Modelling COVID-19 epidemic with confirmed cases-driven contact tracing quarantine

Abstract

The pandemic of novel coronavirus disease 2019 (COVID-19) has been a severe threat to public health. The policy of close contract tracing quarantine is an effective strategy in controlling the COVID-19 epidemic outbreak. In this paper, we developed a mathematical model of the COVID-19 epidemic with confirmed case-driven contact tracing quarantine, and applied the model to evaluate the effectiveness of the policy of contact tracing and quarantine. The model is established based on the combination of the compartmental model and individual-based model simulations, which results in a closed-form delay differential equation model. The proposed model includes a novel form of quarantine functions to represent the number of quarantine individuals following the confirmed cases every day and provides analytic expressions to study the effects of changing the quarantine rate. The proposed model can be applied to epidemic dynamics during the period of community spread and when the policy of confirmed cases-driven contact tracing quarantine is efficient. We applied the model to study the effectiveness of contact tracing and quarantine. The proposed delay differential equation model can describe the average epidemic dynamics of the stochastic-individual-based model, however, it is not enough to describe the diverse response due to the stochastic effect. Based on model simulations, we found that the policy of contact tracing and quarantine can obviously reduce the epidemic size, however, may not be enough to achieve zero-infectious in a short time, a combination of close contact quarantine and social contact restriction is required to achieve zero-infectious. Moreover, the effect of reducing epidemic size is insensitive to the period of quarantine, there are no significant changes in the epidemic dynamics when the quarantine days vary from 7 to 21 days.

1. Introduction

The pandemic of novel coronavirus disease 2019 (COVID-19), caused by novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is continuously threatening public health after 3 years of spreading globally over the world. Nowadays, the new strand Omicron BA.4/BA.5 causes a situation never seen before due to its fast spreading speed. Three years after the outbreak of COVID-19, it remains a challenging issue of how can we control the epidemic spreading with minimal effects on economic development and daily life.

Chinese Mainland has been successful during the last 3 years in the control of the COVID-19 epidemic, and the zero-COVID policy has been proven to be practical under proper control measures (Yuan et al., 2022). The policy of close contract tracing quarantine has been proven to be an effective strategy in controlling the COVID-19 epidemic outbreak, by which close contacts of confirmed cases are traced individually, and the contacts are subjected to quarantine for at least 14 days (shortened to 7 days during late 2022 in accordance with the feature of Omicron strand) in order to break the transmission chain. Now, the COVID-19 epidemic is mostly over after the open policy in China in December 2022, however, it is worth reviewing the control policy in the last three years in order to give valuable advice when the next pandemic event occurs.

The spreading of infectious disease can be described through the susceptible-exposed-infectious-remove (SEIR) model, which has been widely used in forecasting epidemic dynamics. The SEIR model can be applied to describe the epidemic dynamics without control measures. Nevertheless, to predict the epidemic dynamics in terms of close contact tracing of quarantine, we need to know the number of people being traced following confirmed cases every day. However, this number is highly varied and is usually difficult to be formulated as a closed-form expression. In the last two years, a few mathematical models were developed to describe the effect of close contact tracing quarantine. In the existing models, the isolation populations are often assumed to be proportional to the daily newly diagnosed numbers (Yuan et al., 2022), or bilinear functions of daily newly diagnosed and susceptible populations (Li et al., 2020a). However, these functions fail to describe the tracing process and lack data support.

In this study, we intended to develop a mathematical model of the COVID-19 epidemic with confirmed cases-driven contact tracing quarantine. We applied the methods of both model- and data-driven to derive the phenomenological formulation to express the number of quarantine individuals in each data, based on the number of confirmed cases together with susceptible, exposed, and infectious numbers. The model-driven part is derived similarly to the usual routine of deriving the SEIR model, and the data-driven part is implemented through simulated data generated by an individual-based model. Through the proposed model, we further studied the effectiveness of contact tracing and quarantine in epidemic control.

2. Results

2.1. Age-structured model

This study considered the COVID1-19 epidemic with confirmed cases-driven contact trace quarantine. To this end, we extended the classical SEIR model to include the hospitalized compartment (H) for the confirmed infections and the quarantine of closed contacts traced from confirmed cases. Moreover, the quarantine persons are subjected to nucleic acid testing during quarantine, so the confirmed cases are moved to the hospital compartment, otherwise, individuals would become un-quarantine after a certain period (usually 14 days) quarantine. The above model is summarized in Fig. 1.

Fig. 1.

Age-structured model of COVID-19 epidemic with confirmed cases-driven contact tracing followed by quarantine. Susceptible persons become exposed to viruses when contact with infections, exposed persons do not show the ability to infect susceptible persons, and become symptomatic infections at a certain rate. Infections can either transit to the removed compartment due to either recovery or death, or transit to the hospitalized compartment after diagnosis.

To formulate the epidemic dynamics, the whole population is separated into compartments of susceptible (S), exposed (E), infectious (I), recovered (R), hospitalized compartment (H), and quarantine (Q). Susceptible persons become exposed to viruses when contact with infections, with an infection rate β(day⁻¹). Exposed persons do not show the ability to infect susceptible persons, and become symptomatic infections with a rate γ(day⁻¹). Infectious can either transit to the removed compartment due to either recovery or death with a rate κ(day⁻¹), or transit to the hospitalized compartment after diagnosing with a rate η(day⁻¹). Hospitalized persons transit to the removed compartment, due to either recovery or death with a rate μ(day⁻¹). When an infected person is diagnosed, the close contacts in the last 14 days are traced and subjected to quarantine for a period of 14 days. Let q_I(t, a), q_E(t, a), and q_S(t, a) for the numbers of infectious, exposed persons, and susceptible persons in the quarantine compartment at time t and day a after quarantine. All isolated persons need to take nucleic acid testing, confirmed cases are moved to the hospitalized compartment, and the confirmed rates for isolated exposed and infected persons are η₁ and η₂, respectively. Isolated exposed individuals can become the infected state with a rate γ(day⁻¹). Otherwise, all unconfirmed cases and susceptible persons return to the compartments of their corresponding state at the end of quarantine. According to the above strategy, we have the following age-structured model equation

$$\begin{array}{lll}\frac{dS}{dt}& =& -\beta \frac{SI}{N_p}-Q_S(t)+q_S(t,\tau ),\end{array}$$ (1)

$$\begin{array}{lll}\frac{dE}{dt}& =& \beta \frac{SI}{N_p}-\gamma E-Q_E(t)+q_E(t,\tau ),\end{array}$$ (2)

$$\begin{array}{lll}\frac{dI}{dt}& =& \gamma E-\kappa I-\eta I-Q_I(t)+q_I(t,\tau ),\end{array}$$ (3)

$$\begin{array}{lll}\frac{dH}{dt}& =& \eta I+{\eta }_1{\int }_0^{\tau }{q}_E(t,a)da+{\eta }_2{\int }_0^{\tau }{q}_I(t,a)da-\mu H,\end{array}$$ (4)

$$\begin{array}{lll}\frac{dR}{dt}& =& \kappa I+\mu H,\end{array}$$ (5)

$$\begin{array}{lll}\frac{\partial q_S}{\partial t}+\frac{\partial q_S}{\partial a}& =& 0,(0<a<\tau ),\end{array}$$ (6)

$$\begin{array}{lll}\frac{\partial q_E}{\partial t}+\frac{\partial q_E}{\partial a}& =& -{\eta }_1{q}_E(t,a)-\gamma q_E(t,a),(0<a<\tau ),\end{array}$$ (7)

$$\begin{array}{lll}\frac{\partial q_I}{\partial t}+\frac{\partial q_I}{\partial a}& =& -{\eta }_2{q}_I(t,a)+\gamma q_E(t,a),(0<a<\tau ),\end{array}$$ (8)

$$\begin{array}{lll}{q}_S(t,0)& =& Q_S(t),\end{array}$$ (9)

$$\begin{array}{lll}{q}_E(t,0)& =& Q_E(t),\end{array}$$ (10)

$$\begin{array}{lll}{q}_I(t,0)& =& Q_I(t),\end{array}$$ (11)

where N_p = S + E + I + R represents the number of persons that are not isolated and not hospitalized, τ = 14 days represents the duration of quarantine, and Q_S(t), Q_E(t) and Q_I(t) are the traced and isolated susceptible, exposed, and infected populations at time t, respectively.

Equations (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11) give an age-structured model of epidemic dynamics with close contact tracing quarantine. In the equation, the daily isolated populations Q_S, Q_E, and Q_I are dependent on the number of daily newly diagnosed cases and the efficiency of close contract tracing. Obviously, we should have

$$Q_S<S,Q_E<E,Q_I<I$$ (12)

at any time t. Nevertheless, because of the complex contact relationship and the lag time of contract tracing, it is not trivial to derive how the isolation populations depend on other variables in the equations.

2.2. Individual-based model

In the real world, great efforts of epidemiological investigation are required to obtain the data of tracing quarantine, and the obtained data are often incomplete. Here, to obtain the formulations for the isolated population function Q_S, Q_E, and Q_I in the above model, we applied an individual-based model of the COVID-19 epidemic developed in previous studies (Xu et al., 2021) to generate virtual data of tracing quarantine and fitted the data to obtain the final expression of phenomenological functions. The mechanisms for the individual-based model and the previous compartment model are the same, however, the contact relationships in the individual-based model are considered explicitly through a contact network.

In the individual-based model (Xu et al., 2021), the individuals are represented by nodes in a close contact network, in which each node transits among different states as shown in Fig. 1, and the close contact relationships are represented by edges between nodes. We assumed a small-world network to represent the contact relationships among individuals, which is constructed following the Watts-Strogatz algorithm (Watts & Strogatz, 1998). Hence, there are three parameters involved in the contact information, the population size N, the degree of contacts number K, and the probability P of random reconnections.

Each individual in the contact network may be at different states, and transit between the states in accordance with the basic assumptions in Fig. 1 and equations (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11). In the individual-based model, the infectious rate of a susceptible person i at time t is given by

$${\beta }_i(t)=\beta \times \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}i\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}t.$$

Accordingly, the probability that a susceptible individual i is infected within a time interval [t, t + Δt] is expressed as (referred to (Keeling & Rohani, 2008))

$$P_i(t)=1-e^{-{\beta }_i(t)\mathrm{\Delta }t}.$$

After a susceptible individual is infected, the individual is exposed and becomes infectious after an incubation period τ_i, which is assumed to be a gamma distribution random number Γ(τ; a, b). For the original alpha strain, it was assumed that a = 3.07 and b = 2.35 in accordance with the epidemic dynamics of COVID-19 in China in early 2020 (Li et al., 2020b).

The infectious are confirmed with a rate η, and removed with a rate κ, so that probabilities of being either confirmed or removed in a time interval [t, t + Δt] is given by P₂ = 1 − e^ηΔt or P₃ = 1 − e^κΔt, respectively. The hospitalized persons are removed (recovery or death) with a rate μ, hence the probability of being removed in a time interval [t, t + Δt] is P₄ = 1 − e^μΔt.

When an infectious individual is confirmed, we need to trace the close contacts and move the traced contacts to the quarantine compartment. It is usually difficult to carry out 100% close contact tracing, hence we assumed a factor r $(\le 1)$ for the tracing efficiency. In this study, we only considered the first-level contacts. For the individuals during quarantine, there are given rates η₁(day⁻¹) or η₂(day⁻¹) for exposed and infectious individuals to be confirmed, respectively, otherwise the quarantine individuals return to their original compartments.

From the above numerical scheme, and model parameters referred to (Xu et al., 2021), the simulated epidemic dynamics are shown in Fig. 2. In the simulation, we varied the quarantine rate r, and for each value r, randomly simulated the system with initially one infectious for 100 independent runs. Fig. 2a and b shows the dynamics of simulated averaged cumulative infectious number and daily infectious for different values of quarantine rates r. Results show similar dynamics as obtained by the usual SEIR model, and increasing the quarantine rate can significantly reduce the infection size.

Fig. 2.

Simulation results following the individual-based model. (a) Dynamics of cumulative infectious number for different quarantine rates 0 ≤ r ≤ 1. (b) Dynamics of daily infectious number for different quarantine rates 0 ≤ r ≤ 1. (c) Dependence of daily quarantine susceptible individuals versus daily confirmed infectious when the quarantine rate r = 0.1. (d) Dependence of daily quarantine exposed individuals versus daily confirmed infectious when the quarantine rate r = 0.1. (e) Dependence of daily quarantine infectious versus daily confirmed infectious when the quarantine rate r = 0.1. Here, the close contact network was generated as a Watts-Strogatz small-world network N = 4000, K = 20, and P = 0.2. Other parameters are: β = 0.08day⁻¹, κ = 0.02day⁻¹, η = 0.3day⁻¹, μ = 0.1day⁻¹, η₁ = 0.3day⁻¹, η₂ = 0.6day⁻¹, and τ = 14days.

From the simulation results, we examined the dependence of isolation populations with daily confirmed cases and plotted the daily quarantine susceptible (Q_S), exposed (Q_E), and infectious (Q_I) versus daily conformed number (ΔH) obtained from the above simulation (Fig. 2c–e). These results suggest potential nonlinear dependence. The populations Q_S, Q_E, and Q_I are usually dependent on the daily confirmed cases and their close contacts, which are not closed-form functions of ΔH and other population sizes.

To obtain the function Q_I, firstly we examined the relationship between ΔH and I, which shows linear dependence determined by the diagnosis rate (Fig. 3a). Hence, the daily new confirmed number can be represented by the infectious number I.

Fig. 3.

The functionQ_I. (a) Daily confirmed cases versus infectious number for different values of quarantine rates (marked with different color points). (b) Dependence of the isolation population Q_I on the fraction of infectious number I/N_p. Blue dots are obtained from the average of Q_I with given I, and red dots are calculated from the function (13). Here r = 0.1. (c) Dependence of the coefficient k_I on the quarantine rate r, solid line shows the fitting of function (14) with parameters a = 4.29, r₀ = 1.15, ν = 2.

Next, we assumed that Q_I is dependent on I and the fraction of infectious in potential contacts, which is represented by I/N_p (here N_p = S + E + I + R). From the simulated data in Fig. 2e, we assumed power law dependence

$$Q_I=k_{I}I{\left(\frac{I}{N_p}\right)}^{{\alpha }_I}.$$ (13)

Specifically, we took α_I = 1.0 according with simulated data (Fig. 3b), and the coefficient k_I depends on the quarantine rate r through a function of form (Fig. 3c)

$$k(r;a,r_0,\nu )=a\frac{r}{{(r_0-r)}^{\nu }}.$$ (14)

Similar to the above argument, we assumed

$$Q_E=k_{E}I{\left(\frac{E}{N_p}\right)}^{{\alpha }_E},Q_S=k_{S}I{\left(\frac{S}{N_p}\right)}^{{\alpha }_S}.$$ (15)

Fitting the formulation (15) to simulated data, we obtained α_E = 0.5 and α_S = 1, and the coefficients k_E and k_S are dependent on the quarantine rate r through (14) (Fig. 4).

Fig. 4.

The functionsQ_EandQ_S. (a) Dependence of the isolation population Q_E on the fraction of exposed number E/N_p. Blue dots are obtained from the average of Q_E with given I, and red dots are calculated from the function (15). Here r = 0.1. (b) Dependence of the coefficient k_E on the quarantine rate r, the solid line shows the fitting of the function (14) with parameters a = 1.20, r₀ = 1.20, ν = 2. (c) Dependence of the isolation population Q_S on the fraction of susceptible number S/N_p. Blue dots are obtained from the average of Q_S with given I, and red dots are calculated from the function (15). Here r = 0.1. (d) The coefficient k_S as a function of the quarantine rate r through (14), with parameters a = 1.41, r₀ = 1.30, ν = 1. The solid line shows the fitting function (14).

In summary, based on the above individual-based model, we have the isolated population functions

$$\begin{array}{lll}{Q}_I& =& k_I(r)I{\left(\frac{I}{N_p}\right)}^{{\alpha }_I},\end{array}$$ (16)

$$\begin{array}{lll}{Q}_E& =& k_E(r)I{\left(\frac{E}{N_p}\right)}^{{\alpha }_E},\end{array}$$ (17)

$$\begin{array}{lll}{Q}_S& =& k_S(r)I{\left(\frac{S}{N_p}\right)}^{{\alpha }_S},\end{array}$$ (18)

with α_I = 1, α_E = 1/2, α_S = 1, where the coefficients k_I(r), k_E(r) and k_S(r) are taken the form (14) with different parameters a, r₀ and ν.

2.3. Delay-differential equation model

Now, (1)–(11) and (16)–(18) together give close form equations for the epidemic dynamics with confirmed cases-driven contact tracing and quarantine. Moreover, the age-structured model equations (6), (7), (8), (9), (10), (11) can be solved with the method of characteristic line, which gives

$$q_S(t,a)=\left\{\begin{array}{ll}0,& (0<t\le a)\\ Q_S(t-a),& (t>a),\end{array}\right)$$

$$q_E(t,a)=\left\{\begin{array}{ll}0,& (0<t\le a)\\ Q_E(t-a)e^{-({\eta }_1+\gamma )a},& (t>a),\end{array}\right)$$

and

$$q_I(t,a)=\left\{\begin{array}{ll}0,& (0<t\le a)\\ Q_I(t-a)e^{-{\eta }_{2}a}+\frac{\gamma Q_E(t-a)}{{\eta }_1+\gamma -{\eta }_2}(e^{-{\eta }_{2}a}-e^{-({\eta }_1+\gamma )a}),& (t>a),\end{array}\right)$$

Thus, when t < τ,

$$q_S(t,\tau )=q_E(t,\tau )=q_I(t,\tau )=0,$$

and

$$\begin{array}{lll}{\int }_0^{\tau }{q}_E(t,a)da& =& {\int }_0^t{Q}_E(t-a)e^{-({\eta }_1+\gamma )a}da,\\ {\int }_0^{\tau }{q}_I(t,a)da& =& {\int }_0^t\left(Q_I(t-a)e^{-{\eta }_{2}a}+\frac{\gamma Q_E(t-a)}{{\eta }_1+\gamma -{\eta }_2}(e^{-{\eta }_{2}a}-e^{-({\eta }_1+\gamma )a})\right)da.\end{array}$$

When t ≥ τ,

$$q_S(t,\tau )=Q_S(t-\tau ),q_E(t,\tau )=Q_E(t-\tau )e^{-({\eta }_1+\gamma )\tau },$$

$$q_I(t,\tau )=Q_I(t-\tau )e^{-{\eta }_2\tau }+\frac{\gamma Q_E(t-\tau )}{{\eta }_1+\gamma -{\eta }_2}(e^{-{\eta }_2\tau }-e^{-({\eta }_1+\gamma )\tau }),$$

and

$$\begin{array}{lll}{\int }_0^{\tau }{q}_E(t,a)da& =& {\int }_0^{\tau }{Q}_E(t-a)e^{-({\eta }_1+\gamma )a}da,\\ {\int }_0^{\tau }{q}_I(t,a)da& =& {\int }_0^{\tau }\left(Q_I(t-a)e^{-{\eta }_{2}a}+\frac{\gamma Q_E(t-a)}{{\eta }_1+\gamma -{\eta }_2}(e^{-{\eta }_{2}a}-e^{-({\eta }_1+\gamma )a})\right)da.\end{array}$$

Substituting the above equations into (1)–(5), we obtain the following SEIH(Q)R model through delay differential equations

$$\left\{\begin{array}{cc}\frac{dS}{dt}& =-\beta \frac{SI}{N_p}-Q_S(t)+\left\{\begin{array}{ll}0,& (t<\tau )\\ Q_S(t-\tau )& (t\ge \tau )\end{array}\right),\\ \frac{dE}{dt}& =\beta \frac{SI}{N_p}-\gamma E-Q_E(t)+\left\{\begin{array}{ll}0,& (t<\tau )\\ Q_E(t-\tau )e^{-({\eta }_1+\gamma )\tau }& (t\ge \tau )\end{array}\right),\\ \frac{dI}{dt}& =\gamma E-\kappa I-\eta I-Q_I(t)\\ & +\left\{\begin{array}{ll}0,& (t<\tau )\\ \frac{\gamma Q_E(t-\tau )}{{\eta }_1+\gamma -{\eta }_2}(e^{-{\eta }_2\tau }-e^{-({\eta }_1+\gamma )\tau })+Q_I(t-\tau )e^{-{\eta }_2\tau }& (t\ge \tau )\end{array}\right),\\ \frac{dH}{dt}& =\eta I-\mu H+{\int }_0^{t\wedge \tau }({\eta }_1{Q}_E(t-a)e^{-({\eta }_1+\gamma )a}+{\eta }_2{Q}_I(t-a)e^{-{\eta }_{2}a})da\\ & +{\int }_0^{t\wedge \tau }\frac{{\eta }_2\gamma }{{\eta }_1+\gamma -{\eta }_2}{Q}_E(t-a)(e^{-{\eta }_{2}a}-e^{-({\eta }_1+\gamma )a})da,\\ \frac{dR}{dt}& =\kappa I+\mu H,\end{array}\right)$$ (19)

where t ∧ τ = min{t, τ}, N_p = S + E + I + R, and Q_S(t), Q_E(t), Q_I(t) are dependent on S(t), E(t), I(t) through (16)–(18). Assuming that there is 1 exposed individual at time t = 0, which yields the initial condition

$$S(0)=N-1,E(0)=1,I(0)=H(0)=R(0)=0,$$ (20)

where N represents the total population. Thus, the epidemic dynamics can be described by equation (19) with initial condition (20).

2.4. Effectiveness of contact tracing and quarantine

Now, we applied the established delay differential equation model (19) to study the effectiveness of contact tracing and quarantine.

2.4.1. Comparison between the delay differential equation model and the individual-based model

First, to compare the deterministic delay differential equation model with the stochastic individual-based model, we turned the parameters so that the two models give the same epidemic dynamics when the quarantine rate r = 0. We noted that the infection rate β in (19) is not identical to that in the individual-based model, and the transition rate γ should be determined by the average incubation time of the exposed individuals. Hence, we turned the parameters β and γ and fixed other parameter values as those used in Fig. 2, Fig. 3, Fig. 4, so that the solution obtained from (19) is comparable with the average dynamics based the individual-based model (Fig. 5a).

Fig. 5.

Epidemic dynamics based on the SEIH(Q)R model (19). (a) Epidemic dynamics of S(t), E(t), I(t) and R(t) without contact tracing quarantine (r = 0). Solid lines are obtained by solving the delay differential equation (19), and circles are average dynamics based on the individual-based model. (b) Dependence of the final infection size on the quarantine rate r. (c) dependence of the infectious peak value on the quarantine rate r. In (b) and (c), gray bars show the data from deterministic equation (19), blue dots are data obtained from stochastic simulation based on the individual-based model, with mean and standard deviation shown by red error bars. Here, in the deterministic equation, β = 0.810, γ = 0.148, and other parameters are the same as those used in Fig. 2, Fig. 3, Fig. 4.

Next, we varied the quarantine rate r from 0 to 1.0 and compared the final infection size and the peak of the infectious number obtained from the two types of models, results are shown in Fig. 5b and c. From Fig. 5b and c, the deterministic model (19) can capture the average responses with respect to different quarantine rates. However, individual-based model simulation gives rise to stochastic responses, so the same quarantine rate may yield diverse responses. For instance, when r = 0, an epidemic outbreak may occur in most cases, however, there are rare cases that both the final size and the infectious peak are very low, which means that the epidemic outbreak would not happen (Fig. 5b and c). These results suggest that the proposed delay differential equation model (19) can describe the average epidemic dynamics, however, it is not enough to describe the diverse responses due to the stochastic effect.

2.4.2. Effects of contact tracing and quarantine with infection features

Since the outbreak of the COVID-19 pandemic, there are several virus strains, including Alpha, Beta, Delta, Gamma, and Omicron, each virus strain may cause different infection features and symptoms. Now, we study how the control policy of contact tracing and quarantine may affect the epidemic dynamics for different virus strains with various infection features. To this end, we varied the infection rate β and the diagnosed rate η to represent different infection features and solved the model equation with different quarantine rates.

Fig. 6a shows the final size of the infection number with different values β and η (the quarantine rate r = 0.7). The final size obviously decreases with the decreasing of the infection rate β, and the increase of the diagnose rate η. Furthermore, to compare the effects of contract tracing quarantine, we calculated the reduction ratio of final sizes, which is given by the relative reduction number in final size with respect to that without quarantine policy. The results are given in Fig. 6b, which show that the reduction ratio increases with the increase of the diagnose rate η and the decrease of the infection rate β.

Fig. 6.

Effects of contact tracing and quarantine with various parameters of the infection rateβand the diagnose rateη. (a) Dependence of the final size on (β, η). (b) Dependence of the final size reduction ratio on (β, η). (c) Dynamics of total infections (I_total = E + I + H) with (β, η) = (0.6, 0.4) and various quarantine rates. (d) Dynamics of total infection number with r = 0.7 and different parameters (β, η). Other parameters are the same as in Fig. 5.

We further compared the dynamics of total infections number for different parameter values. The results are given in Fig. 6c and d. Results show that increasing the quarantine rate r can obviously reduce the peak value of the total infections number, however, the timing to reach the peak value does not change with the quarantine rate. Moreover, we also noted that increasing the quarantine rate may result in the long tail effect in the later stage (Fig. 6c). We further fixed the quarantine rate r = 0.7 and changed the parameter (β, η) to compare the latter stage dynamics. Results show that for different parameters (β, η), the dynamics in the early stages may be different, however, all three situations give the same prolonged dynamics in the later stage (Fig. 6d). These results indicate that for infection diseases with a large infection rate, the policy of contact tracing quarantine can reduce the total infection number, however, may not be enough to eliminate the infections in a short time. This result is consistent with simulation results based on the individual-based model (Xu et al., 2021).

2.4.3. Effects of contact tracing and quarantine with different incubation time and quarantine duration

In the control strategy of epidemic contact tracing and quarantine, the period of quarantine is important for the overall effect of minimizing the effects on daily life and controlling the epidemic dynamics. To study the effect of quarantine days, we fixed the quarantine rate r = 0.7, changed the transition rate γ from exposed state (E) to infectious state (I), and the quarantine days (τ), and solved the model equation (19) to t = 360 days. Fig. 7a shows the dependence of final size on γ and τ. The final size decreases with the decreasing of γ, i.e., increasing the incubation period. Nevertheless, when we changed the quarantine days from 7 to 21 days, the final size does not sensitively dependent on the changes in the quarantine days (Fig. 7a). Fig. 7b shows the dynamics of total infectious number with different combinations of (γ, τ), which confirms that changes in the quarantine days do not yield significant changes in the epidemic dynamics. Particularly, there is no obvious benefit to the extension of quarantine days from 14 to 21.

Fig. 7.

Effects of contact tracing and quarantine with different incubation periods and quarantine duration. (a) Dependence of the final size on (γ, τ). (b) Dynamics of total infectious number with various parameters (γ, τ). Other parameters are the same as in Fig. 5.

2.4.4. Effectiveness of close contact quarantine and social contact restriction

In the above results, we have seen that the policy of close contact quarantine alone is not enough to eliminate the infectious in a short time. Therefore, further control measures with social contact restriction could be required in order to achieve the outcome of zero-infectious. Here, we examined how social contact restriction may affect epidemic dynamics.

In simulations, we took the default infection rate β = 0.8 and assumed the quarantine rate r = 0.7 that starts when the occurrence of the first infectious, however, the policy of social contact restriction may be implemented at a different time point, which was represented by the decreasing of the infection rate β. Firstly, we assumed that social contact restriction begins at day 50, with different values β from 0.1 to 0.8 (no restriction), smaller β means more strict restriction. The dependence of the total infectious number on the parameter β is shown in Fig. 8a. From Fig. 8a, the infectious number obviously decreases after the implementation of social contact restriction. However, the long-time dynamics are obviously dependent on the infection rate β. There is a threshold on β (black line in Fig. 8a), so that when β is smaller than the threshold, the total infection number achieves zero in the short time, otherwise, there is a long tail of infectious numbers when β is larger than the threshold (Fig. 8a). These results suggest that strict restriction of social contacts so that the reproduction number is less than 1 is required to achieve the aim of the zero-COVID policy.

Fig. 8.

Effects of control measure with contact quarantine and social contact restriction. (a) Dynamics of total infectious number with quarantine rate r = 0.7 and various strengths of the social contact restriction. Here, the social contact restriction starts from day 50, and the infection rate β takes values 0.8 (no restriction), 0.6, 0.4, 0.3, 0.2, and 0.1. The black line shows the dynamics corresponding to the threshold β = 0.3. (b) Dynamics of total infectious number when social contract restriction starts from different days. Here, we take the quarantine rate r = 0.7, and β = 0.1 during social contact restriction. Inset shows the relation between the duration of social contact restriction to achieve zero social infectious and the day of starting the restriction. Other parameters are the same as in Fig. 5.

Next, we examined the effect when the control measure of social contact restriction (β = 0.1) begins to implement at different time points (Fig. 8b). The total infectious number rapidly decreases after we set β = 0.1. To examine how the timing of policy implementation may affect the timing of re-open when the social infectious number reaches zero, we calculated the time length (ΔT) of restriction, which is given by the duration from the starting point of β = 0.1 to the time point when E + I < 1. The result is shown by the inset in Fig. 8b. From the simulation result, starting the restriction early may need less time to control the epidemic situation, with 22 days restriction starting from day 30, compared with the duration of 26 days when starting from day 45. However, if the restriction starts from the latter stage (after day 45 in the simulation), there is no difference in the duration of the restriction, no matter when the control measure starts (Fig. 8b). Simulations show that 4 weeks restriction is in general required to achieve zero social infectious, which is in agreement with real-world data of local COVID-19 outbreaks in various cities of China during 2021–2022.

2.4.5. Comparison between the delay differential equation model and the ordinary differential equation model

Finally, we note that the age-structured model (1)–(11) yields the delay differential equation model (19), which is mathematically different from the ordinary differential equation model (e.g., the SEIR model) in most studies of epidemic dynamics. The age-structured compartment is straightforward for modeling the quarantine situation, however, is mathematically more complicated. Thus, we asked how much difference between the delay differential equation model and the ordinary differential equation model.

To compare the two types of models, we ignored the age-structured in the model (1)–(11) and obtained an ordinary differential equation model. To this end, we write

$${\tilde{q}}_S(t)={\int }_0^{\tau }{q}_S(t,a)da,{\tilde{q}}_E(t)={\int }_0^{\tau }{q}_E(t,a)da,{\tilde{q}}_I(t)={\int }_0^{\tau }{q}_I(t,a)da$$

that represent the number of persons in the quarantine compartment. Next, we integrated equations (6), (7), (8) with a over [0, τ], and replaced q_i(t, τ)(i = S, E, I) in (1), (2), (3) with ${\tilde{q}}_i(t)/\tau$, and obtained the following ordinary differential equation model

$$\left\{\begin{array}{cc}\frac{dS}{dt}& =-\beta \frac{SI}{N_p}-Q_S(t)+\frac{1}{\tau }{\tilde{q}}_S\\ \frac{dE}{dt}& =\beta \frac{SI}{N_p}-\gamma E-Q_E(t)+\frac{1}{\tau }{\tilde{q}}_E\\ \frac{dI}{dt}& =\gamma E-\kappa I-\eta I-Q_I(t)+\frac{1}{\tau }{\tilde{q}}_I\\ \frac{dH}{dt}& =\eta I+{\eta }_1{\tilde{q}}_E+{\eta }_2{\tilde{q}}_I-\mu H\\ \frac{dR}{dt}& =\kappa I+\mu H\\ \frac{d{\tilde{q}}_S}{dt}& =Q_s(t)-\frac{1}{\tau }{\tilde{q}}_S\\ \frac{d{\tilde{q}}_E}{dt}& =Q_E(t)-\frac{1}{\tau }{\tilde{q}}_E-{\eta }_1{\tilde{q}}_E-\gamma {\tilde{q}}_E\\ \frac{d{\tilde{q}}_I}{dt}& =Q_I(t)-\frac{1}{\tau }{\tilde{q}}_I-{\eta }_2{\tilde{q}}_I+\gamma {\tilde{q}}_E\end{array}\right)$$ (21)

Here, N_p = S + E + I + R, and Q_S(t), Q_E(t), Q_I(t) are defined as (16)–(18).

To compare the two models, we select the same parameter values as above and solve the model equations. Results are shown in Fig. 9. When the quarantine rate r = 0, the two models give the same results (Fig. 9a). When the quarantine rate is non-zero, there are slight differences between the results obtained from the two models, the ordinary differential equation model gives a slightly higher number of infected persons (Fig. 9b and c).

Fig. 9.

Comparison between the delay differential equation model (solid lines) and the ordinary differential equation model (dots), with different values of the quarantine rate: (a) r = 0, (b) r = 0.2, (c) r = 0.7. Parameters are the same as in Fig. 5.

3. Discussion

Because of the fast spreading of COVID-19, the policy of close contact tracing quarantine has been an effective measure to control the epidemic dynamics. Here, we developed a mathematical model of the COVID-19 epidemic with confirmed cases-driven contact tracing quarantine. In model formulation, a key issue is to establish the relationship of how the number of quarantine individuals depends on the confirmed number. To address this issue, we applied the individual-based model to simulate the virtual epidemic dynamics with close contact tracing quarantine, from which the formulations of quarantine numbers are derived through simulated data-driven modeling. Finally, a closed-form mathematical model of epidemic dynamics with close contact tracing quarantine was established as a system of delay-differential equations given by equations (16), (17), (18), (19).

Through the established model, we investigated the effectiveness of the control measure of contact tracing quarantine. In general, the policy of contact tracing and quarantine can obviously reduce the epidemic size, however, may not be enough to achieve zero-infectious in a short time. Hence, to achieve zero-infectious, the combination of close contact quarantine and social contact restriction is suggested.

In mathematical modeling, the epidemic dynamics in the real world are often stochastic, as seen in individual-based model simulations. However, the delay-differential equation model (19) is a deterministic model and hence fails to describe the randomness of epidemic control. Particularly, the quarantine functions (16)–(18) can only give a phenomenological formulation of the mean number of quarantine individuals. In real-world situations, the exact quarantine number is often a random number and unpredictable. Hence, the quarantine number should be extended to a random number based on the phenomenological expressions in (16), (17), (18), which yield a random differential equation model for the epidemic dynamics.

Mathematical models have been widely used in predicting epidemic dynamics. However, since experiments on epidemic dynamics are in general banned, it is difficult to discover the detailed process of epidemic transmission through the complex social contact networks. Thus, mathematical models are often developed based on general assumptions on transmission rules. When non-pharmaceutical interventions (NPIs) are applied to control epidemic transmission, it is often difficult to formulate the effectiveness of different NPIs through traditional mathematical models. In this study, we applied the individual-based model to generate virtual data for the epidemic dynamics and established the model formulation with reference to the simulated data. Through the individual-based model, we are able to discover details of NPIs at the individual level and hence can establish a connection between macroscopic epidemic dynamics data and individual-level detail transmission process. This strategy may provide a novel way for the development of epidemic dynamics models with more realistic information.

Declaration of competing interest

The authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Handling Editor: Dr. Jianhong Wu