Page 67 - GHES-1-1

Global Health Econ Sustain Implication of close contact

transmission rate for contact with symptomatic and To model the search for new cases, we assume first
asymptomatic individuals, respectively. δ is the mean time that the number of close contacts tested is proportional to
-1
between infection and the onset of infectiousness, p is the βI. Some of them may be infected, asymptomatic, or just
proportion of new asymptomatic (1-p for symptomatic) before the onset of symptoms. The proportion of close
individuals leaving the exposed population, and β is contact tests performed on symptomatic or asymptomatic
-1
the mean duration for a symptomatic individual to be individuals can be obtained by multiplying the number
tested and then be placed in quarantine. Asymptomatic of tests by I/N or A/N, respectively. Finally, we absorb
individuals were typically not tested for COVID-19, being all constants into ω to simplify the expression. Then, the
α the mean duration of their infectiousness (Table 1 for a terms θ(t)AI and θ(t)I represent those close contacts,
-1
2
reference list of parameters). asymptomatic or symptomatic, respectively, of the infected
The new terms introduced include the function θ(t), individuals put into quarantine previously, found by
which represents the success rate of identifying new the health services after testing. Then, the parameter ω
COVID-19 cases through testing of close contacts of represents the efficiency of the health system in searching
individuals who have tested positive. We represent this for new infected individuals among close contacts due to
function using a time-dependent sigmoidal function the proportion between the number of tests performed
(Figure 1): and βI. Higher values of ω indicate that more individuals
have been tested among those who have had contact with a
ω positive case. For ω going to zero, the health system is not
θ t () = (6)
1 + ξe − γ t−( τ ) taking any testing measures at all.
It is important to mention that we are assuming that, As mentioned before, population R includes those
at the beginning of the epidemic, the health-care system individuals who have been tested by the health system and
is unprepared and may not be actively searching for placed in quarantine, those who have recovered from the
secondary cases. However, as time passes, health-care disease, and those who have been hospitalized.
facilities may improve, and the effort in contact tracing Figure 1 shows the evolution of the different populations
to identify secondary cases may increase, eventually considering several values for the parameter ω. We can
reaching an optimum level. In Equation 6, ω represents the observe that the infected populations (E, I, and A) reach
maximum effort put into searching for secondary cases, ξ a quasi-steady state (QSS) when we introduce the terms
represents a proportion between the initial and final effort that represent the testing of close contacts. The flux of
of the healthcare system, γ is a rate that indicates how fast newly infected individuals in this QSS decreases as we
measures are implemented, and τ is the starting time of increase the efficiency of contact tracing (by increasing ω).
close contact testing. It is possible that the effort put into However, it is not a formal steady state because we have
the search for secondary cases may eventually vary, for assumed a closed population, and there is a continuous flux
instance, due to a decrease in the number of positive cases, of individuals from the susceptible to the removed class. As
resulting in a variation of ω. the susceptible population is large but finite, it reduces its

Table 1. Definition of the model parameters and their values for each city
Parameter Definition Value References
South Korea New York city
µ Symptomatic transmission rate 1.8 2 Fitted
v Asymptomatic transmission rate 0.45 0.5 Fitted
δ Rate of exposed becoming infectious 0.5 0.5 (Thompson et al., 2020)
p Probability of exposed to become asymptomatic 0.8 0.8 (Liu et al., 2021)
α Recovery rate of asymptomatic individuals 1/15 1/15 (Streeck et al., 2020)
β Rate at which a symptomatic individual is placed in quarantine 1/4 1/4 (Streeck et al., 2020)
N Total population 4.8×10 7 8.1×10 6 (NYS Dept Labor, 2022; KSIS, 2022)
ω Final rate of secondary cases testing ability per infected cases - - Fitted
ξ Relation between initial and final secondary cases search effort - - Fitted
γ Optimization rate of secondary cases search - - Fitted
τ Starting time of massive testing - - Fitted

Volume 1 Issue 1 (2023) 4 https://doi.org/10.36922/ghes.0873