IEEE Computational Intelligence Magazine - August 2022 - 56
dt
dI
dt
dI
dt
q
=+ -- -=+
+-()
1
iE
dI id I
id I
rq rqrr pq
qrrp q
dR id Ii() ()pqrr
qerq lr
dRq
dt
=+ -+ -+
+- -rdpqr
=+
+- -
dD =+id Ii dI
dt
rr pqrr pq
+
iE dI id Ii dI1
rr rr prrp
()
(6)
(7)
pR iR cc IR pR
rr11
dI vS
pq r
(8)
cc IR pR pR iR
rdpqrqer qlrq
(9)
(10)
The Equations (2) to (10) contain
the following parameters: cr indicates the
contact rate. The transmission probability
is shown by tp. The incubation rate (ir)
specifies how long it takes from exposure
to becoming infectious. The preventive
quarantine rate (pqr) determines
the rate with which susceptible individuals
are quarantined preventively, and
the preventive quarantine end rate (pqer)
determines how long this confinement
lasts. The contact detection probability
(cdp) defines the chance that the contact
of a susceptible individual with an
infected individuals is successfully traced,
and that the susceptible individuals can
be sent to quarantine based on this
information. The diagnosis rate (dr)
describes the rate with which infected
individuals are detected and isolated.
The infected individuals' recovery rates
are given by the infected recovery rate
(irr) and infected quarantined recovery
rate (iqrr). The immunity loss rate (ilr)
determines how long-lasting immunity
from the virus is. The vaccination rate
(vr) determines how frequently susceptible
individuals transition to immunity
without requiring infection. Finally, the
death probability is indicated by dp. The
presented pandemic model
integrates
control policies, by allowing the policies
to alter these parameters during simulation
(see Section III-B).
The equations are built from individual
terms, where each term describes
the flow of individuals from one compartment
to another. Every term appears
exactly twice, once as an addition to a
compartment, once as a subtraction, thus
guaranteeing that the total population
remains constant. The first group of
terms is concerned with the contact
between susceptible, recovered and
infected: ()
ct cIS represents sus-
rp1
dp
ceptible
individuals who had a traced
contact with someone who was infected,
without becoming exposed. They are
sent into preventive quarantine (S to Sq).
Similarly, ct cISrp dp
represents susceptible
individuals with traced contacts that
were exposed and are sent into quarantine
(S to Eq). The term ()ct cIS
rp 1 dp
-
represents exposure, but without being
detected (S to E). Individuals in the E
compartment do not know of their
exposure and therefore do not adapt
their behavior. For instance, they still go
to work and eat at restaurants. Next,
pS
qr
are susceptible individuals ordered
to stay in quarantine for prevention purposes
(S to Sq), and pS
qerq represents
individuals returning from such quarantine
orders (Sq to S). It is assumed that
even recovered individuals must obey
quarantine rules, despite their immunity.
Therefore, there are analogous terms for
individuals entering and leaving preventive
quarantine for the exposed and
recovered compartment pairs. The next
term,
cc IRrdp
, describes
successfully
traced contacts between infected individuals
and already recovered individuals,
leading to quarantine afterwards
(R to Rq). The term vSr
captures the
vaccination of susceptible individuals
(S to R), while iR
lr
lr
time, the exposed individuals
symptoms and transition to I: iEr
and iRq capture
the loss of immunity (R to S, respectively
Rq
to Sq). After the incubation
show
(E
to I). The same occurs for the quarantined
versions of the compartments:
iE
rq (Eq to Iq). A fraction of individuals
that carry an infection get tested
and receive a positive diagnosis,
dI ,r
which consequently puts them in
quarantine (I to Iq). Finally, individuals
either succumb to the pathogen, or
recover and become immune. This
applies to both I and Iq. Therefore,
id Irr p
and id Iqrrp q
rr 1 p
-
while () and ()
represent the fraction of individuals
id I
qr 1rp q
-
56 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
that
recover
from the virus
become immune (I/Iq to R).
A. The Economy Compartment
The relative economic predictor of the
proposed model is introduced via a virtual
compartment, which is unidirectionally
influenced by the pandemic
compartments (see Figure 1). Hence, it
is assumed that the pandemic's course
shapes the economy, but not vice versa.
This virtual compartment is called GDP
(Gross Domestic Product). It is to be
understood as a rough indicator for the
state of the economy relative to the
starting point of the pandemic:
dGDP
dt
=+ -- -
-- -
gi1
ii
()
iqiq
bD pp Sp eE
pi Ip iI pp R
iqiq
qi qi qi q
(11)
The GDP is modeled by the characteristic
of economic growth bg of capitalist
economies. Usually, an economy's
GDP consists of the Consumption, Investment,
Government Spending, and Net
Exports-in this approach an overly complex
representation of the economy. In
the presented model, the GDP fundamentally
grows linearly at a constant rate,
although it scales with the population in
this model (economic growth weakens as
deaths accumulate). The remaining
parameters quantify the influence of negatively
associated compartments on economic
health: Sq, Eq, I, Iq, and Rq. For
each of these compartments, which
counteract the fundamental economic
growth, there is a scaling factor that
describes the strength of its negative
impact on the economy: preventive quarantine
impact (pqi), exposed quarantine
impact
(eqi), infected impact
(ii), and
describe the rate of
individuals succumbing (I/Iq to D),
id I
infected quarantine impact (iqi). Such
scaling factors are required to represent
different economies. For instance, hightech
economies are less impacted by a
pandemic than those economies strongly
built upon tourism. In addition, the
model includes a general pandemic influence
parameter (pi) that serves as a simple
scaling factor for the previously mentioned
individual impacts. This way, overall
impact can be adjusted easily without
altering relative impacts, which proved
useful in preliminary experiments.
and
IEEE Computational Intelligence Magazine - August 2022
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