According to existing research, there are various options for COVID-19 prevention and control policies. These include social distancing, tracking exposed populations, nucleic acid or antigen testing, social lockdowns, and isolation of infected populations. Countries around the world choose different policy combinations to manage public health emergencies. The differences in these epidemic control policy combinations ultimately manifest as differences in the intensity of social activity control. Thus, research on the choice of COVID-19 prevention and control policies explores how to optimise the degree of control over social activities. Among the economic models for policy options, the major ones are cost benefit analysis (CBA), cost effectiveness analysis (CEA), cost utility analysis (CUA), social return on investment (SROI), cost consequence analysis (CCA) [30, 31]. Among them, CBA can express costs and benefits in monetary terms, showing decision makers a direct correlation between cost inputs and project outcomes. It is also able to calculate the net benefits to the economy and society as a whole, which can be used to assess whether or not a policy should be implemented in the whole society. Therefore, compared to other economic models, CBA will be more appropriate for the research objectives of this paper.

Our study constructs a cost-benefit analysis framework for the optimal policy based on the theory of epidemic economics and accounting for factors such as the severity of control and control costs. The framework is mainly based on a counterfactual study: The difference in epidemiologic loss of life and cost of disease between policy interventions and no interventions is treated as a benefit of prevention and control, and the medical resources and economic losses of prevention and control are treated as costs. The difference between prevention and control benefits and costs is the net benefit. If the epidemiological characteristics of the virus change (under conditions such as decreased mortality rate and increased infection rate), how would the intensity of control policies evolve to ensure the maximisation of net benefits?

Therefore, we first need to construct a dynamic model of COVID-19 epidemiology based on the epidemiological characteristics of the virus and social activity control policies. We calculate the number of infections, quarantined individuals, exposed individuals, and deaths with non-pharmacological interventions and no interventions, respectively. A cost-benefit model is then established based on these calculations. Finally, by setting different condition ranges, we simulate the intensity of control when the net benefit of the cost-benefit model is maximised to help us analyse the evolution of optimal control policies.

The SEQIRD modelAssumption 1

The populations involved in the transmission include susceptible (S), exposed (E), extra-quarantined (Q), infected (I), deceased (D), and recovered (R) individuals, where (1) the extra quarantined population (Q) mainly refers to the susceptible population who might have been in contact with the identified infected individuals, including sub-close and general contacts. Q refers to those who have a chance of exposure but are not infected. For ease of simulation and calculation, we assume that those in the exposed population who become infected are quarantined for treatment and are not part of the extra-quarantined population. (2) The exposed population refers to those in close contact with the infected population, who will proportionally become infected. Due to the application of big-data-based epidemiological tracing, close contacts can be accurately identified and isolated either by self-isolation or in specific locations. Therefore, even in the absence of strict containment policies, this group bears isolation costs and is not considered part of the extra-quarantined population. (3) The infected population includes asymptomatic individuals, mild cases, and cases of moderate severity and above. (4) The recovered population includes those who recovered after infection and those who received the vaccine from the susceptible population.

The composition of the population at a certain period is as follows: \(_=_+_+_+_+_+_\). The flow between the different populations is shown in Fig. 1.

Fig. 1

Population dynamics of the SEQIRD model

Assumption 2

According to existing research, current policy types can be divided into two categories based on the intensity of intervention and the degree of strictness of control over social activities: suppression policies aimed at eliminating the virus by lockdown or social distancing, and mitigation policies focused on flattening the curve [7]. Hence, this paper sets the intervention intensity as M, where M is in the range of [0,1]. If policymakers prefer to adopt mitigation policies, then M approaches zero. If policymakers prefer to implement strict control policies, then M approaches one.

Assumption 3

The virus transmission rate \(_\) depends on the virus reproduction number \(_\) and the reciprocal of infection duration \(}\), \(_=_\gamma\) [32]. This involves several parameters: (1) the virus reproduction number \(_\) is the basic reproduction number, which usually refers to the natural reproduction rate of the virus without policy control. The time-variant reproduction number \(_\) is affected by the strictness of control policies and vaccination. (2) γ is the average rate of recovery or death of the infected population, which is the reciprocal of the infection period.

Assumption 4

Changes in the extra quarantined population depend on the quarantine release rate δ. When infection cases are identified, relevant authorities will quarantine susceptible people who may have come into contact with them. These extra quarantined, susceptible individuals can be classified into secondary contacts and general contacts. Close contacts belong to the exposed group. The broader the definition of close contacts in the policy, the higher the proportion of the population that will be quarantined, which may even exceed the infected or exposed population. We assume that the quarantined population comes from the susceptible population, and the population outside quarantine will produce new exposed and infected people based on the increase in the infected population. The main purpose of quarantine is to decrease the rate at which susceptible individuals become exposed and infected by reducing population mobility.

$$d_/_=-\beta __-M*__$$

(1)

$$d_/_=M*__ -\delta _$$

(3)

The transmission rate between those six groups depends on virus epidemiological characteristics, control policies, and the degree of social distancing implementation [12]. Changes in the susceptible population \(_\) mainly include the emergence of exposed individuals (\(\beta __\)), and the extra quarantine population (\(M*__\)) created based on the epidemiological investigation of the infected population. \(_\) represents the rate of symptoms appearing in the exposed population or the patients who test positive in universal testing (these could be asymptomatic patients, mild cases, or symptomatic individuals).

The severity of control measures directly affects the number of extra quarantined individuals. Under these policies, the direct exposure group consists of close contacts who, if effectively identified, will be isolated. The primary group spreading the virus is the unidentified asymptomatic individuals within the exposed group. However, in stringent control policies, some countries also isolate susceptible individuals who may have been in contact with infected people indirectly. This minimises the chance of the unidentified exposed population spreading the virus. The size of this group typically directly correlates with the number of new infections \(__\), represented as \(M*__\). If the control policy requires a broader range of contacts to be quarantined, such as spatial and temporal companions, the number of quarantined individuals could be one, two, or three times \(__\) depending on the quarantine policy design. Under a policy of complete social relaxation where M = 0, the number of additional quarantined individuals would be zero.

According to the aforementioned epidemic model, this study calculates the cumulative number of infections, quarantines, and deaths over a specific period; this provides a foundation for subsequent simulations and analyses of the cost-benefit of control policies.

The cost-benefit model

The theoretical approach of economic epidemiology provides a fundamental framework for analysing epidemic control policies. In economic epidemiology, the cost-benefit analysis of epidemic control includes two aspects: the costs of disease control and the benefits of disease eradication or control. The results of cost-benefit analysis therefore depend on the difference between policy benefits and costs. Hence, the net benefits \(NV\) of control policies can be expressed as the difference between the total benefits \(T_\) and total costs \(T_\) generated by epidemic control policies: \(NV=T_-T_\).

Total health and economic benefits of pandemic control policy

The total benefit \(T_\) refers to the potential losses avoided by control policies. These include the difference in health economic losses and disease costs of infected individuals under both full freedom of movement and control states. The main reason for considering avoidable potential health economic losses as benefits is that COVID-19 infection not only brings disease costs but also results in economic losses due to the depreciation of human capital (manifested as the reduction of quality-adjusted life years for infected individuals). Different policy choices have differentiated impacts on human capital depreciation; this is also one of the key factors in our analysis of the cost-benefits of COVID-19 governance models.

$$\beginT_=\left(_-TDAL_\right]}\right)_+\left(T_-T_\right]}\right)_\\ =_+YLLs*T_\right)}__-_+YLLs*T_\right)}_\right]}_+\left(T_-T_\right]}\right)_,\end$$

(11)

Where TDALY refers to the years of life lost due to death and illness from COVID-19 [33]. TDALY is an increasing function of infection rate and infection mortality rate. The measurement unit is disability-adjusted life years (DALYs),Footnote 2 which can be expressed as the sum of years of life lost (YLLs) and years lost due to disability (YLDs) among infected individuals [10].

\(_-T_)+YLLs*T_\right)}__\) represents the health economic loss of the infected population under a state of complete non-control; \(_-T_)+YLLs*T_\right)}_]}_\) represents the health economic loss of all DALYs in the infected population under some degree of control; \(_\) is the health economic loss per person per DALY - the economic output corresponding to a year of life.Footnote 3

\(\left(T_-T_\right]}\right)_\) represents the epidemiological cost that can be avoided after the implementation of control policies. Generally, the costs associated with an epidemic contain both epidemiological costs and excess burden. Epidemiological costs refer to the cost of treatment, lost wages, and physical and mental suffering of the infected population. Excess burden refers to the costs related to disease prevention, such as self-protection and vaccination costs. Regarding excess burden, as long as the epidemic exists, regardless of control, vaccine costs and self-isolation costs will exist for each person. Therefore, in this formula, \(_\) only represents epidemiological costs. It should be noted that the infected population includes not only patients with different degrees of infection severity but also those who died after infection. Here, we do not distinguish the structure of epidemiological costs among the infected population.

Total intervention cost of pandemic control policy

The intervention cost of control policy (\(}}}_}}\)) includes direct control costs and indirect costs brought about by control. Direct costs refer to the medical and social resources required for control social activity, such as additional isolation sites, medical staff, testing costs, medical observation, and so on. Indirect costs refer to the economic slowdown or stagnation brought about by control, such as economic losses caused by people being restricted from conducting productive activities. The expression is as follows:

$$T_=\left(_+_\right)T_\right]}*\mathrm,$$

(12)

where \(T_\right]}\) is the total number of people quarantined during the pandemic, \(_\) is the per capita direct cost due to control policy, \(_\) is the per capita indirect cost, and Days refers to work time loss because of quarantine.

Net value of pandemic control policy

$$\underset}\, NV=T_-T_$$

(13)

$$=_+YLLs*T_\right)}__-_+YLLs*T_\right)}_\right]}_i\\ \end}+\left(T_-T_\right]}\right)_-(_+_)T_\right]}*\mathrm$$

(14)

s.t. Eqs. (1)–(10)

Based on Eqs. (7)–(13), Eq. (14) can be derived, in which the net benefit is expressed as a function of parameters such as infection rate, mortality rate, number of exposed people, and others. Cost estimation can be done without considering discounting. Given the relatively short timeframe of the pandemic outbreak, a social discount rate between 3 and 5% makes little difference within a year [34].

In accordance with the principle of maximising net value, we use the GEKKO proposed by Beal et al. [35] to conduct numerical simulations to solve for the optimised M when maximising the net benefit.

Parameter setting

For the above SEQIRD model and cost-benefit model, this paper uses the IPOPT solver in GEKKO to calculate the optimal solution for linear programming [35]. For the parameters of the SEQIRD model, we refer to the research of Berger et al. [19], mainly adopting conclusions that are as consistent as possible in most studies. Of course, there are multiple variants of the coronavirus, so its epidemiological characteristics will change with each variation. To account for this, we categorise the virus variants into three major types: the initial outbreak of COVID-19, the Delta variant series, and the Omicron variant series. There are significant characteristic differences between these three series.

In addition, for the parameters of the cost-benefit model, different countries face different cost levels, which is one of the key factors driving various economic entities to adopt different control policies. For this, we divide the economic entities into high-cost and low-cost economies. The high-cost economy mainly includes high-income economic entities, while the low-cost economy mainly includes middle- and low-income economic entities.

We have summarised and collated the parameters estimated and measured in the existing literature and set the model parameters in this article based on these. The definitions and numerical ranges of the parameters in the model are as follows:

(1)

Basic reproduction number \(}}_\): according to existing research, we set the range of \(}_\) to be between 1.4 and 24. This is mainly based on the large number of estimates for the basic reproduction number of the coronavirus made by scholars during the early outbreak in 2020. For example, some scholars estimated the average \(}_\) in countries like the US and Japan to be between 3 and 5 [36]. Estimates for \(}_\) during the COVID-19 pandemic in Wuhan, China, range from 3.11 to 6.47 [37,38,39], or 1.4 to 6.47 [40]. With the virus mutations, some scholars have estimated the transmission characteristics of the Omicron variant in five countries including India, Indonesia, Malaysia, Bangladesh, and Myanmar, and found that the range of reproduction number is between 0 and 9 [41]. The range of basic reproduction number for each series of viruses is shown in Table 1.

(2)

Infection rate \(}}_}}\): the infection rate can be defined as the proportion of the exposed population that becomes infected, which is equal to the reciprocal of the average incubation period. Regarding the infection rate of the virus during the uncontrolled period in the early days in Wuhan, some studies found that the range of \(_\) is between 1/5 [28] and 1/3 [42]. With the mutation of the virus, the average combined incubation period is 6.57 days, and the average incubation days vary among different variants. For example, the average incubation period for cases caused by the Alpha variant is 5.00 days, 4.50 days on average for the Beta variant, 4.41 days on average for the Delta variant, and 3.42 days on average for the Omicron variant [43]. Some research has indicated that the incubation period for Delta is 4.16 ± 2.03 days, 4.85 ± 2.37 days for Omicron BA.1, and 4.17 ± 1.94 days for Omicron BA.2 [44]. Accordingly, the infection rate of the virus is the reciprocal of the above incubation days. The range of incubation periods for virus is detailed in Table 1.

(3)

Reciprocal of infection duration \(}\): this is the proportion of the infected population that becomes the recovered and dead population, represented as the reciprocal of the average duration of disease. Existing research estimates the range of the reciprocal of disease duration to be between 1/18 [28] and 1/5 [42]. When the infection incubation period is around 5 days, the duration of the disease is correspondingly set at 18 days [28]. The range of disease duration is detailed in Table 1.

(4)

Infection fatality rate \(}}_}}\): there are two kinds of measures for infectious disease fatality rate, including case fatality ration (CFR) and infection fatality ratio (IFR) [45, 46]. When all infection cases can be fully identified, the value of CFR and IFR would be the same; otherwise, CFR will overestimate IFR. Some scholars have estimated IFR, finding that the IFR of the early virus is approximately between 0.5% and 1% [45, 46]. The infection fatality rate of the Omicron variant in India, Indonesia, Malaysia, Bangladesh, and Myanmar is estimated to be between 0.016% and 0.136% [41]. The range of IFR is detailed in Table 1.

(5)

Recovery rate \(}}_}}\): the recovery rate can be expressed as \(}_}=\gamma -_\) because the population faces two scenarios after a certain course of the disease: recovery or death.

(6)

Rate of release from quarantine \(}\): since the virus incubation period is between 5 and 18 days, during the early outbreak of COVID-19 in Wuhan, the isolation period was 14 days. In this study, the contact isolation ratio during the containment period is calculated as 1/14.

(7)

Disability-adjusted life years \(}}}}}\): according to existing research, different degrees of disease symptoms cause different losses of life. According to [22], the YLLs caused by COVID-19 would be 14.24 years per case. YLDs are summed for mild, severe, and critical illnesses. To compare the results more intuitively in this paper, we calculated the weighted average YLDs for the above three symptoms according to the parameters in table 1 of Zhao et al. [22], namely\(\sum proportion\, of\, cas_*disability\, weigh_*duration\, of\, cas_=0.815*0.01*0.04+\left(0.138+0.047\right)*0.53*0.12=0.01\).Footnote 4 Furthermore, some research has indicated that COVID-19 may leave certain sequelae, and the life years loss caused by sequelae of the disease per person in Zhao et al. [22] research is \(\sum disability\, weigh_*duration\, of\, cas_=0.17*0.25=0.0425\). Therefore, the YLDs are set as 0.0525 per case in the simulation data.

(8)

Per capita output \(}}_}}\): this indicator refers to the economic loss of disability-adjusted life years, measured here by gross domestic product (GDP) per capita [22]. Since the measurement of the value of life is more controversial as well as influenced by different values, this paper only analyses the economic loss due to health loss from the economic perspective, which also reflects the relative evaluation of the value of life. Different economies have different GDP per capita, which can be classified as high income, upper-middle income, lower-middle income, and low income according to World Bank statistics. The global average per capita GDP was $12,236 in 2021. This study selects high-income economies and upper-middle-income economies both above and below this average as the two typical objects for comparison in the cost-benefit model.

(9)

Per capita epidemic cost \(}_}\):\(_\) is defined as the direct medical costs per case. Wage loss due to illness is already expressed in \(_\), so it is not double-counted here. High-income economies usually have different direct medical costs of epidemics with other income-level economies, i.e. upper-middle income, low income economies. For instance, existing research suggests that the range of average medical costs for different COVID-19 symptoms (general care, inpatient care, critical care patients) in the United States is approximately $9,763–$61,168 per case [47], with a national average medical cost $3,045 per case [15]. For other upper-middle-income economies like China, the weighted treatment cost for severe, moderate, and mild confirmed COVID-19 cases is ¥22,061.94 (USD $3,192.76) per case [48].

(10)

Direct cost per capita of quarantine \(}_}\): For upper-middle-income economy, existing research by Jin et al. [48] found that the average direct medical cost of managing the quarantined population is ¥584.08 (USD $84.53) per person, which includes case identification, testing, and medical observation during isolation. The direct non-medical cost includes fees for isolation sites. The per capita isolation cost for those testing negative is ¥150 (USD $21.4) per person per day [48]. Therefore, the per capita direct cost of isolation is the sum of the direct medical cost and the direct non-medical cost, averaging ¥734.08 (USD $104.87) per person per day. In high-income countries like the United States, the direct non-medical cost of managing the isolated population in alternative care sites is $304 per person per dayFootnote 5 [49], and the testing cost is $51 [49], which totals $355.

(11)

Indirect cost per capita of quarantine \(}_}\): there is a significant difference in indirect costs between high-income and lower-middle-income countries. We use the average daily wage per capita to express the indirect losses of isolated personnel. It is important to note that the indirect losses experienced by isolated individuals may differ depending on their occupation and income level.

(12)

Quarantine duration per capita \(}}}}\): the number of days of quarantine varies at different stages of the epidemic. The government mandates an average of 14 days of isolation for quarantined people [48].

Table 1 Parameter settings of the SEQIR model and cost-benefit model

COVID-19 has gone through variants like D614G, Beta, Delta, and Omicron. We list the features of virus strains at different periods below according to existing studies (see Table 1), such as reproduction number, incubation period, duration of illness (proportion corresponding to the transition to the recovered and deceased population), and infection fatality rate.

In accordance with Panel A of Table 1, the initial virus had less ability to spread relative to mutant strains such as Delta and Omicron, but the mortality rate was much higher than that of other mutant strains; this is due to both the decrease in lethality during the mutation of the virus itself and the implementation of vaccination programmes in various countries. The trends in the evolution of epidemiological characteristics can be summarised as follows. Firstly, the basic reproduction number increases with virus variation and the incubation days become relatively shorter, which indicates that the infection rate is increasing with variation and the transmission rate is also increasing. Secondly, the disease duration gradually decreases. Thirdly, the mortality rate of infection decreases with variation, vaccine, and treatment.

In addition, various economies face different cost levels for prevention and control, especially for economies with different income levels. In this regard, this paper establishes two sets of initial values (see Panel B): One group consists of high-income economies with high COVID-19 epidemic prevention and control costs, represented by the United States in Panel B; the other group consists of upper-middle-income economies with low COVID-19 epidemic prevention and control costs, represented by China in Panel B. We investigate how the optimal intensity of COVID-19 control policies in economies with different cost levels changes with the evolution of virus transmission characteristics.

All economies face changes in virus transmission characteristics. We limit these changes to two scenarios. The first is based on the initial virus transmission characteristics of relatively low reproduction numbers, a long incubation period, a long infection period, and a high mortality rate. The second scenario is based on the transmission characteristics of variants like Delta and Omicron. This scenario is characterised by relatively high reproduction numbers, a short incubation period, a short infection period, and a low mortality rate.

For the first scenario (Scenario 1), which has relatively low reproduction numbers, a long incubation period, a long infection period, and a high mortality rate, we set the initial values as follows: a basic reproduction number of 3, an incubation period of 6.5 days, an infection period of 18 days, and a mortality rate of 0.01. Based on this, while keeping other parameters constant, we adjust one parameter within the variable range set by Table 1. We then analyse how the degree of pandemic control in different economies changes with that parameter.

For the second scenario (Scenario 2), characterised by relatively high reproduction numbers, a short incubation period, a short infection period, and a low mortality rate, we set the initial values as follows: a basic reproduction number of 9.5, an incubation period of 3 days, an infection period of 11 days, and a mortality rate of 0.0001. Similarly, while keeping other parameters constant, we adjust one parameter within the variable range set by Table 1. We then analyse how the degree of pandemic control in different economies changes with that parameter.