### Kinks and Inflections as Temporal Markers in Flattening an Epidemic Curve

Emmanuel T. Rodulfo

#### Abstract:

A certain amount of time after the implementation of intervention to control an outbreak, a kink develops in the epidemic curve. These discontinuities in the gradient of the curve should be treated as observables as they confirm the effectiveness of interventions that began in the past. As a retarded response time of an epidemic to intervention, this may be used to parametrize the effectiveness of the intervention in flattening the curve. To construct a model that recognizes how this kink heralds the flattening of the curve, we draw upon the qualitative features of the S-?-? model, but in recognition of the role of the kink, we forego smoothness of the curve at this transitional stage. With these in mind, we formulate an analytic expression for the fraction of the population infected by a contagious disease spreading according to some power law, and responsive to a delayed intervention. Up to the time when the effect of the intervention manifests, instead of using the conventional exponentially increasing function known to over-estimate the data, the outbreak is modeled by a real power function which more closely describes the accelerated stage of epidemics. The moment the intervention takes effect, a decaying exponential function with a characteristic time defined by the viral power and the time of the appearance of the kink is introduced to flatten the power curve. The model provides a calculation for the peak of the epidemic as well as the inflection times when rate of change in the number of infectives is extremum. The recent Philippine May 23, 2020 data on COVID-19 pandemic is used in sample calculations.

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