A Pragmatic Model of Fear for Pandemic Policy-makers Andrew Brown andb87@gmail.com October 22, 2020 1 Introduction How frightened should one be of Covid-19? When and how should policy-makers implement policy to mitigate this fear and how should they balance this fear against the social costs of fear-reducing policy? In this paper we model the fear F an individual should feel at time t given the disease spread characteristics at that time, C ( t ). We model fear, F ( C ), as a function of disease characteristics like the transmission rate, R 0 , and number of active cases, A After creating and interpreting this model, we conclude with a pragmatic approach for policy-makers. 2 Notation and Assumptions We will represent time discretely in units of days. We will model disease spread in the regime of low cumulative historical in- fections: that is, we assume the total number of already-infected (and therefore possibly-immune) individuals is low This is an accurate as- sumption in most areas and will allow us to keep our model simple. We will also ignore the possibility of reinfection and assume that the fear one ought to feel is proportional to the probability of becoming infected at any time in the future. The disease-spread characteristics one might consider are those readily avail- able (or inferable) for many counties: Parameter Description A ( t ) Number of active cases on day t N ( t ) Number of new cases on day t D ( t ) Number of deaths on day t R 0 ( t ) R-value for disease on day t 1 We will also use the probability mass function f ( t ) to represent the probabil- ity of becoming infected on day t and we assume this probability is non-negligible ( f ( t ) > ) while the disease exists. 3 Modeling Fear The probability of infection between t 0 and t 1 is the sum of the probability mass function f ( t ) in that interval: P infected | t 1 t 0 = t 1 ∑ t = t 0 f ( t ) The fear one ought to feel at t now is proportional to the cumulative proba- bility of future infection: F ( t now ) ∝ ∞ ∑ t now f ( t ) Without introduction of a vaccine, f ( t ) remains non-negligible, the sum diverges, and this model fails. This motivates the need for two new assumptions: 1. At some point t vacc , a vaccine will be made available. 2. Once available, the probability of infection vanishes. f ( t ) = 0 ∀ t > t vacc These assumptions allow us to bound the sum to a tractable domain. F ( t now ) ∝ t vacc ∑ t now f ( t ) (1) 3.1 Modeling f ( t ) Today Clearly, our model will be sensitive to both t vacc and the PMF f , whose depen- dence on time we model through the disease characteristics C ( t ). Recall the four disease characteristics enumerated in the table above. The simplest (and likely best) model for probability of infection on a par- ticular day is as follows: an individual’s probability of infection on day t is proportional to the number of new cases in their region on that day. f ( t ) ∝ N ( t ) (2) This allows us to calculate the probability of infection today , but we still need to forecast future probabilities in order to compute the full fear summation given in equation 1. 2 3.2 Forecasting f ( t ) in the Future Forecasting tomorrow’s probability of infection requires answering the question: Given the disease characteristics today C ( t now ), what is f ( t now+1 )? Using equation 2, this question reduces to: Given C ( t now ) what is N ( t now+1 )? The simplest model predicts that the number of infections remains constant day-to-day. That is: N ( t + 1) = N ( t ) But this ignores useful disease characteristics, like R 0 and A , and fails to acknowledge the agency that local policy has on the fear one ought to feel. This simple model is, in a sense, hopeless. We can improve the model by recognizing that the number of new infections today is proportional to both the number of active infections and the disease spread rate. That is: N ( t ) ∝ A ( t ) R 0 ( t ) (3) Combining equations 1, 2 and 3, our model becomes: F ( t now ) ∝ t vacc ∑ t now A ( t ) R 0 ( t ) (4) We spend the remainder of the article interpreting this model. 4 Interpreting the Model The model given in equation 4 is highly sensitive to: 1. When a vaccine becomes available. 2. The forecasted values of A and R 0 (and the product of those values). The arrival date of a vaccine is generally out of the control of individuals and local policy makers. So while its value is important, it doesn’t help policy- makers manage fear. Fortunately, we do have agency in altering both the number of active cases, A , and the transmission rate, R 0 3 4.1 Policy Manipulations of A and R 0 The transmission rate R 0 is a ”twitchy” parameter: policy changes implemented today will be felt today. For example, by closing down restaurants a municipal government can almost instantly reduce R 0 The number of active cases is also manipulable, but not as twitchy. The number of active cases tomorrow will be equal to the number of active cases today minus the number of newly-closed cases plus the number of new infec- tions. Following the math of the model, we see that R 0 ( t ) is really the only ”lever” that local governments can pull. Pulling this lever comes at a cost. Restaurants, after all, are nice. Fortu- nately, governments have tight control over when and how strongly they choose to pull it. 5 Conclusion: A Pragmatic Approach to Policy When should the R 0 lever be pulled? Policy interventions should be made to balance a region’s justifiable fear against the costs of reduced social interaction (and reduced R 0 ). Since R 0 is twitchy and its relevance to fear ought only be felt through the forecasted product of A ( t ) R 0 ( t ) the author cautions against local governments overreact- ing to sudden changes in R 0 The model defined in equation 4 suggests a pragmatic approach to balancing fear and socialization through policy. This approach requires recognizing a few key facts: 1. Policy makers have access to good information about the current ”state” of the disease in their region. That is, they generally know the current values of the disease’s characteristic parameters. 2. Policy changes can impact R 0 quickly. 3. While policy changes can also impact future values of A , these changes will occur with a lag which is set by the ≈ two week timescale of the disease. The pragmatic approach suggested by these facts can be stated as: Policy makers should seek to balance the tradeoffs of fear and so- cial isolation which are associated with high- R 0 and low- R 0 policies, respectively. They should do this by keeping a close eye on both the total number of active cases, A , and the transmission rate, R 0 They should avoid sudden policy decisions which are based exclu- sively on R 0 values. Instead, they should consider the current value of R 0 in context with A as they seek policy which allows social inter- action while preventing the high-risk and high-fear scenarios which accompany a large number of active cases. 4