A Pragmatic Model of Fear for Pandemic Policymakers Andrew Brown andb87@gmail.com October 22, 2020 1 Introduction How frightened should one be of Covid19? When and how should policymakers implement policy to mitigate this fear and how should they balance this fear against the social costs of fearreducing 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, R0 , and number of active cases, A. After creating and interpreting this model, we conclude with a pragmatic approach for policymakers. 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 alreadyinfected (and therefore possiblyimmune) 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 diseasespread 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 R0 (t) Rvalue 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 nonnegligible (f (t) > ) while the disease exists. 3 Modeling Fear The probability of infection between t0 and t1 is the sum of the probability mass function f (t) in that interval: t1 X Pinfected tt10 = f (t) t=t0 The fear one ought to feel at tnow is proportional to the cumulative proba bility of future infection: ∞ X F (tnow ) ∝ f (t) tnow Without introduction of a vaccine, f (t) remains nonnegligible, the sum diverges, and this model fails. This motivates the need for two new assumptions: 1. At some point tvacc , a vaccine will be made available. 2. Once available, the probability of infection vanishes. f (t) = 0 ∀ t > tvacc These assumptions allow us to bound the sum to a tractable domain. tX vacc F (tnow ) ∝ f (t) (1) tnow 3.1 Modeling f (t) Today Clearly, our model will be sensitive to both tvacc 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(tnow ), what is f (tnow+1 )? Using equation 2, this question reduces to: Given C(tnow ) what is N (tnow+1 )? The simplest model predicts that the number of infections remains constant daytoday. That is: N (t + 1) = N (t) But this ignores useful disease characteristics, like R0 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)R0 (t) (3) Combining equations 1, 2 and 3, our model becomes: tX vacc F (tnow ) ∝ A(t)R0 (t) (4) tnow 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 R0 (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, R0 . 3 4.1 Policy Manipulations of A and R0 The transmission rate R0 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 R0 . 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 newlyclosed cases plus the number of new infec tions. Following the math of the model, we see that R0 (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 R0 lever be pulled? Policy interventions should be made to balance a region’s justifiable fear against the costs of reduced social interaction (and reduced R0 ). Since R0 is twitchy and its relevance to fear ought only be felt through the forecasted product of A(t)R0 (t) the author cautions against local governments overreact ing to sudden changes in R0 . 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 R0 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 highR0 and lowR0 policies, respectively. They should do this by keeping a close eye on both the total number of active cases, A, and the transmission rate, R0 . They should avoid sudden policy decisions which are based exclu sively on R0 values. Instead, they should consider the current value of R0 in context with A as they seek policy which allows social inter action while preventing the highrisk and highfear scenarios which accompany a large number of active cases. 4
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