 (2019) 1059–1063.
  * [39] B. Hou, Time parameters shape the controllability of temporally switching networks, IEEE Transactions on Automatic Control 68 (2022) 2064–2078.
  * [40] N. Perra, A. Baronchelli, D. Mocanu, B. Gonçalves, R. Pastor-Satorras, A. Vespignani, Random walks and search in time-varying networks, Physical Review Letters 109 (2012) 238701.
  * [41] L. E. Rocha, N. Masuda, Random walk centrality for temporal networks, New Journal of Physics 16 (2014) 063023.
  * [42] J. Petit, R. Lambiotte, T. Carletti, Classes of random walks on temporal networks with competing timescales, Applied Network Science 4 (2019) 72.
  * [43] G. P. Harmer, D. Abbott, Game theory: Losing strategies can win by Parrondo’s paradox, Nature 402 (1999) 864.
  * [44] J. M. Parrondo, G. P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Physical Review Letters 85 (2000) 5226–5229.
  * [45] G. P. Harmer, D. Abbott, A review of Parrondo’s paradox, Fluctuation and Noise Letters 2 (2002) R71–R107.
  * [46] J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order?, Physica D 200 (2005) 124–132.
  * [47] J. Cánovas, A. Linero, D. Peralta-Salas, Dynamic Parrondo’s paradox, Physica D 218 (2006) 177–184.
  * [48] J. S. Canovas, M. Munoz, Revisiting Parrondo’s paradox for the logistic family, Fluctuation and Noise Letters 12 (2013) 1350015.
  * [49] M.-F. Danca, M. Fečkan, M. Romera, Generalized form of Parrondo’s paradoxical game with applications to chaos control, International Journal of Bifurcation and Chaos 24 (2014) 1450008.
  * [50] M.-F. Danca, W. K. Tang, Parrondo’s paradox for chaos control and anticontrol of fractional-order systems, Chinese Physics B 25 (2015) 010505.
  * [51] S. A. Mendoza, E. W. Matt, D. R. Guimarães-Blandón, E. Peacock-López, Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems, Chaos, Solitons & Fractals 106 (2018) 86–93.
  * [52] A. Cima, A. Gasull, V. Mañosa, A dynamic Parrondo’s paradox for continuous seasonal systems, Nonlinear Dynamics 102 (2020) 1033–1043.
  * [53] Y. Ye, K. H. Cheong, Y.-w. Cen, N.-g. Xie, Effects of behavioral patterns and network topology structures on Parrondo’s paradox, Scientific Reports 6 (2016) 37028.
  * [54] J. W. Lai, K. H. Cheong, Social dynamics and Parrondo’s paradox: a narrative review, Nonlinear Dynamics 101 (2020) 1–20.
  * [55] Y. Ye, X.-R. Hang, L. Liu, L. Wang, N.-g. Xie, The impact of network degree correlation on Parrondo’s paradox, Complex Networks and Their Applications VIII (2020) 483–494.
  * [56] Y. Ye, X.-s. Zhang, L. Liu, N.-G. Xie, Effects of group interactions on the network Parrondo’s games, Physica A 583 (2021) 126271.
  * [57] Y. Ye, N.-g. Xie, L. Wang, Y.-w. Cen, The multi-agent Parrondo’s model based on the network evolution, Physica A 392 (2013) 5414–5421.
  * [58] R. Spurgin, M. Tamarkin, Switching investments can be a bad idea when Parrondo’s paradox applies, Journal of Behavioral Finance 6 (2005) 15–18.
  * [59] A. Chakrabarti, Switching as an investment strategy: Revisiting Parrondo’s paradox, Analytical Issues in Trade, Development and Finance (2014) 231–239.
  * [60] D. C. Osipovitch, C. Barratt, P. M. Schwartz, Systems chemistry and Parrondo’s paradox: computational models of thermal cycling, New Journal of Chemistry 33 (2009) 2022–2027.
  * [61] D. Heath, D. Kinderlehrer, M. Kowalczyk, Discrete and continuous ratchets: from coin toss to molecular motor, Discrete and Continuous Dynamical Systems Series-B 2 (2002) 153–168.
  * [62] F. A. Reed, Two-locus epistasis with sexually antagonistic selection: A genetic Parrondo’s paradox, Genetics 176 (2007) 1923–1929.
  * [63] D. Abbott, Asymmetry and disorder: A decade of Parrondo’s paradox, Fluctuation and Noise Letters 9 (2010) 129–156.
  * [64] K. H. Cheong, J. M. Koh, M. C. Jones, Paradoxical survival: Examining the Parrondo effect across biology, Bioessays 41 (2019) 1900027.
  * [65] J.-P. Capp, A. M. Nedelcu, A. M. Dujon, B. Roche, F. Catania, B. Ujvari, C. Alix-Panabières, F. Thomas, Does cancer biology rely on Parrondo’s principles?, Cancers 13 (2021) 2197.
  * [66] K. H. Cheong, J. M. Koh, M. C. Jones, Multicellular survival as a consequence of Parrondo’s paradox, Proceedings of the National Academy of Sciences of the United States of America 115 (2018) E5258–E5259.
  * [67] N. Masuda, N. Konno, Subcritical behavior in the alternating supercritical Domany-Kinzel dynamics, European Physical Journal B 40 (2004) 313–319.
  * [68] M.-F. Danca, D. Lai, Parrondo’s game model to find numerically stable attractors of a tumor growth model, International Journal of Bifurcation and Chaos 22 (2012) 1250258.
  * [69] K. H. Cheong, T. Wen, J. W. Lai, Relieving cost of epidemic by Parrondo’s paradox: A COVID-19 case study, Advanced Science 7 (2020) 2002324.
  * [70] S. E. Page, The many model approach, in: G. Manzo (Ed.), Research Handbook on Analytical Sociology, Edward Elgar Publishing, Northampton, MA, USA, 2021, Ch. 25, pp. 474–489.
  * [71] M.-F. Danca, Controlling the dynamics of a COVID-19 mathematical model using a parameter switching algorithm, Mathematical Methods in the Applied Sciences 46 (2023) 8746–8758.
  * [72] M. I. Sejunti, Parrondo’s paradox in the susceptible-infectious-susceptible (SIS) model on temporal networks, github repository, https://github.com/Sejunti/Parrondos_paradox_in_the_susceptible-infectious-susceptible_SIS_model_on_temporal_networks, accessed: 12-05-2022.
  * [73] P. V. Mieghem, J. Omic, R. Kooij, Virus spread in networks, IEEE/ACM Transactions on Networking 17 (2008) 1–14.
  * [74] P. V. Mieghem, Performance Analysis of Communications Networks and Systems, Cambridge University Press, Cambridge, UK, 2006.
  * [75] P. V. Mieghem, Epidemic phase transition of the SIS type in networks, EPL 97 (2012) 48004.
  * [76] G. F. D. Duff, D. Naylor, D. A. McQuarrie, Differential equations of applied mathematics, American Journal of Physics 34 (1966) 1205–1206.
  * [77] A. Ganesh, L. Massoulié, D. Towsley, The effect of network topology on the spread of epidemics, Proceedings of the 24th IEEE Conference on Computer Communications 2 (2005) 1455–1466.
  * [78] M. Ogura, V. M. Preciado, Stability of spreading processes over time-varying large-scale networks, IEEE Transactions on Network Science and Engineering 3 (2016) 44–57.
  * [79] B. C. Hall, B. C. Hall, Lie groups, Lie algebras, and representations, Springer, Cham, Switzerland, 2013.
  * [80] C. D. Meyer, Matrix analysis and applied linear algebra, SIAM, PA, USA, 2023.
  * [81] E. Valdano, M. R. Fiorentin, C. Poletto, V. Colizza, Epidemic threshold in continuous-time evolving networks, Physical Review Letters 120 (2018) 068302.
  * [82] N. Masuda, C. L. Vestergaard, Gillespie Algorithms for Stochastic Multiagent Dynamics in Populations and Networks, Cambridge University Press, Cambridge, UK, 2022.
  * [83] E. Abbe, Community detection and stochastic block models: Recent developments, Journal of Machine Learning Research 18 (2018) 6446–6531.
  * [84] C. Lee, D. J. Wilkinson, A review of stochastic block models and extensions for graph clustering, Applied Network Science 4 (2019) 122.
  * [85] K. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, Hoboken, NJ, USA, 1991.
  * [86] S. Mauras, V. Cohen-Addad, G. Duboc, M. Dupré la Tour, P. Frasca, C. Mathieu, L. Opatowski, L. Viennot, Mitigating COVID-19 outbreaks in workplaces and schools by hybrid telecommuting, PLOS Computational Biology 17 (2021) e1009264.
  * [87] O. Karin, Y. Bar-On, T. Milo, I. Katzir, A. Mayo, Y. Korem, et al., Cyclic exit strategies to suppress COVID-19 and allow economic activity. medRxiv doi: https://doi.org/10.1101/2020.04.04.20053579 (2020).
  * [88] H. Kaygusuz, A. N. Berker, The effect of weekend curfews on epidemics: a Monte Carlo simulation, Turkish Journal of Biology 45 (2021) 436–441.
  * [89] D. Meidan, N. Schulmann, R. Cohen, S. Haber, E. Yaniv, R. Sarid, B. Barzel, Alternating quarantine for sustainable epidemic mitigation, Nature Communications 12 (2021) 220.
  * [90] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Annales Scientifiques de l’École Normale Supérieure 12 (1883) 47–88.
  * [91] C. Gökçek, Stability analysis of periodically switched linear systems using Floquet theory, Mathematical Problems in Engineering 2004 (2004) 521989\.