During the COVID-19 pandemic, many policymakers are usually facing two separated sources of information: economic models that usually predict an economic collapse [15] and epidemic models that focus on death counts [12] .
However, both the economic and mortality figures are key policy variables during a pandemic but few articles integrate both approaches [9, 16] . In particular, no research (to our knowledge) has analyzed both the fiscal and mortality impact of different mitigation measurements. In this article we strive to fill that gap by approximating the impact of physical distancing and patient care on the death toll and government budget, in a attempt to find the optimal conditions to balance it all.
Vaccination or therapeutics can eradicate epidemics from the population, like the case of smallpox [2, 3] but when a newly discovered virus hits the population, the entire world is at risk because everyone is susceptible as in the case of the novel SARS-CoV-2 that is impacting us in 2020 [13] . In the case of an imported infection (i.e. not an endemic epidemic), the first-best strategy would be to control borders, identify, treat and isolate infected individuals. This occurred in the U.S. with the Ebola virus, which never became an epidemic [7] . But when a virus is already circulating in a territory and there is no antidote or massive testing and contact tracing available, social or physical distancing is an alternative to mitigate a pandemic and provide the scientific community time to research and find alternative measures such as an effective treatment or a vaccine. Also, physical distancing measure gives fragile healthcare systems the leverage to take care of chronically ill patients without saturation of existing capacity. What are the fiscal and human costs of all these measurements in the short and long run?
Thus, two research questions drive this study: What is the optimal physical distancing policies in a country and what are the implications of these policies for both the government budget and loss-of-life? We constructed an enhanced mathematical SIR (Susceptible, Infected, Recovered) epidemic model [5] to simulate the COVID-19 epidemic in the US in an attempt to estimate the fiscal impact and the optimal conditions to mitigate this ongoing pandemic. We found that a policy of no physical distancing or a race towards herd immunity is not the optimal policy choice when both human and fiscal costs are considered.
In Section 2 we lay out our methodology. In Section 3 we show the dynamics associated to our calibrated system of differential equations. In Section 4 we discuss our results and in Section 5 we conclude and recommend public policies.
We first describe a simple economy with three sectors; businesses, government, and a household sector with two actors. In the second part of this section we describe our epidemic model.
In this economy, the household sector is mobile within the country and is composed of L workers and U individuals that are not working. Thus, employment is less than full. This characterization allows us to consider the supply shocks associated to the COVID-19 pandemic [10] , where laborers are impeded to work fully because of lock-downs or infections affecting members of the household sector.
Firms produce goods and services i, which require X amount of L. A fixed amount of total output y is predetermined to be produced in period t=0 and is given by, y = X i * L i . However, firms are able to adjust its output when external changes hit the labor stock. The total output that considers the impact of such external changes is observed in, Y t = yH t where H t = dL/dt.
We hold the following assumptions over H:
• if physical distancing is implemented at t=1, H t = −0.3 during the physical distancing. When the physical distancing ends in t=n, and H t=n = 0.1
This setting let us capture the V-shape growth that is being projected [11] in the post-COVID-19 period.
• if no physical distancing is implemented, the pandemic ends in t=n+j, H t=n+j+1 = 0, and H t<j = −0.1. This decline is lower than the lower bound estimate of fourteen percent decline in output constructed by JP Morgan for the U.S., a country that did not declare a general lock-down [15] . In our case, n+j+1 equals 500 days.
• if physical distancing is implemented late at t=i, with j<i, H t = −0.3 during the physical distancing, H t=n = 0.1 when the physical distancing gets relaxed, the pandemic ends in t=n+i. In our case, n+i equals 600
On the other hand, the government sector has a fixed level of expenditures predetermined in period t=0, that includes money transfers to low-income U.
Tax revenues C are governed by,
where τ is the tax rate. If there is a pandemic, the government spends M v in the treatment of each infected individuals.
In order to assess the fiscal impact that a pandemic such as COVID-19 will convey by the epidemic dynamics we construct an epidemic model, with the addition of a fiscal impact differential equation based on the previous section that quantifies the impact in the budget of the country. The epidemic model is a typical SIR-type model with the epidemiological classes needed for the evolution of COVID-19 within the population. There are six epidemiological classes defined Susceptible (S) individuals that progress to incubate the virus in the Exposed (E) class when they have an effective contact with an infected symptomatic individual (I) with probability βSI N using mass action or if a Susceptible individual had contact with an Asymptomatic (A) individual with a lower probability of infection than a contact with a symptomatic governed by the factor βSµA N . Treated individuals (T) are assumed to be quarantined, hence, they are not considered to be able to transmit the diseases in the context of this modelling approach. After 14 days of incubating the virus a proportion of individuals becomes symptomatic and the rest are asymptomatic (estimated here at 35% [1] ). Infected individuals that are asymptomatic recover without further complications, symptomatic individuals that get critically ill seek treatment while the mild cases recover. Following the related literature cited in Table 1 below, symptomatic and treated individuals have a probability of dying due to health complications estimated at 3%. In Table 1 below there is a description of the parameters used.
The system of differential equations for COVID-19, is now presented in Equations 1 to 6. In particular, the system is described as:
next couple of sections. pandemic.
The parameter values in our table were taken from a recollection of the events developing on COVID-19 and the literature. Perturbations around the means showed in Table 1 may affect the quantitative magnitude of our figures, but results are qualitatively similar after such perturbations. In other words, our findings for the second-best scenarios are robust to deviations of our parameters:
such deviations will largely scale up or down the quantitative aspects of each scenario.
The parameters of the model were fitted or estimated to maintain a basic reproductive number relatively close to 3.8. For comparison purposes lets note that the 0 of seasonal influenza ranges approximately within 1.7 to 2.1 [21] .
For the 2009 influenza A-H1N1 the 0 was estimated to be between 1.2 to 1.6.
In particular, the basic reproductive number 0 is the number of secondary cases a single infectious individual generates during the period of infectivity on a completely susceptible population. We assume that the entire population is susceptible such that S ≈ N and that the epidemic has not started in t=0. The individuals that can potentially infect the population in our model are infected individuals that are either symptomatic or asymptomatic. Treated individuals are assumed to be quarantined. Following the related literature [22] , we use the next generation operator to compute the 0 . Let the vector F be the rate of new infections flowing to the latent compartment and the vector V to be the rate of transfer of individuals out of the compartment that are able to transmit the disease. Then, using our SIR system of equations we define
In order to compute 0 , let the gradient of F be defined as F = ∂F ∂E , ∂F ∂A , ∂F ∂I and let the gradient of V be define as V = ∂V ∂E , ∂V ∂A , ∂V ∂I then we get:
Then 0 is the spectral radius of the second generation operator ρ(FV −1 )
also know as the dominant eigenvalue of the matrix FV −1 . Hence,
, which means that the basic reproductive number is:
Our intention in this article is to study the impact of the COVID-19 epidemic in an attempt to estimate the optimal conditions to mitigate the fiscal and mortality impact associated to this pandemic. Thus, the stability of the system and equilibrium points will not be addressed, only the 0 was computed. We focused our efforts on simulating scenarios, which are presented in the next set of sections.
In order to go through the methodology of our simulations we lay out simple In the long run, herd immunity has a higher death toll and implies more government expenditures than the alternated intensity of physical distancing, as shown in Table 2 . In particular, with no physical distancing would die 10,689 more individuals than in the alternated scenario and the government would lose $1.16 trillion more than in the varying physical distancing. Note that under the alternated scenario scientists have approximately 400 days to find an antidote with a very low number of victims, vis-a-vis 150 days under the no distancing case. Figure 2 : Varying physical distancing starting two days after the epidemic: lowering the infection rate 20% for four-week intervals and increasing it between 50% to 75% for eight weeks intervals as described on Table 2 . Figure 1 , 2, and 3
On Figure 2 , we illustrate the simulated figures that are obtained when we shorten the length of the cycles showed previously. Here we are also holding the assumption that the pandemic started in day two. The variation in the probability of infection is shown in the gray line, which goes first to an extreme reduction of 80% for four weeks, then is relaxed to 50% for eight weeks, then two cycles of extreme measurement for four weeks followed by a relaxation of Table 2 . We observed that the strict alternated scenario of Figure 1 would save the federal government $142.9 billion more than in the relaxed scenario of Figure 2 , while mortality is virtually the same. In other words, the distancing cycles shown in Table 2 appear to be less optimal than in the more restrictive case of Table 2 . Table 3 . Since there is no infection curve in the restrictive and permanent case, the budget of the government is mostly affected by the decline in tax revenues caused by lowered economic output. However, when infection rate is lowered just by 40%, the government budget becomes highly impacted after day 400: once infection starts to increase, the public sector is affected by both spending more in treatment and by the reduction in tax revenues caused by the economic collapse. In the case of no physical distancing, even though economic output decreased, the federal budget is highly reduced by the treatment cost. At the end, if there is an opportunity to implement a persistent and early public health policy, it would be better to do it intensely (e.g. by combining a strong physical distance with massive testing): otherwise, the relatively human and fiscal cost would increase significantly.
How would the figures change if the persistent measure is implemented late?
The evolution of infection and mortality is similar, but there are differences in the long run for the associated fiscal costs. When the infection is reduced by 40% or 80%, the associated cost of the pandemic to the government decreased by $51.4 billion than when the public health policy is implemented earlier like in Figure 4 . This difference arises because the economy was not halted during the first weeks of the pandemic. That is, even in the case of these late decisions the government is better off in implementing the strict measure: death toll does not change while fiscal costs perform much either in day 2 or day 20 after the pandemic (See Appendix B).
In Figure 5 we show the effects of lowering the probability of infection within the whole U.S. population for different time intervals. When infection is reduced by 90% for eight weeks, mortality is 10,713 lower than in the herd immunity case and the fiscal cost of the pandemic is the minimum with respect to any of the scenarios presented in this article. That is, this short and strong distancing policy would have a lower death toll (24 deaths) than in the alternated scenario of Figure 1 , but with a lowered fiscal cost ($180.3 billion less). Figure 5 If the infection rate is reduced by 50% for twenty four weeks, the death toll is a bit higher than in the case of a strong policy for eight continuous weeks. In either case, the pandemic would kill at least 10,705 fewer individuals than in the herd immunity case. In terms, of budget, the longer period under distancing induces a larger fiscal decline than in the case of the eight-week distancing.
However, the herd immunity is again the most costly option in terms of both death and government budget.
These results reveal that if a policymaker is going to implement distancing measures, overall better results would be found with a strict measure for a short period than with a weak measure for a long period. In fact, in terms of human and fiscal costs, we can state that this strong policy for eight continuous weeks is the preferred option after the case of Figure 6 . We acknowledge that our model may not be exported to developing countries where the lock-down can also result in deaths of individuals from starvation:
given the low safety nets and salaries in many poor countries, lack of employment can severely reduce dietary intake resulting in other serious health-related issues or death. In that case, we recommend adding a death variable associated to forced unemployment. Such an approach exceeds the scope of this paper.
We like to thanks Dr. Ricardo González-Méndez from the University of Puerto Rico School of Medicine as well as Dr. Ricardo J. Cordero-Soto from California Baptist University for their insights and suggestions. The usual disclaimers apply.
• Both authors contributed equally in the designed, codification and preparation of the article. All authors have approved the final article.
• This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
• Declarations of interest: none vaccine or effective treatment is not implemented before the physical measures are lifted, the epidemic will raise again.
A policymaker would like to find an easier mechanism than a cyclical regime to combat the pandemic. In Figure B .8 we show the case where just one restrictive policy that declines the infection rate by 80% is applied for 250 days. The probability of infection is then kept by a permanent reduction of 25% to account for people being more careful and taking measures to avoid being infected. In this case the peak of the distancing curve arrives much later than before at approximately day 600. As expected, more lives (26 persons) are saved than in the alternated cycles of Figure 2 but now the federal budget is almost halved.
Note that in any of these scenarios, no physical distance or herd immunity is an optimal solution, either in terms of human or fiscal cost. Figure B .8