A benefit–cost analysis of different response scenarios to COVID‐19: A case study

Abstract Background This paper compares the direct benefits to the State of Western Australia from employing a “suppression” policy response to the COVID‐19 pandemic rather than a “herd immunity” approach. Methods An S‐I‐R (susceptible‐infectious‐resolved) model is used to estimate the likely benefits of a suppression COVID‐19 response compared to a herd immunity alternative. Direct impacts of the virus are calculated on the basis of sick leave, hospitalizations, and fatalities, while indirect impacts related to response actions are excluded. Results Preliminary modeling indicates that approximately 1700 vulnerable person deaths are likely to have been prevented over 1 year from adopting a suppression response rather than a herd immunity response, and approximately 4500 hospitalizations. These benefits are valued at around AUD4.7 billion. If a do nothing policy had been adopted, the number of people in need of hospitalization is likely to have overwhelmed the hospital system within 50 days of the virus being introduced. Maximum hospital capacity is unlikely to be reached in either a suppression policy or a herd immunity policy. Conclusion Using early international estimates to represent the negative impact each type of policy response is likely to have on gross state product, results suggest the benefit–cost ratio for the suppression policy is slightly higher than that of the herd immunity policy, but both benefit–cost ratios are less than one.

enforce quarantine and self-isolation measures to contain spread. This response, termed "suppression," is consistent with policies implemented by governments around the world aimed at slowing and eventually reversing epidemic growth, reducing case numbers to low levels, and maintaining that situation indefinitely. 3 As an alternative, the Western Australian government could have chosen to follow a different management strategy based on the principle of "herd immunity." A herd immunity response policy assumes the likelihood of an infected individual coming into contact with a susceptible individual is lessened with a proportion of the population (but not all) being immune. 4,5 This effect may soon be achieved through widespread vaccination, but given the disease-induced herd immunity level for COVID-19 is relatively low, 6 it could be achieved by allowing infections and recovery to occur in less-vulnerable sections of the population. Sweden has officially adopted this natural science herd immunity approach in its COVID-19 mitigation strategy. 7 This paper describes an S-I-R (susceptible-infectious-resolved) model that is used to estimate the likely benefits of a suppressed COVID-19 response compared to a herd immunity alternative.

| Model
A simple S-I-R model is used to simulate the spread of COVID-19 through the Western Australia population. Its components are: I t ¼ I tÀ1 þ R 0 βI tÀ1 S tÀ1 S t0 À γ 1 À δ I A À Á I tÀ1 À γδ I AI tÀ1 ð2Þ In Equations (1)-(4), S t is the number of susceptible individuals within a population N in time period t after the initial introduction of the virus who have not been infected; I t is the number of people within the population in period t who are infected and can transmit the virus; R t is the number of resolved cases in period t who are no longer capable of transmitting the virus, including those who have returned to health and those who have died; R 0 is the average number of people that one infectious person will go on to infect; β is the final outbreak size expressed as a proportion of the total susceptible population; γ is the resolution rate of infections; δ I is the case fatality rate among people with a high risk of severe infection; and A is the proportion of the population at high risk of severe infection.
Equation (1) Here, D t is number of deceased individuals within a population N in time period t after the virus is introduced; η is the proportion of infected individuals requiring hospitalization; B t is the number of hospital beds available for COVID-19 patients on day t after virus introduction; and δ E is the case fatality rate among people with a high risk of severe infection who are turned away from hospitals when no beds are available (ie, δ E > δ I ).
Equation (5)  By simulating values for I t and D t from Equations (2) and (4), costs imposed by COVID-19 can be estimated as it moves through the Western Australia population. The costs related to nonfatal infections t days after virus introduction, C I t , are: Here, H is the cost of that hospitalization; ω is the proportion of infected individuals in need of sick leave; W is the average fortnightly wage rate (ie, assuming two working weeks are lost as a result of illness); and E is the extra costs society pays to treat those turned away from hospitals in other locations.
Equation (6) states that if the number of cases is less than the number of available hospital beds, the costs related to nonfatal infections will depend on the number of infected individuals requiring hospitalization, the cost of hospitalization, the number of people who require time off work to recover from the virus, and the average fortnightly wage. If the number of cases is greater than the capacity of the State's hospitals, the costs related to nonfatal infections depend on the maximum number of beds and the cost of hospitalization, the number of people who require time off work to recover from the virus, and the average fortnightly wage, plus the societal costs involved in providing care for those patients turned away from hospitals.
The cost of fatal infections at time t days after the introduction of the virus, C D t is calculated as: Here, D t is the number of fatal infections occurring t days after the virus is introduced to the population; and L is the value of a statistical life-a measure of the willingness of individuals to pay for a reduction in mortality risk sufficient to lower the expected number of fatalities by one over a given period of time.
Equation (7) states that the cost of fatal infection costs will be determined by the number of fatal infections and the statistical value of lives lost.
With simulated values for C I t and C D t , the combined total nonfatal and fatal infection costs of the virus (C T ) over n days are: In the results section, C I t , C D t , and C T are reported for: (a) the counterfactual "do nothing" policy in which the virus is permitted to spread throughout the Western Australian population without special measures to slow infection, (b) the suppression policy reflecting the policy currently in place, and (c) a herd immunity scenario in which the virus is allowed to spread through the nonvulnerable portion of the population while vulnerable portion is protected through isolation.
Specifically, the do nothing scenario involves no restrictions to people movements or behavior being enforced by the Western Australian government. This does not mean people will not take personal decisions to minimize risks associated with virus spread, such as social distancing, self-isolating when ill, or wearing face masks in The total benefit achieved by pursuing either the suppression policy (B T S ) or herd immunity policy (B T H ) over n days is measured by avoided costs and, therefore, is calculated as the difference in total costs incurred in these scenarios and in the do nothing scenario: Here, C T D , C T S , and C T H are the total nonfatal and fatal infection costs likely to occur over n days under do nothing, suppression, and herd immunity response policies, respectively.
Based on the parameter values specified in Section 2.2, the virus simulation is run for 365 days following an initial introduction. Using the Monte Carlo method, 10 000 iterations of the input model defined in this section are run to generate probability distributions of possible outcomes. The results reveal all possible events that could happen according to the model's structure and parameters, and the probability of each outcome occurring.

| Parameters
Model parameters and their assumed values, drawn from the relevant literature, appear in Table 1 and are discussed below. Pert distributions are preferred when evidence and expert opinions on parameter values are mixed, 12 and uniform distributions are used to represent highly uncertain parameters.
The total population of Western Australia is currently 2 590 290. 13 This value is used for the parameter N.
Preliminary estimates of the R 0 for COVID-19 indicate a broad range of values between 1.4 and 4.0. 11,[14][15][16] It is specified in the model using the pert distribution pert(1.5,2.5,3.5) under the do nothing scenario. Following Ferguson et al, 3 it is assumed actions taken in the suppression policy option (ie, social distancing, case isolation, household quarantine, and school and university closures) will reduce R 0 to close to one, although a range of possibilities is considered. R 0 is assumed to change by uniform (À75%,À30%) under the suppression policy and by uniform(À15%,À5%) under the herd immunity policy.
The final outbreak size expressed as a proportion of the total susceptible population, β, is estimated to be between 5% and 40%. 11 This wide range reflects differing accounts of its breadth in different countries. It is specified in the model using a narrower distribution, pert (0.25,0.3,0.35).
The resolution rate, γ, is the inverse of the infectious period for an average person. Early indications are that the γ is around 33%. 17 It is specified in the do nothing scenario as pert(0.3,0.325,0.35). Under the herd immunity policy, the average infected person is likely to be of a younger age than in either the do nothing or suppression scenarios. Therefore, a higher resolution rate of pert(0.33,0.358,0.385) is assumed, representing a 10% increase.
Similarly, the case fatality rate, δ I , is likely to fall as the composition of infectives changes according to the scenario. For the population as a whole, it is specified as pert(0.007,0.01,0.014) [18][19][20] and is assumed constant over the do nothing and suppression scenarios. As the average age of infectives is younger in the herd immunity scenario, the case fatality rate is reduced by 10% to pert (0.006,0.009,0.013). The importance of this specification is discussed in the sensitivity analysis in Section 3.4.
The proportion of the population at greatest risk, A, is approximated using age demographic data for Western Australia. 13 This is shown in Figure 1, below. Assuming people between the ages of 70 and 100+ to be at the highest risk, 3,21 this accounts for approximately 10% of the Western Australian population. Allowing variability around this mean value, A is specified in the model as pert (5%,10%,15%).

Using Ferguson et al 3 as a broad indication of social distancing effects,
its parameter value is changed by uniform (À90%,À75%) and uniform (À75%,À50%) under the suppression and herd immunity policies, respectively.
Nguyen-Van-Tam et al 22  The cost of hospitalization, H, is specified as $4600 per day in today's dollars, 23 with an average length of hospital stay of 3 to 9 days with a most likely duration of 5 days. 24 The proportion of infections requiring time off work, ω, is assumed to be between 3% and 13% [ie, uniform(0.03,0.13)]. This distribution is estimated on the basis that approximately 80% of people infected with COVID-19 show mild symptoms, while 20% exhibit more severe symptoms, 21,25 and around 64% of people infected are likely to be of working age. 13 When infected workers do stay at home, it is assumed they are absent for two full weeks of work. Hence, the average fortnightly wage for Western Australian workers is used to approximate the parameter W. A value of $2660/fortnight is used. 26 Given the majority of 70+ year olds are retired from the workforce, this parameter is assumed constant across the three scenarios.
It is difficult to estimate the cost society incurs as a result of additional care for excess patients turned away from hospitals when capacity is reached, E. These patients require adequate nursing availability, 24-hour on-call medical advice and home support, patientcentered planning, daily nursing review and adjustment of individual care plan, professional multidisciplinary team support (eg, occupational therapy, physiotherapy, social work), and a discharge hand-over to ongoing support services. 27,28 Without economies of scale, the marginal cost of these services tends to be higher in makeshift facilities or homes than in hospitals, and the duration of health episodes can be longer. 29 The cost of providing services to infected patients turned away hospitals is assumed to be double the hospitalization costs, H. 28 The case fatality rate of patients turned away from hospitals, δ E , is also difficult to estimate. Various studies have shown a negligible difference in treatment outcome between patients utilizing home hospital care and those treated in hospitals. 28,[30][31][32][33] Notably, Vianello et al 34 also found no statistical difference in treatment failure rate for patients treated for respiratory tract infections at home or in hospital.
However, no COVID-19-specific estimates are available. In the absence of empirical evidence and to allow for a range of possibilities, it is assumed δ E > δ I by a factor of uniform (0%,50%).
The age-adjusted value of a statistical life, L, represents the marginal rate of substitution between wealth and mortality risk corrected for the age of the population studied. 35 In a review of empirical estimates relevant to Australia, Abelson 36 44 Assuming that approximately one-third of these beds will be occupied by patients with other needs, this leaves around 6 600 beds available at any one time for COVID-19 patients.

| Benefits
If the number of infected individuals requiring hospitalization exceeds this capacity, excess patients will be unable to access appropriate care. As stated in Equation (6), the model assumes these cases have a higher likelihood of resulting in death.

| Sensitivity analysis
To determine the effect of uncertain parameter values on model output, each parameter is sampled across its specified range while  24,45 Hospitalization cost has a positive relationship with total benefits, and because the input distribution is right-skewed, the righthand-side sensitivity bar is longer than the left. Changing hospitalization costs from $23 000 in the base case to $13 800 (a change of À40%) lowers total benefit from $4.3 billion to $3.6 billion (À16%) in the herd immunity scenario and from $9.2 billion to $5.5 billion (À40%) in the suppression scenario. Conversely, increasing its value to $41 400 (80%) increases the total benefit to $5.0 billion (16%) in the herd immunity scenario and $11.3 billion (23%) in the suppression scenario.
The age-adjusted value of a statistical life is also a highly sensitive parameter in both scenarios. Lowering its value from the mid-point value of $422 500 to $280 000 (a change of À34%) changes total benefits of the herd immunity policy by À16% (to $3.6 billion) and by À9% in the suppression scenario (to $8.4 billion). Likewise, increasing it to $565 000 (a change of +34%) increases the total benefit by approximately 14% (to $4.9 billion) in the herd immunity scenario and by 10% (to $10.1 billion) in the suppression scenario.
In view of its sensitivity, problems with the age-adjusted value of a statistical life used in this study need to be recognized. Its derivation assumes a linear relationship between age and the value of a statistical life, but in practice, this is highly uncertain. 42 Evidence suggests the relationship may in fact follow lifetime consumption patterns, being low early and late in life and high in the middle. 46 Moreover, quality of life is not captured in the age-adjusted value of a statistical life, meaning the value of life years spent in discomfort due to poor health is considered the same as those spent in good health. However, methods that apply a discount to years of ill-health or disability, termed "quality adjusted life years," are problematic, particularly in terms fairness. Negative social perceptions of medical conditions inflate the perceived social benefit of interventions to address them, unfairly raising the value of quality adjusted life years for worse conditions. 47 Similarly, the more treatable a condition, the higher the perceived social benefit of sufferers receiving treatment before those suffering from less-treatable conditions. 48

| DISCUSSION OF BENEFITS RELATIVE TO COSTS
To put the total benefits of each policy into perspective, they can be represented as a proportion of the Gross State Product (GSP) of the Western Australian economy. This is equivalent to the national Gross of implementing a policy that will exactly offset the benefits it is likely to generate. The break-even policy cost can be estimated as follows: Here, P S and P H are the break-even economic costs for suppression and herd immunity policies expressed as a percentage of GSP.
Given the 2018/19 GSP of Western Australia was $260.6 billion, 49 histograms of P H and P S are show in Figure 6. Results indicate mean P H is approximately 1.6% of GSP over the 365-day period simulated in the model (ie, $4.3 billion), whereas estimated mean P S is 3.5% of GSP (ie, $9.2 billion).
It is too early to have rigorous estimates of the negative economic impact of both types of policy response in terms of decline in GSP with which to compare these estimates of benefits. However, early estimates of the impact of Sweden's herd immunity approach to virus response suggest an overall negative GDP impact for 2020 of between 4% and 6.7% of Sweden's GDP. 50,51 Early estimates of the broadly suppression type of response for the EU as a whole suggest an overall negative GDP impact for 2020 of between 5% and 12% of EU GDP. 52 The mid-points of each of these sets of estimates put the eco-  58 More research is needed on these broader societal impacts of response actions before a comprehensive policy assessment can be made.

ACKNOWLEDGMENTS
The authors would like to thank an anonymous reviewer for helpful comments and suggestions that greatly improved the paper.

CONFLICT OF INTEREST
Rob W. Fraser and Simon J. McKirdy certify that they have no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this manuscript. David C. Cook declares that he has an affiliation with the Western Australian Department of Primary Industries and Regional Development.

AUTHOR CONTRIBUTIONS
Conceptualization: David C. Cook, Rob W. Fraser, and Simon J.

McKirdy
Formal Analysis: David C. Cook Methodology: David C. Cook, Rob W. Fraser F I G U R E 6 Break-even economic costs of implementing response policies. The total benefit of each response policy is shown as a proportion of the Gross State Product (GSP) of the Western Australian economy