# Modeling and Controlling the Spread of Epidemic with
Various Social and Economic Scenarios

S. P. Lukyanets<EMAIL_ADDRESS>I. S. Gandzha<EMAIL_ADDRESS>O. V.
Kliushnychenko<EMAIL_ADDRESS>Institute of Physics, National
Academy of Sciences of Ukraine, Prosp. Nauky 46, Kyiv 03028, Ukraine

(September 3, 2024)

###### Abstract

We propose a dynamical model for describing the spread of epidemics. This
model is an extension of the SIQR (susceptible-infected-quarantined-recovered)
and SIRP (susceptible-infected-recovered-pathogen) models used earlier to
describe various scenarios of epidemic spreading. As compared to the basic SIR
model, our model takes into account two possible routes of contagion
transmission: direct from the infected compartment to the susceptible
compartment and indirect via some intermediate medium or fomites. Transmission
rates are estimated in terms of average distances between the individuals in
selected social environments and characteristic time spans for which the
individuals stay in each of these environments. We also introduce a collective
economic resource associated with the average amount of money or income per
individual to describe the socioeconomic interplay between the spreading
process and the resource available to infected individuals. The epidemic-
resource coupling is supposed to be of activation type, with the recovery rate
governed by the Arrhenius-like law. Our model brings an advantage of building
various control strategies to mitigate the effect of epidemic and can be
applied, in particular, to modeling the spread of COVID-19.

###### keywords:

Spreading process , Epidemic , SIR model , COVID-19 , Economic resource ,
Arrhenius law

##  1 Introduction

The spread of contagions, deceases, information, rumors, ideas, or concepts
share many similarities. In most cases, such spreading processes are governed
by similar models [1, 5, 6, 4, 3, 2], which can vary in terms of complexity or
approximations. The simplest dynamical models (see, e.g., Refs. [7, 8, 9, 10,
11, 12, 13, 14]) have their origin in classical works [15, 16, 17, 18, 19].
More complex models include stochastic effects [20, 21, 22], spatial flows
(e.g., diffusion [23, 24]), or allow for nontrivial spatial structure or
topology [25, 26, 27, 28, 29, 30]. One common feature in the majority of these
models is the presence of kinetic coefficients or parameters that characterize
the probability of elementary processes (reactions) per unit time.

For instance, contagion, infection, or decease spreading is associated with
scattering of infected individuals on the spatiotemporal fluctuations of
population density. The probability of being infected is not a monotonous
function of population density [31, 32]. The probability of scattering of
infected individuals on susceptible individuals is lower at low densities,
which results in a smaller total scattering cross-section. At higher
densities, the mobility of spreaders decreases suppressing the spreading
process. Moreover, the scattering of spreaders on local population-density
fluctuations should determine the critical concentration of secondary infected
cases sufficient for the initiation of the collective process, i.e., an
epidemic. The kinetic description of spreading is generally quite a
complicated problem that should take account of the internal state of
spreaders, which changes in the course of collisions, the presence of spatial
inhomogeneities, and non-equilibrium properties of the system.

The dynamics of any spreading process such as an epidemic is determined by
individual peculiarities of people in a selected social group, by the type and
mechanism of infection, etc. It can also be affected by the ways the process
is controlled and influenced. A dramatic example is the spread of COVID-19
(SARS-CoV-2 coronavirus), when different countries and governments resorted to
different control strategies and quarantine measures [33, 34]. Such measures
can be controlled by the choice of certain parameters accessible to society
(e.g., working day duration, average density of people in public places,
frequency of disinfection, intensity of transport communications, etc.). The
problem of strategy selection reduces to problems of optimal control theory
for feedback systems [35, 36, 37] or to the theory of games in a more general
case [38].

On the other hand, kinetic coefficients or probabilities are not the proper
control parameters that are readily available, and they can only be estimated
indirectly in terms of other parameters. For example, the recovery rate is
generally determined by the quality of provision with medical services and
food, apart from the individual peculiarities of the given member of
population (see, e.g., Refs. [39, 40, 41, 42, 43, 44]). The quickest recovery
depends on the cost of medical services and the bare subsistence level of
consumption $E$, as well as on the availability of collective economic
resource $\rho$. Since the cost of services is fixed, the service is
terminated if there is no sufficient resource ($\rho\ll E$). In other words,
the parameter $E$ serves as the height of some energy barrier peculiar to the
given system. Therefore, the recovery rate can be supposed to have an
activation-type (Arrhenius-like) dependence, $\propto\exp(-E/\rho)$, similar
to the temperature dependence of common activation processes with activation
energy $E$ [45, 46].

In this case, the economic resource $\rho$ formally plays the role of
effective market temperature and the minimum level of resource consumption is
associated with activation energy $E$ [47]. This observation follows from the
fact the effective market temperature can be associated with the average
amount of money or average income per individual (or economic agent) and the
equilibrium distribution of income or money for single economic agents is
governed by the exponential Boltzmann law (at least for low and middle income
classes) [48, 49, 50, 51].

The activation process implies that the system can exhibit the so-called
explosive (or catastrophic) instability [52]. For example, when a chemical
reaction occurs with the release of heat and has an activation character, it
goes faster at higher temperatures. This leads to yet greater temperature
increase and ultimately to a thermal explosion, which is described in the
framework of the Zel’dovich-Frank-Kamenetskii theory [53, 54, 55].

The fight against the epidemic involves similar catastrophic processes. The
spread of epidemic and the associated quarantine measures result in the
reduction of the collective resource $\rho$. When the resource is depleted,
the quality of medical services drops and the recovery rate goes down. As a
result, the number of active members in population decreases. This, in turn,
leads to a further reduction of the collective resource, with the level of
income needed for the basic survival being lower and lower. Such a scenario
can finally result in the ultimate collapse of the system—the effect opposite
to thermal explosion [47].

In this paper, we address some of the points mentioned above. In other words,
we are interested not only in modeling the spread of epidemic as an example of
a spreading process but also in identifying the ways it can be controlled by
socially available parameters and which consequences such control measures can
lead to. To demonstrate a number of possible effects, we turn to a quite
simple dynamical feedback model. This model has the following two important
features.

(I) To take into account the fact that an individual can stay in different
environments or public places (e.g., transport, shop, work) characterized by
different densities of surrounding individuals, it is convenient to consider
some social group and its average daily cycle ($T$). For instance, we can
introduce the average daily time spans for which the individual stays at home
($T_{1}$), at shop or other public places ($T_{2}$), in transport ($T_{3}$),
at work ($T_{4}$), etc. and define (at least under normal conditions) the
characteristic density $c_{j}$ of individuals or the average distance
$\overline{\ell_{j}}$ between them in each of the locations $j$. We also make
a qualitative assessment of the dependence of the transmission rate on the
density of individuals or average distance between them in order to clearly
identify the set of control parameters $\\{T_{j},\,\overline{\ell_{j}}\\}$.
Finally, we take into consideration a possibility of indirect transmission
through the medium (e.g., contaminated water [56, 57]) or via the so-called
fomites [58, 59]. The indirect transmission rate depends on the same set of
control parameters $\\{T_{j},\,\overline{\ell_{j}}\\}$.

(II) The interplay between the epidemic and the economic resource available
for the selected social group is taken into account by means of the
activation-type dependence of characteristic recovery rates. To this end, we
use the simplest form of the equation for the dynamics of the collective
resource $\rho$.

This paper is organized as follows. In Section 2, we estimate the direct and
indirect transmission rates of contagion in terms of basic social control
parameters such as the average distance between the individuals and the
average time spent in a certain location. In Section 3, we consider an
extended suspected-infected-quarantine-recovered (SIR/SIQR) model that takes
into consideration the above-discussed points (I) and (II). In Section 4, we
provide several examples of numerical modeling. In particular, we demonstrate
the effect of indirect transmission, the effect of quarantine measures, and
the effect of limited resource. Conclusions are drawn in Section 5.

##  2 Direct and indirect transmission rates

As mentioned above, the spread of contagion such as infection can be
controlled only by the parameters accessible to society. For example, the
average distance $\overline{\ell_{j}}$ between the individuals and the average
time $T_{j}$ spent in a certain location $j$ can be such control parameters.
Here we estimate the infection transmission rates in terms of these parameters
in the case of direct transmission (infected-to-susceptible) and indirect
transmission (through the environment).

Table 1: Social control parameters and transmission rates for $\ell_{0}=1$ m, $\ell_{c}=4$ m, and $\tau_{is}=T/3$ ($T=1$ day). Location | Description |  $T_{j}$, h |  $\overline{\ell_{j}}$, m | $\omega_{j}$ |  $T_{j}$, h |  $\overline{\ell_{j}}$, m | $\omega_{j}$ |  $T_{j}$, h |  $\overline{\ell_{j}}$, m | $\omega_{j}$
---|---|---|---|---|---|---|---|---|---|---
$j=1$ | Home | 11.5 | 3.5 | $0.06\,T^{-1}$ | 12.5 | 3.5 | $0.06\,T^{-1}$ | 19.5 | 3.5 | $0.06\,T^{-1}$
$j=2$ | Shopping | 1.5 | 1.5 | $1.15\,T^{-1}$ | 1.5 | 2.25 | $0.41\,T^{-1}$ | 0.5 | 2.25 | $0.41\,T^{-1}$
$j=3$ | Transport | 3 | 1 | $2.81\,T^{-1}$ | 2 | 1.5 | $1.15\,T^{-1}$ | 0 |  |
$j=4$ | Work | 8 | 3 | $0.15\,T^{-1}$ | 8 | 3.25 | $0.10\,T^{-1}$ | 4 | 3.25 | $0.10\,T^{-1}$
| $\beta$ | Casual | $0.5\,T^{-1}$ | Soft quarantine | $0.18\,T^{-1}$ | Strict quarantine | $0.07\,T^{-1}$

###  2.1 Direct transmission

It can naturally be assumed that the probability of being infected, which
generally depends on the distance between the infected and susceptible
individuals, is described by a damped function with a characteristic
correlation radius $\ell_{c}$ (decay length). For simplicity, we suppose that
the susceptible individual and the infected individual can interact only when
they fall into the vicinity of radius $\ell_{c}$ with each other and that the
probability of elementary transmission per unit time in this case does not
depend on distance and equals $\omega$. We also assume that for any
susceptible individual the transmissions from two different infected
individuals are independent events. Note that in our simplified consideration
the correlation radius $\ell_{c}$ can be associated with a socially safe
distance.

Next we suppose that for each social location $j$ the average population
density $\overline{c_{j}}$ is not affected by any social conditions and is
solely determined by the average distance between individuals,
$\overline{c_{j}}\approx(\sqrt{\pi}\overline{\ell_{j}})^{-2}$. The average
number of individuals falling into the circle of radius $\ell_{c}$ around the
susceptible individual in each of the locations $j$ is
$\overline{N}_{j}\approx\overline{c_{j}}\pi\ell_{c}^{2}$. Then the probability
for the susceptible individual to be infected over time interval $\Delta t$
can be estimated as $\Delta P_{j}=\omega\kappa(\overline{N_{j}}-1)\Delta t$.
Here $\kappa\approx i=I/N$ is the probability that the given individual in the
vicinity of the susceptible individual is infected, $N$ being the total number
of individuals, $I$ being the total number of infected individuals, and $i$
being the number density of infected individuals. Summing over all the
susceptible individuals $S$, we obtain an increment of the number of infected
individuals, $\Delta I=S\Delta P_{j}$. Dividing by the total population $N$
and the time interval $\Delta t$, we get a characteristic growth rate of the
number density of infected individuals:

$\frac{\Delta i}{\Delta t}\approx\omega(\overline{N_{j}}-1)is,$ (1)

where $s=S/N$ is the number density of susceptible individuals. Thus, the
direct (infected-to-susceptible) transmission rate of infection in the
selected area with population density $\overline{c_{j}}$ is estimated as

$\omega_{j}=\omega(\overline{N_{j}}-1)\approx\omega\left(\left(\frac{\ell_{c}}{\overline{\ell_{j}}}\right)^{2}-1\right).$

The probability $\omega$ of elementary transmission per unit time can be
estimated in terms of the minimum possible distance $\ell_{0}$ between two
individuals (the so-called close contact distance). Assume that the suspected
individual is infected with probability equal to unity if he stays at the
distance $\ell_{0}$ from the infected individual for some characteristic time
$\tau_{is}$. Then the probability of becoming infected at the minimum distance
per unit time is proportional to $\tau_{is}^{-1}$, so that
$\omega=a\,\tau_{is}^{-1}$. The parameter $a$ can be estimated as the ratio of
the contact area between the susceptible individual and the infected
individual to the total area determined by the correlation radius $\ell_{c}$:
$a\approx(\ell_{0}/\ell_{c})^{2}$. Thus, we have

$\omega_{j}\approx\tau_{is}^{-1}\left(\frac{\ell_{0}}{\ell_{c}}\right)^{2}\left(\left(\frac{\ell_{c}}{\overline{\ell_{j}}}\right)^{2}-1\right).$
(2)

Let us assume that each individual can stay in $N_{\omega}$ different
locations during some typical period $T$ (e.g., one day). Each location is
characterized by its own average distance $\overline{\ell_{j}}$. Then the
integral transmission rate $\beta$ for all locations is given by the following
formula:

$\beta=\sum_{j=1}^{N_{\omega}}\omega_{j}\frac{T_{j}}{T},$ (3)

where $T_{j}$ are the characteristic daily time spans spent in each of the
locations $j$, with

$\sum_{j=1}^{N_{\omega}}T_{j}=T.$ (4)

As an example, consider four different social locations ($N_{\omega}=4$).
Location 1 refers to staying at home (limited social contacts), location 2
refers to shopping and other social contacts during a day, location 3 refers
to staying in public transport (where the transmission probability is the
highest), location 4 refers to staying at work. Table 1 gives the transmission
rates $\omega_{j}$ and $\beta$ calculated for various sets of social control
parameters $T_{j}$ and $\overline{\ell_{j}}$. We considered three possible
scenarios: casual (no restrictions), soft quarantine (social distancing in
place), and strict quarantine (restricted public transport, limited social
contacts, partial cutting of economic activity). The soft quarantine measures
reduce the integral transmission rate $\beta$ by a factor of 3, and the strict
quarantine measures reduce it by a factor of 7. It can be seen that our rough
estimates given by formulas (2) and (3) allow the transmission rates to be
controlled by the proper choice of parameters $T_{j}$ and
$\overline{\ell_{j}}$ in different social scenarios. Our $\beta$ estimates in
the case of casual scenario fall in the range ($\approx\,$$0.6\,T^{-1}$)
derived from the statistical data for the COVID-19 epidemic in the Wuhan city
[60].

Note that the $\omega_{j}$ estimate given by formula (2) is very rough. In
fact, the transmission rate depends on fluctuations in the number of
individuals falling into the circle of radius $\ell_{c}$. The maximum number
$N_{m}$ of individuals that can fall into the circle of radius $\ell_{c}$ is
determined by the closest packing of the region $r<\ell_{c}$ by hard spheres
with radius equal to the the minimum possible distance $\ell_{0}$ between two
individuals. For example, considering a triangular lattice with lattice
constant $\ell_{0}$, we have

$N_{m}\approx
1+6\left(\frac{\ell_{c}}{\ell_{0}}\right)\left(\frac{\ell_{c}}{\ell_{0}}+1\right).$

Let $\vartheta$ be the probability that the lattice node is occupied. Then the
average number of individuals falling into the region $r<\ell_{c}$ is
$\overline{N}=\vartheta N_{m}$, and its mean-square fluctuation is estimated
as

$\overline{\Delta N^{2}}=\overline{N^{2}}-\overline{N}^{2}=\vartheta
N_{m}+\vartheta^{2}N_{m}(N_{m}-1)\approx\vartheta N_{m}\approx\overline{N}.$

Finally, the relative error introduced by such fluctuations in the
$\omega_{j}$ estimate can therefore be estimated as

$\frac{\delta\omega_{j}}{\omega_{j}}=\frac{\sqrt{\overline{\Delta
N^{2}_{j}}}}{\overline{N_{j}}}\approx\overline{N_{j}}^{-\frac{1}{2}}\approx\frac{\overline{\ell_{j}}}{\ell_{c}}.$

It should also be noted that the characteristic scattering cross-section given
by relation (1) can have a more complex dependence on density,
$\sim\overline{c}^{\alpha}$ ($1\leq\alpha\leq 3$), and be governed by a
nonlinear incidence function other than $is$ [31, 61, 62, 63, 64, 32].

###  2.2 Indirect transmission

In addition to the direct transmission of infection by direct contact of
susceptible individuals with infected individuals, the infection caused by a
pathogen (e.g., virus) can be transmitted indirectly either through the medium
(e.g., contaminated water [56, 57]) or via intermediate objects (like hands,
counters, doorknobs, etc.) often called fomites [58, 59]. Such an intermediate
medium or fomites will further be referred to as ‘‘cloud’’. We associate a
separate cloud with each social location $j$.

Let $d_{j}$ be the pathogen density per individual in the cloud $j$. Then a
characteristic growth rate of the number density of infected individuals
attributed to the indirect transmission of the infection from the cloud $j$ to
susceptible individuals can be calculated in the same way as in the case of
direct transmission, namely

$\frac{\Delta i}{\Delta t}\approx\Omega_{j}d_{j}s.$ (5)

Here the parameter $\Omega_{j}$ describes the typical (indirect) rate of
pathogen transmission from the cloud $j$ to a susceptible individual. The
estimates of indirect transmission rates for various sets of control
parameters $T_{j}$ and $\overline{\ell_{j}}$ are provided in A.

##  3 Dynamical model

###  3.1 Mathematical formulation

Figure 1: A schematic diagram of the dynamical model given by Eqs. (6) that
depicts transitions between different compartments. The estimates of the
transition rate constants $\omega_{j}$, $\Omega_{j}$, $\Gamma_{is}$,
$\Gamma_{ir}$, $\Gamma_{iq}$, $\Gamma_{qs}$, $\Gamma_{qr}$, and $\Gamma_{qf}$
between the compartment levels are given in Table 1.

We subdivide the selected social group into five compartments: susceptible
(S), infected (I), quarantined (Q), recovered (R), and deceased (F).
Susceptible individuals (S) are those who are at risk to be infected. Infected
individuals (I) are those who have been infected and pass through the virus
incubation period. Then they either recover (with or without the acquired
immunity) or pass to the severe form of decease (like fever or organ
disfunction) when they need to stay isolated either at home or at hospital
getting a medical help. Such individuals are referred to as quarantined
individuals (Q). The quarantined individuals either recover (with or without
the acquired immunity) or die. The corresponding mathematical model is given
by the following system of ODEs:

$\displaystyle\partial_{t}s$
$\displaystyle=-s\sum_{j}\Bigl{(}\omega_{j}\,i+\Omega_{j}\,d_{j}\Bigr{)}\frac{T_{j}}{T}+\Gamma_{is}\,i+\Gamma_{qs}\,\mathrm{e}^{-E/\rho}\,q,$
(6a) $\displaystyle\partial_{t}i$
$\displaystyle=s\sum_{j}\Bigl{(}\omega_{j}\,i+\Omega_{j}\,d_{j}\Bigr{)}\frac{T_{j}}{T}-\Gamma_{i}\,i,$
(6b) $\displaystyle\partial_{t}q$
$\displaystyle=\Gamma_{iq}\,i-\Gamma_{q}\,q,$ (6c)
$\displaystyle\partial_{t}r$
$\displaystyle=\Gamma_{ir}\,i+\Gamma_{qr}\,\mathrm{e}^{-E/\rho}\,q,$ (6d)
where the operator $\partial_{t}$ stands for the derivative with respect to
time $t$. The functions $s(t)$, $i(t)$, $q(t)$, and $r(t)$ describe the number
densities of the susceptible, infected, quarantined, and recovered
compartments, respectively. Here the index $j=1,\ldots,N_{\omega}$ runs
through all the social locations, and $N_{\omega}$ is the total number of such
locations taken into consideration (see Section 2). The direct transmission
rates $\omega_{j}$ in each of the locations $j$ are defined by formula (2).
The characteristic daily time spans $T_{j}$ in each of the locations $j$ are
given in Table 1, the total time spent in all the locations being constant
each day and equal to the day duration $T$ [see formula (4)].

Each of the functions $d_{j}(t)$ describes the pathogen density per individual
in the cloud $j$ and contributes to the indirect transmission of the infection
from the cloud to susceptible individuals. The pathogen dynamics in the cloud
$j$ is given by the following equation:

$\partial_{t}d_{j}=\sigma_{j}\,i-\gamma_{j}\,d_{j}.\\\ $ (6e)

The first term, $\sigma_{j}\,i$, describes the pathogen shedding by the
infected individuals into the cloud, and the second term, $\gamma_{j}\,d_{j}$,
describes the pathogen decay in the cloud due to natural inactivation,
decontamination, or other routes.

The function $\rho(t)$ represents the average resource associated with the
average amount of money or income per individual in the selected social group.
The resource balance equation is written as follows [47]

$\partial_{t}\rho=G(s+i+r)-\Gamma_{\rho\,}\rho+\Lambda.$ (6f)

The acquisition of this resource per unit time is proportional to the number
density of working (active) individuals (the quarantined individuals are
assumed to be not working). The parameter $G$ formalizes the resource amount
acquired by them per unit time. In line with Ref. [65], we refer to this
parameter as acquisition rate. It is proportional to the average working time
$T_{4}$ for the given social group (see Table 1). The second term,
$\Gamma_{\rho}\,\rho$, formally describes the collective expenses or taxes.
Roughly speaking, the expenses are assumed to be proportional to earnings.
Thus, the coefficient $\Gamma_{\rho}$ represents the resource consumption
rate. The parameter $\Lambda$ represents a resource source (constant resource
inflow into the system from some external reservoir) or a resource sink
(constant resource outflow from the system). When $\Lambda>0$, resource is fed
into the system (e.g., in the form of subsidies) from some external source,
e.g., a central bank or central government. When $\Lambda<0$, resource flows
out from the system, e.g., in the form of infrastructure expenses,
depreciation, rent, interest payments, or other fixed expenses.

The recovery process is governed by the general economic situation
characterized by a certain minimum level of resource consumption $E$, which
reflects the cost of medical and other essential life services. It defines the
minimum amount of the consumed resource needed for the recovery of individuals
in the selected social group, therefore implying the existence of some
‘‘energy’’ barrier for their recovery. The recovery rates, i.e., the
coefficients at the function $q(t)$ in Eqs. (6a) and (6d), are supposed to
have an activation-type (Arrhenius-like) dependence, $\propto\exp(-E/\rho)$,
similar to the temperature dependence of common activation processes with
activation energy $E$.

The number density of fatal cases is described by the function $f(t)$ that is
determined by the equation

$\partial_{t}f=\left(\Gamma_{q}+(\Gamma_{qf}-\Gamma_{q})\,\mathrm{e}^{-E/\rho}\right)q.$
(6g)

The above equations are supplemented with the following initial conditions

$\begin{split}s(0)&=1-i_{0},\;i(0)=i_{0},\\\
q(0)&=r(0)=f(0)=d_{j}(0)=0,\;\rho(0)=\rho_{0},\end{split}$ (6h)

$i_{0}$ being the initial number density of infected individuals and
$\rho_{0}$ being the initial resource value.

The total number density of individuals is assumed to be constant:

$s(t)+i(t)+q(t)+r(t)+f(t)=1.$ (7)

Effectively, our model is a combination or extension of SIR-like (susceptible-
infected-recovered) epidemic models such as SIQR (susceptible-infected-
quarantined-recovered) [66], SIWR/SIVR/SIRP (susceptible-infected-recovered-
pathogen) [56, 67, 68, 69], SIRD (susceptible-infected-recovered-deceased)
[70, 71], and EITS (environmental infection transmission system) [58].

The extension to multiple groups is given in B.

###  3.2 Assumptions

  1. 1.

The population subsystem is closed (no migration outside the selected
population group and no one is added to the population).

  2. 2.

Natural demography is ignored.

  3. 3.

A uniform spatial distribution of people is assumed. The pathogen distribution
in each cloud is assumed to be uniform.

  4. 4.

The latent period from exposure to the onset of infectiousness is ignored,
i.e., all the exposed individuals are assumed to be infected and can infect
others. The SEIR (susceptible-exposed-infected-recovered) model was
demonstrated to have no practical advantage as compared to the SIR model [60].

  5. 5.

Quarantined individuals do not infect others.

  6. 6.

There is no pre-existing immunity in susceptible individuals.

  7. 7.

There is no loss of immunity by recovered individuals.

###  3.3 Parameters

The full description and indicative values of the parameters of Eqs. (6) are
given in Table 1 (see A). Figure 1 shows a schematic diagram that depicts
transitions between different compartments and identifies the corresponding
transition rates.

In particular, the parameter $\Gamma_{is}$ describes a rate at which the
infected individuals recover without acquiring the immunity and come back to
the susceptible compartment. The parameter $\Gamma_{iq}$ describes a rate at
which the infected individuals develop a severe condition and pass to the
quarantine compartment, where they become isolated either at home or at
hospital. The parameter $\Gamma_{ir}$ describes a rate at which the infected
individuals recover without complications and acquire the immunity. It is
proportional to the probability $\mu$ of acquiring the immunity (see Table 1).
The rate constant $\Gamma_{i}$ is defined as follows:

$\Gamma_{i}=\Gamma_{iq}+\Gamma_{ir}+\Gamma_{is}=\tau_{i}^{-1},$

where $\tau_{i}$ is the characteristic pathogen incubation period.

The parameters $\Gamma_{qr}$ and $\Gamma_{qs}$ describe the rates at which the
quarantined individuals recover with or without the acquired immunity. The
parameter $\Gamma_{qf}$ describes the fatality rate in the case of unlimited
resource ($E\ll\rho$). In this case, the quarantined individuals all get the
necessary medical care and the fatalities are only attributed to insuperable
health complications (such as concomitant diseases or age factor). In the
opposite case, when $E\gg\rho$ (no resource to fight against the epidemic),
the fatality rate is at its maximum and is equal to the rate constant

$\Gamma_{q}=\Gamma_{qr}+\Gamma_{qs}+\Gamma_{qf}=\tau_{q}^{-1}$

for the quarantined individuals. Here $\tau_{q}$ is the mean
quarantine/hospitalization period. This case refers to the full collapse of
the medical system when the quarantined individuals get no medical help or
treatment.

Each of the parameters $\Omega_{j}$ describes the typical rate of pathogen
transmission from the cloud $j$ to susceptible individuals. Two other cloud-
related parameters, $\sigma_{j}$ and $\gamma_{j}$, are the pathogen shedding
rate (infected-to-cloud) and decay rate in the cloud, respectively. Their
estimates are provided in A.

Note that the instantaneous number density of quarantined (ill) individuals
$q(t)$ is not often a convenient indicator for practical applications. The
integral number density of quarantined individuals for a certain period of
time can be used instead. It is calculated as follows

$q_{\Sigma}=\Gamma_{iq}\int_{0}^{t}i\,\mathrm{d}\tau.$ (8)

In the case of unlimited resource ($E=0$), the integral number density of
quarantine individuals in the end of epidemic ($t\rightarrow\infty$) is
proportional to the total number density of fatal cases,

$(q_{\Sigma})_{\infty}=\eta\,f_{\infty},$ (9)

where $\eta$ is the probability of the fatal scenario for a quarantined
individual.

###  3.4 Analysis

For our further analysis, we first find the stationary solution to Eq. (6e)
for the cloud $j$:

$d^{*}_{j}=\frac{\sigma_{j}}{\gamma_{j}}\,i^{*}.$ (10)

This relation allows us to introduce the dimensionless pathogen concentration
in the cloud $j$, namely

$p_{j}=\frac{\gamma_{j}}{\sigma_{j}}\,d_{j},$ (11)

so that $p^{*}_{j}=i^{*}$. Then Eqs. (6) can be rewritten as

$\displaystyle\partial_{t}s$
$\displaystyle=-\Bigl{(}\beta\,i+\sum_{j}\beta_{j}\,p_{j}\Bigr{)}s+\Gamma_{is}\,i+\Gamma_{qs}\,\mathrm{e}^{-E/\rho}\,q,$
$\displaystyle\partial_{t}i$
$\displaystyle=\Bigl{(}\beta\,i+\sum_{j}\beta_{j}\,p_{j}\Bigr{)}s-\Gamma_{i}\,i,$
(12) $\displaystyle\partial_{t}q$
$\displaystyle=\Gamma_{iq}\,i-\Gamma_{q}\,q,$ $\displaystyle\partial_{t}r$
$\displaystyle=\Gamma_{ir}\,i+\Gamma_{qr}\,\mathrm{e}^{-E/\rho}\,q,$
$\displaystyle\partial_{t}p_{j}$
$\displaystyle=\gamma_{j}\left(i-p_{j}\right).$

Here $\beta$ is the direct transmission rate (infected-to-susceptible) given
by formula (3). The parameters

$\beta_{j}=\nu_{j}\frac{T_{j}}{T}$ (13)

are defined in terms of the scaled indirect transmission rates (infected-
cloud-susceptible)

$\nu_{j}=\Omega_{j}\frac{\sigma_{j}}{\gamma_{j}}$ (14)

via each of the clouds $j$.

Figure 2: Effect of indirect transmission for fixed $\beta=0.5\,T^{-1}$ ($T=1$ day). (a) No cloud: $\beta_{p}=0$, $i_{0}=10^{-7}$ and $i_{0}=10^{-5}$. (b) With cloud: $\beta_{p}=0.2\,T^{-1}$, $i_{0}=10^{-5}$. (c) With cloud: $\beta_{p}=0.5\,T^{-1}$, $i_{0}=10^{-5}$. Table 2: Main parameters of numerical solutions shown in Figs. 2 and 4. $\beta$ | $\beta_{p}$ | $\mathcal{R}_{0}$ | $i_{0}$ | $i_{\max}$ |  $t_{i_{\max}}$, days | $q_{\max}$ |  $t_{q_{\max}}$, days | $(q_{\Sigma})_{\infty}$ | $s_{\infty}$
---|---|---|---|---|---|---|---|---|---
$0.5\,T^{-1}$ | 0 | 2.5 | $10^{-7}$ | 0.245 | 55.4 | 0.081 | 63.7 | 0.193 | 0.132
$0.5\,T^{-1}$ | 0 | 2.5 | $10^{-5}$ | 0.245 | 40.0 | 0.081 | 48.3 | 0.193 | 0.132
$0.5\,T^{-1}$ | $0.2\,T^{-1}$ | 3.5 | $10^{-5}$ | 0.337 | 30.5 | 0.097 | 38.2 | 0.210 | 0.059
$0.5\,T^{-1}$ | $0.5\,T^{-1}$ | 5 | $10^{-5}$ | 0.425 | 23.8 | 0.107 | 31.0 | 0.216 | 0.030
Quarantine scenario | $10^{-5}$ | 0.212 | 73.7 | 0.071 | 81.6 | 0.205 | 0.079

Equations (3.4) supplemented with Eqs. (6f) and (6g) possess two equilibrium
points. The first one is the disease-free equilibrium

$\displaystyle s^{(0)}=1,\;\rho^{(0)}=\frac{G+\Lambda}{\Gamma_{\rho}},$ (15)
$\displaystyle i^{(0)}=q^{(0)}=r^{(0)}=f^{(0)}=p_{j}^{(0)}=0.$

The second one is the endemic equilibrium

$s^{*}=\mathcal{R}_{0}^{-1},\quad i^{*}=q^{*}=p_{j}^{*}=0,$ (16)

where the parameter

$\mathcal{R}_{0}=\tau_{i}\sum_{j}\left(\omega_{j}+\nu_{j}\right)\frac{T_{j}}{T}=\frac{\beta+\beta_{p}}{\Gamma_{i}}$
(17)

is the basic reproduction number and

$\beta_{p}=\sum_{j}\beta_{j}$ (18)

is the integral indirect pathogen transmission rate via all the clouds.

In general, the basic reproduction number $\mathcal{R}_{0}$ defines the
average number of transmissions one infected individual makes in the entire
susceptible compartment during the entire time of being infected. When
$\mathcal{R}_{0}\leqslant 1$, the disease-free equilibrium is stable, and
there is no epidemic outbreak. When $\mathcal{R}_{0}>1$, the disease-free
equilibrium is unstable, and the system evolves to the state of endemic
equilibrium.

The first two equations of system (3.4) are strongly nonlinear. There are no
analytical solutions known for the general form of these equations. However,
in one particular case, when $\Gamma_{is}=\Gamma_{qs}=0$ ($\mu=1$, no loss of
immunity) and $E=0$ (unlimited resource), one can get the following
asymptotics at $t\rightarrow\infty$:

$\log\left(\frac{s(0)}{s_{\infty}}\right)=\mathcal{R}_{0}\left(1-s_{\infty}\right),\;i_{\infty}=0,\;q_{\infty}=0.$
(19)

Equation (19) can be used to control the accuracy of the numerical integration
of system (3.4). In the examples considered in the next Section, we used the
fourth-order Runge-Kutta method to integrate Eqs. (3.4) numerically with step
$\Delta t=T/10$ sufficient to achieve the reasonable accuracy. In the case of
$\mu=1$ (no loss of immunity), our numerical estimate of $s_{\infty}$
coincided with the value given by Eq. (19) to an accuracy of $10^{-10}$.

##  4 Examples

Now we consider particular examples to demonstrate various effects described
by our model. First we focus on the case when there is no resource depletion
($\rho\gg E$), so that the resource activation barrier could be ignored
($E=0$). We illustrate the ‘‘patient zero’’ phenomenon, demonstrate the effect
of indirect transmission, and model a quarantine scenario. Next we consider an
example of a social group with limited resource ($E\neq 0$).

###  4.1 Effect of $i_{0}$

Figure 2a shows the number densities of susceptible, infected, quarantined,
and recovered individuals as functions of time in the case of two different
initial number densities of infected individuals $i_{0}$ for the fixed basic
reproduction number ($\mathcal{R}_{0}=2.5$). The case $i_{0}=10^{-7}$
corresponds to an initial density of one per 10 million, and the case
$i_{0}=10^{-5}$ corresponds to an initial density of one per 100 thousand. The
number density of infected individuals exhibits a typical peak and then drops.
The peak has the same height for the both initial densities, but in the case
of larger $i_{0}$ it is reached much faster (see Table 2). This example serves
as an illustration of the ‘‘patient zero’’ phenomenon. When
$\mathcal{R}_{0}>1$, the epidemic spreads even when it starts only from one
infected individual (patient zero). Then it reaches the same intensity in a
certain period of time, which is shorter when the initial number of infected
individuals is larger. This effect is also clearly seen in the phase portraits
$i(s)$ at different $i_{0}$ (Fig. 3).

Figure 3: Susceptible-infected phase plane portrait for $\mathcal{R}_{0}=2.5$
and various initial number densities of infected individuals $i_{0}$.

###  4.2 Effect of indirect transmission

Figure 2 shows the number densities of susceptible, infected, quarantine, and
recovered individuals in the cases when there is no indirect transmission
[panel (a)] and when there is such a transmission [panels (b,c)]. The model
parameters were selected according to Table 1 (see A). The inclusion of the
cloud increased the basic reproduction number $\mathcal{R}_{0}$, so that the
peaks of infected ($i_{\max}$) and quarantine ($q_{\max}$) densities might
become larger and shift to shorter times (Table 2).

###  4.3 Quarantine scenario

The epidemic dynamics is governed by the basic reproduction number
$\mathcal{R}_{0}$. The epidemic starts to spread when $\mathcal{R}_{0}>1$. The
greater $\mathcal{R}_{0}$, the larger are $i_{\max}$, $q_{\max}$, and
$(q_{\Sigma})_{\infty}$. Thus, to reduce the epidemic peak and to slow the
epidemic down, one should reduce $\mathcal{R}_{0}$. According to formula (17),
this can be achieved by reducing the transmission rates $\omega_{j}$ and,
therefore, the integral transmission rate $\beta$. Such measures are usually
referred to as quarantine. To model the quarantine scenario, we assumed the
following form of the transmission rates:

$\beta=\left\\{\begin{array}[]{ll}\beta(0),&t<t_{1},\\\
\beta^{\prime},&t_{1}\leqslant t<t_{2},\\\ \beta(0),&t\geqslant
t_{2},\end{array}\right.\beta_{p}=\left\\{\begin{array}[]{ll}\beta_{p}(0),&t<t_{1},\\\
\beta^{\prime}_{p},&t_{1}\leqslant t<t_{2},\\\ \beta_{p}(0),&t\geqslant
t_{2},\end{array}\right.$ (20)

where $t_{1}$ is the moment when the quarantine starts, $t_{2}$ is the moment
when the quarantine ends, $\beta(0)$ and $\beta_{p}(0)$ are the direct and
indirect transmission rates during the period when there is no quarantine, and
$\beta^{\prime}$ and $\beta^{\prime}_{p}$ are the transmission rates during
the quarantine period.

Figure 4 demonstrates an example of the quarantine scenario when the initial
basic reproduction number $\mathcal{R}_{0}=3.5$ was reduced to
$\mathcal{R}_{0}=1.1$ by quarantine measures at the moment $t_{1}=20$ days. In
particular, this can be achieved by increasing the average distance
$\overline{\ell_{j}}$ between the individuals in transport and other social
locations and by reducing the average time $T_{j}$ spent in these locations
(see Table 1 in Sect. 2). The number density of quarantine individuals
continued to grow during the quarantine period but at much lesser rate and
acquired a local peak at $t\approx 48$ days. Then the quarantine was
terminated at the moment $t_{2}=60$ days. The number densities of the infected
and quarantined individuals immediately started to grow again and reached the
new peaks that were much larger than those during the quarantine (row 5 in
Table 2). As compared to the ‘‘no quarantine’’ scenario (Fig. 2b, row 3 in
Table 2), the absolute heights of the peaks decreased, but the integral number
densities of quarantine individuals $q_{\Sigma}$ and, therefore, fatal cases
remained nearly the same. Thus, the quarantine scenario allows one to win time
but does not seriously affect the total number of ill and deceased people by
the end of epidemic, in the case when the mortality rate remains to be
constant.

The second (post-quarantine) peak in the number densities of infected and
quarantined individuals clearly illustrates the effect known as the second
wave of the epidemic, which has in particular been observed in the case of
COVID-19 epidemic in many countries [72].

Our results are in line with the results of modeling presented in Ref. [33].
The greater the reduction in transmission, the longer and flatter is the
epidemic curve, with the risk of resurgence when interventions are lifted to
mitigate economic impact. The similar results were obtained when modeling the
COVID-19 quarantine scenario for the Wuhan city, with a stochastic SEIR model
fitted to the available statistical data [73]. The pre-quarantine
$\mathcal{R}_{0}$ value equal to 2.35 (the median estimate) dropped to
$\mathcal{R}_{0}\approx 1.05$ after the start of the quarantine.

Figure 4: Number densities of susceptible, infected, quarantined, and
recovered individuals in the case of quarantine scenario with $t_{1}=20\,T$,
$t_{2}=60\,T$, $\beta(0)=0.5\,T^{-1}$, $\beta_{p}(0)=0.2\,T^{-1}$
($\mathcal{R}_{0}=3.5$) and $\beta^{\prime}=0.18\,T^{-1}$,
$\beta^{\prime}_{p}=0.04\,T^{-1}$ ($\mathcal{R}_{0}=1.1$). Figure 5: Effect of
limited resource $\varrho=\rho/\rho^{(0)}$ on the number density of fatal
cases $f$ in the case of $\mathcal{E}=E/\rho^{(0)}=0.1$. The number density of
quarantined individuals $q$ is the same in the cases of limited
($\mathcal{E}=0.1$) and unlimited ($\mathcal{E}=0$) resource. The model
parameters are the same as in the example shown in Fig. 2b. Figure 6: Effect
of two different quarantine scenarios on the number density of fatal cases $f$
in the case of limited resource ($\mathcal{E}=0.1$). The model parameters for
the quarantine scenarios (see Table 1) with $t_{1}=20\,T$ and $t_{2}=60\,T$
are $\beta(0)=0.5\,T^{-1}$, $\beta_{p}(0)=0.2\,T^{-1}$ (no quarantine,
$\mathcal{R}_{0}=3.5$); $\beta^{\prime}=0.18\,T^{-1}$,
$\beta^{\prime}_{p}=0.04\,T^{-1}$ (soft quarantine,
$\mathcal{R}^{\prime}_{0}=1.1$); $\beta^{\prime}=0.07\,T^{-1}$,
$\beta^{\prime}_{p}=0.02\,T^{-1}$ (strict quarantine,
$\mathcal{R}^{\prime}_{0}=0.45$).

###  4.4 Effect of limited resource

Here we demonstrate the effect of nonzero resource activation parameter $E$.
Let us rewrite resource equation (6f) in terms of dimensionless resource
$\varrho=\rho/\rho^{(0)}$,

$\partial_{t}\varrho=\frac{\Gamma_{\rho}}{1+s_{\Lambda}}\bigl{(}k(s+i+r)-(1+s_{\Lambda})\,\varrho+s_{\Lambda}\bigr{)},$
(21)

where $\rho^{(0)}=(G^{(0)}+\Lambda)/\Gamma_{\rho}$ is the equilibrium resource
value when there is no epidemic (so that $\varrho^{(0)}\equiv 1$),
$s_{\Lambda}=\Lambda/G^{(0)}$ is the number density of active individuals that
would have to be working to acquire the resource amount equal to $|\Lambda|$
(per unit time), $k=G/G^{(0)}\leqslant 1$ is the time-dependent coefficient
that allows for variations in the resource acquisition rate during the spread
of the epidemic, and $G^{(0)}$ is the resource acquisition rate when there is
no epidemic and the system is in equilibrium. For the sake of simplicity, we
will limit our consideration to the case $s_{\Lambda}=0$.

Resource equation (21) is coupled to Eqs. (3.4), where the exponential factor
$\exp(-E/\rho)$ needs to be identically rewritten as
$\exp(-\mathcal{E}/\varrho)$. The parameter $\mathcal{E}\equiv E/\rho^{(0)}$
is the resource activation level $E$ normalized to the equilibrium resource
value. Nonzero $\mathcal{E}$ reduces the recovery rates of quarantine
individuals, $\Gamma_{qr}\exp(-\mathcal{E}/\varrho)$ and
$\Gamma_{qs}\exp(-\mathcal{E}/\varrho)$, so that there would be more fatal
cases as compared to the case of unlimited resource ($\mathcal{E}=0$).

Figure 5 demonstrates the effect of limited resource in the case of
$\mathcal{E}=0.1$, with all other parameters selected such as in the example
shown in Fig. 2b. Nonzero $\mathcal{E}$ has a profound effect on the number of
fatal cases, which is nearly 6 times larger than in the case of unlimited
resource (Fig. 5a). Such a manyfold increase in the number of fatal cases is
caused only by a 5% drop of resource $\varrho$ (Fig. 5b). Resource began to
decline as the number of quarantined individuals grew up (Fig. 5a), since the
quarantined individuals are supposed to be passive and not acquiring the
resource [see Eq. (21)]. After the number of quarantined individuals passed
through its peak, resource passed through its minimum and started to increase
towards its initial value. This example illustrates a scenario when the
economic subsystem has a limited capacity to support the medical
infrastructure, so that seriously ill (quarantined) individuals could not get
the necessary medical help to overcome the infection.

The resource acquisition rate $G$ was assumed to be constant ($k\equiv 1$) in
the above example. The parameter $G$ is determined by the average number of
working hours per working individual and by the working efficiency. The
average number of working hours is proportional to the control parameter
$T_{4}$, which can be reduced in the case of quarantine, as discussed in Sect.
2 (see Table 1). Figure 6 demonstrates the effect of two different quarantine
scenarios on the number density of fatal cases. When there is no quarantine,
the number density of fatal cases is the same as in Fig. 5a for the case of
$\mathcal{E}=0.1$. In the case of soft quarantine, the transmission rate
$\beta$ goes down (Table 1), the basic reproduction number $\mathcal{R}_{0}$
becoming smaller and the epidemic spreading being less intensive. The soft
quarantine scenario does not affect the average number of working hours per
working individual, and the resource acquisition rate $G$ remains the same as
in the case of no quarantine ($k=1$). The number of fatal cases grows much
slower during the quarantine, but it gradually goes back towards its value in
the case of no quarantine after the quarantine is terminated.

In the case of strict quarantine, the transmission rate $\beta$ is yet smaller
(Table 1), mainly because this quarantine scenario also affects the total
number of working hours, which is twice as small as compared to the case of
soft quarantine. As a result, the resource acquisition rate goes down
($k=0.5$) during the quarantine. In the short run, the number of fatal cases
in the case of such strict quarantine measures substantially declines as
compared to the cases of soft quarantine and no quarantine (Fig. 6). However,
it starts to rapidly increase after the quarantine is terminated and
eventually becomes larger than it was in the case of no quarantine. This
example demonstrates that strict quarantine measures that affect the general
economic situation may have serious negative social outcomes in the case of
limited economic resource.

##  5 Conclusion

We proposed a dynamical model for describing the spread of epidemics. The
spreading process within a selected social group is governed by the equations
that explicitly take into account the dependence of characteristic
transmission rates on the local population density in various social zones.
The indirect channel of transmission via an intermediate environment or the
so-called fomites was also taken into consideration. A negative feedback
between the infected population size and a collective economic resource
associated with the average amount of money or income per individual was
introduced to describe the socioeconomic interplay. The epidemic spread and
the use of quarantine measures was demonstrated to be connected with economic
losses, which in turn could aggravate the negative outcomes of the epidemic.

The model presented in this work can be used to model the COVID-19 epidemic
for particular social groups and regions. It can also be applied to describing
other spreading processes, such as the spread of information, rumors, ideas,
or concepts.

## Acknowledgments

O.K. was partially supported by a grant for research groups of young
scientists from the National Academy of Science of Ukraine (Project No.
0120U100155). We thank Prof. B.I. Lev for fruitful discussions.

##  Appendix A Model parameters (extended)

Table 1: Model parameters.

| Description | Our model value | Literature data
---|---|---|---
$\omega_{j}$ |  Infected-to-susceptible transmission rate for location $j$ |  see Table 1 |
$\beta$ | Integral direct transmission rate (infected-to-susceptible) | $0.5\,T^{-1}$ |  $0.6\,T^{-1}$ [60]†, $0.15\,T^{-1}$ [74]†
$\Omega_{j}$ |  Indirect transmission rate (infected-cloud-susceptible) in cloud $j$ |  see Eq. (24) |
$\nu_{j}$ |  Scaled indirect transmission rate in cloud $j$ |  see Eq. (24) |
$\beta_{j}$ |  Scaled indirect transmission rate in cloud $j$ corrected for time span $T_{j}$ |  see Table 2 |
$\beta_{p}$ | Integral indirect transmission rate (infected-cloud-susceptible) | $0.2\,T^{-1}$ |  $0.35\,T^{-1}$ [67]‡
$\chi_{j}$ |  Infected-cloud-susceptible transmission efficiency in cloud $j$ | $5\times 10^{-6}$ |
$\sigma_{j}$ |  Average pathogen shedding rate in cloud $j$ |  see Eq. (23) |
$\gamma_{j}$ |  Average pathogen decay rate in cloud $j$ | $\tau_{p}^{-1}$ |
$\Gamma_{js}$ |  Average contact rate (pickup) of susceptible individuals with cloud $j$ | $240\,T^{-1}$ |  24–480 $T^{-1}$ (fomites) [58]
$\Gamma_{ij}$ |  Average contact rate (shedding) of infected individuals with cloud $j$ | $360\,T^{-1}$ |  $360\,T^{-1}$ (coughs) [75]§
$\Gamma_{iq}$ | Infected-to-quarantine rate constant | $(1-\xi)\,\tau_{i}^{-1}$ |
$\Gamma_{ir}$ | Infected-to-recovered rate constant (with immunity) | $\xi\,\mu\,\tau_{i}^{-1}$ |
$\Gamma_{is}$ | Infected-to-susceptible rate constant (no immunity) | $\xi\,(1-\mu)\,\tau_{i}^{-1}$ |
$\Gamma_{i}$ |  Integral rate constant for infected individuals, $\Gamma_{iq}+\Gamma_{ir}+\Gamma_{is}$ | $\tau_{i}^{-1}$ |
$\Gamma_{qr}$ | Quarantine-to-recovered rate constant (with immunity) | $\mu\,(1-\eta)\,\tau_{q}^{-1}$ |
$\Gamma_{qs}$ | Quarantine-to-susceptible rate constant (no immunity) | $(1-\mu)\,(1-\eta)\,\tau_{q}^{-1}$ |
$\Gamma_{qf}$ | Fatality rate in the case of unlimited resource | $\eta\,\tau_{q}^{-1}$ |
$\Gamma_{q}$ |  Integral rate constant for quarantined individuals, $\Gamma_{qr}+\Gamma_{qs}+\Gamma_{qf}$ | $\tau_{q}^{-1}$ |
$\Gamma_{\rho}$ | Resource consumption rate | $\tau_{\rho}^{-1}$ |
$T$ | Unit of time | 1 day |
$T_{j}$ |  Average time spent in location $j$ |  see Table 1 |
$\ell_{0}$ | Minimum possible distance between two individuals | 1 m |
$\ell_{c}$ | Correlation radius (the maximum transmission distance) | 4 m |
$\overline{\ell_{j}}$ |  Average distance between individuals in location $j$ |  see Table 1 |
$\tau_{is}$ | Characteristic time of becoming infected at close contact | $T/3$ |
$\tau_{p}$ | Average pathogen decay time outside the host | $2\,T$ |  0.1–14 $T^{*}$ [76]†
$\tau_{i}$ | Average pathogen incubation period | $5\,T$ |  3–10 $T$ [77]†, $\approx 5\,T$ [78, 79]†
$\tau_{q}$ | Average quarantine/hospitalization time | $14\,T$ |  $(12.4\pm 5)\,T$ [74]†, 14.5 $T$ [80]†
$\tau_{\rho}$ | Characteristic resource consumption time | $30\,T$ |
$\xi$ | Probability for the infected individual to recover without quarantine | 0.8 |  0.8∗∗ [33]†
$\mu$ | Probability of acquiring the immunity | 0.9 |
$\eta$ | Probability of the fatal scenario for the quarantined individual | 0.015 |  0.014 [81, 82]†
$E$ | Minimum level of resource consumption (activation energy) | 0 |
$G$ | Resource acquisition rate | $G(0)$ |
$\Lambda$ | Resource inflow or outflow per unit time | 0 |
$i_{0}$ | Initial number density of infected individuals | $10^{-5}$ |

† for COVID-19
‡ for cholera outbreak
§ for pandemic influenza
∗ depends on medium, ambient temperature, and surface type
∗∗ 80% of COVID-19 cases are mild or asymptomatic

Table 1 in Sect. 2 gives the direct transmission rates $\omega_{j}$ and
$\beta$ calculated by formulas (2) and (3) for various sets of social control
parameters. Table 1 gives the full description of the model parameters and
their estimates used in our computations.

In particular, the cloud-related parameters can be estimated as follows. The
typical (indirect) rate $\Omega_{j}$ of pathogen transmission from the cloud
$j$ to a susceptible individual can roughly be estimated as

$\Omega_{j}\approx\Gamma_{js}\frac{\upsilon_{c}}{\Delta}\,\theta,\quad[\,\mathrm{time}^{-1}\times\mathrm{mass}^{-1}\times\mathrm{volume}\,]$
(22)

where $\Gamma_{js}$ is the average rate the susceptible individual contacts
the cloud $j$, $\upsilon_{c}$ is a typical volume of the pathogen transferred
from the cloud to the susceptible individual per one contact, $\Delta$ is some
characteristic weight of one pathogen specimen that can be interpreted as the
minimum portion (‘‘quant’’) of the pathogen that can be transferred per one
contact, and $\theta$ is the probability the transmission of this quant
results in infection. The smaller the pathogen, the smaller is the quant
$\Delta$ and the more intensive is the transmission ($\Omega_{j}$ is higher).
The parameters $\upsilon_{c}$, $\Delta$, and $\theta$ are assumed to be
independent of the particular cloud.

The pathogen shedding rate $\sigma_{j}$ (infected-to-cloud) can roughly be
estimated as

$\sigma_{j}\approx\Gamma_{ij}\frac{n\Delta}{V_{j}},\quad[\,\mathrm{time}^{-1}\times\mathrm{mass}\times\mathrm{volume}^{-1}\,]$
(23)

where $\Gamma_{ij}$ is the average rate the infected individual contacts the
cloud $j$ (number of coughs, sneezes, touches, etc. per unit time), $n$ is the
typical number of the pathogen quants $\Delta$ transferred by the infected
individual to the cloud per one contact, and $V_{j}$ is the total volume
(capacity) of the cloud $j$. The parameter $n$ is assumed to be independent of
the particular cloud.

Then the scaled indirect transmission rates $\nu_{j}$ can be estimated as

$\nu_{j}=\Omega_{j}\frac{\sigma_{j}}{\gamma_{j}}\approx\frac{\Gamma_{ij}\,\Gamma_{js}}{\gamma_{j}}\,\chi_{j}.$
(24)

The dimensionless parameter

$\chi_{j}=\frac{\upsilon_{c}}{V_{j}}\,n\,\theta$ (25)

defines the transmission efficiency from infected individuals to susceptible
individuals through the cloud $j$. Although the parameter $\Delta$ was
eliminated by scaling (11), the expression for $\chi_{j}$ still contains the
parameters that are hard to estimate from some physical principles. Therefore,
this parameter can rather be estimated by fitting the model to some real
statistical data. In practice, it is selected by assuming that the indirect
and direct routes of transmissions have approximately the same likelihood,
i.e. $\nu_{j}\approx\omega_{j}$ [56].

Table 2 gives the indirect transmission rates $\nu_{j}$ and $\beta_{j}$
calculated by formulas (24) and (13) for a particular set of social control
parameters corresponding to the casual scenario in Table 2. The infected-to-
cloud contact rates $\Gamma_{ij}$, pathogen decay rates $\gamma_{j}$, and
transmission efficiencies $\chi_{j}$ are assumed to be the same for each cloud
and listed in Table 1. The cloud-to-susceptible contact rate $\Gamma_{js}$ is
assumed to be inversely proportional to the squared average distance
$\overline{\ell_{j}}$ between the individuals [as in Eq. (2)].

Table 2: Scaled indirect (infected-cloud-susceptible) transmission rates $\nu_{j}$ and $\beta_{j}$ in the case of the casual epidemic scenario (see Table 1). The aggregate indirect transmission rate $\beta_{p}$ is given by formula (18). Cloud |  $T_{j}$, h |  $\overline{\ell_{j}}$, m | $\Gamma_{js}$ | $\nu_{j}$ | $\beta_{j}$
---|---|---|---|---|---
$j=1$ | 11.5 | 3.5 | $20\,T^{-1}$ | $0.07\,T^{-1}$ | $0.03\,T^{-1}$
$j=2$ | 1.5 | 1.5 | $107\,T^{-1}$ | $0.38\,T^{-1}$ | $0.02\,T^{-1}$
$j=3$ | 3 | 1 | $240\,T^{-1}$ | $0.86\,T^{-1}$ | $0.11\,T^{-1}$
$j=4$ | 8 | 3 | $27\,T^{-1}$ | $0.10\,T^{-1}$ | $0.03\,T^{-1}$
$\beta_{p}$ |  |  |  |  | $0.20\,T^{-1}$

Table 3 lists some estimates of the basic reproduction number
$\mathcal{R}_{0}$ derived from the statistical data on COVID-19 (literature
data).

Table 3: Available $\mathcal{R}_{0}$ estimates for COVID-19. $\mathcal{R}_{0}$ | Reference | Data source
---|---|---
1.5–3.5 |  Imai et al. [83] | Wuhan
2.4–4.1 |  Read et al. [84] | Wuhan
2.2–3.6 |  Zhao et al. [85] | Wuhan
1.4–3.9 |  Li et al. [79] | Wuhan
2.5–2.9 |  Wu et al. [86] | Wuhan

##  Appendix B Extension to multiple groups

Similarly to the basic SIR model [7, 9], our model can easily be extended to
the multigroup formulation, e.g., with subdivision by age. The dynamics of
each group $n$ is governed by the following equations:

$\displaystyle\partial_{t}s^{(n)}=$
$\displaystyle-\Bigl{(}\sum_{m}\beta^{(m)}\,i^{(m)}+\sum_{j}\beta_{j}\,p_{j}\Bigr{)}s^{(n)}$
$\displaystyle+\Gamma_{is}^{(n)}\,i^{(n)}+\Gamma_{qs}^{(n)}\,\mathrm{e}^{-\mathcal{E}/\varrho}\,q^{(n)},$
$\displaystyle\partial_{t}i^{(n)}=$
$\displaystyle\;\Bigl{(}\sum_{m}\beta^{(m)}\,i^{(m)}+\sum_{j}\beta_{j}\,p_{j}\Bigr{)}s^{(n)}-\Gamma_{i}^{(n)}\,i^{(n)},$
$\displaystyle\partial_{t}q^{(n)}=$
$\displaystyle\;\Gamma_{iq}^{(n)}\,i^{(n)}-\Gamma_{q}^{(n)}\,q^{(n)},$
$\displaystyle\partial_{t}r^{(n)}=$
$\displaystyle\;\Gamma_{ir}^{(n)}\,i^{(n)}+\Gamma_{qr}^{(n)}\,\mathrm{e}^{-\mathcal{E}/\varrho}\,q^{(n)},$
$\displaystyle\partial_{t}f^{(n)}=$
$\displaystyle\;\left(\Gamma_{q}^{(n)}+(\Gamma_{qf}^{(n)}-\Gamma_{q}^{(n)})\,\mathrm{e}^{-\mathcal{E}/\varrho}\right)q^{(n)},$

with index $m$ running over all the groups. The direct and indirect
transmission rates are defined as

$\beta^{(m)}=\sum_{j=1}^{N_{\omega}}\omega_{j}^{(m)}\frac{T_{j}}{T},\quad\beta_{j}=\Omega_{j}\frac{\sum_{m}\sigma_{j}^{(m)}}{\gamma_{j}}\frac{T_{j}}{T}.$

The pathogen dynamics in the cloud $j$ is given by an equation

$\partial_{t}p_{j}=\gamma_{j}\left(\frac{\sum_{m}\sigma_{j}^{(m)}i^{(m)}}{\sum_{m}\sigma_{j}^{(m)}}-p_{j}\right).$

The resource equation is

$\partial_{t}\varrho=\frac{\Gamma_{\rho}}{1+s_{\Lambda}}\Bigl{(}k\sum_{m}(s^{(m)}+i^{(m)}+r^{(m)})-(1+s_{\Lambda})\,\varrho+s_{\Lambda}\Bigr{)}.$

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