Situation of COVID-19 in Brazil in August 2020: An Analysis via Growth Models as Implemented in the ModInterv System for Monitoring the Pandemic

Abstract

In this work we introduce a novel methodology to classify the dynamical stages of an epidemic, based on the different acceleration regimes of the corresponding growth curve. Our classification scheme is implemented by fitting the empirical data with a general class of mathematical growth models, from which we compute not only the growth acceleration but also its jerk and jounce (i.e., the first and second derivatives of the acceleration, respectively), thus allowing for a finer distinction of the epidemic stages. Using this methodology, we analyze the cumulative curves of deaths attributed to COVID-19 in the 26 Brazilian States and the Federal District, up until August 21, 2020. The online application ModInterv COVID-19, which automatically implements the classification scheme and which can be accessed via an internet browser or a mobile app, was used to investigate the epidemic stages in each of the Brazilian federal units. The analysis revealed that almost all states in the Northern and Northeastern regions were already in the saturation phase, meaning that the epidemic was relatively under control, whereas in all Southern states and in most states in the Midwest the epidemic was still accelerating or showed only a slight deceleration. The Southeastern region presented a great diversity of epidemic stages, with each state being found at a different stage, ranging from acceleration to saturation. It is argued that understanding this heterogeneous geographical distribution of the epidemic is relevant for public health authorities, as it may help in devising more effective strategies against the COVID-19 pandemic in a continental country like Brazil.

ModInterv COVID-19: An online platform to monitor the evolution of epidemic curves

“…Equation (A.1) must be supplemented with the initial condition for some given value of C 0 . The BLM admits an analytic solution [27] in implicit form given by where with 2 F 1 ( a, b ; c ; x ) being the Gauss hypergeometric function. Equation (A.3) describes a sigmoidal curve, whose inflection point is located at the time t 2 = t c given by For completeness, we also quote here the characteristic points t 1 and t 3 , corresponding to the points of zero jerk, , of the BLM, which are given by t 1,3 = f ( Kx 1,3 ), where [25]: with θ and Δ being given by θ = 2 q (−1+2 q ) and Δ = 4 pq (−1+2 q )+ p 2 (1−2 α + α 2 +8 αq ), respectively. One can also compute the point t 4 of maximum jerk, i.e., , but in this case the expression is rather long and so it is given separately in Appendix C. The BLM described above is one of the most general growth models, from which many other known models emerge as special cases [23, 27].…”

“…For epidemiological reasons, it is sensible to restrict the values of α to the range 0 < α < 1 [28], in which case the epidemic curve bends slower towards the plateau than the logistic curve. Note, however, that within the above allowed range of α , the higher the α the sharper the bending. One major difference of the RM with respect the two previous models (BLM and GRM) is that the RM admits an explicit solution in the following form [28]: where the inflection point t 2 = t c can be obtained in terms of the initial condition C 0 via the relation: C 0 = K/ [1 + α exp ( αrt c )] 1 /α or, alternatively, One can also obtain explicit expressions for the points of zero jerk for the RM [25]: The expression for the point t 4 of maximum jerk for the RM can be obtained from the corresponding result for the BLM, given in Appendix C, after setting q = p = 1. The RM has three free parameters, namely ( r, α, K ), to be numerically determined from a fitting to the empirical data. It is therefore the simplest model that can describe an asymmetric sigmoidal curve, meaning that it has the least number of free parameters (among the ones implemented in the ).…”

“…At best, one can make short term estimates, like the doubling time, T d , which corresponds to the number of days (counted from the date of the last data point) that it will take until the number of cases or deaths reaches the double of the present value, assuming that the curve will continue to follow the q -exponential trend. It is not difficult to show [25] that T d is given by Thus, the doubling time in the q -exponential model grows linearly in time for q < 1; whereas it remains constant, i.e., T d = (ln 2)/ r , for the purely exponential growth ( q = 1). The fact that T d increases linearly in time (for q < 1) is a direct manifestation of the subexponential growth.…”

“…One major difference of the RM with respect the two previous models (BLM and GRM) is that the RM admits an explicit solution in the following form [28]: where the inflection point t 2 = t c can be obtained in terms of the initial condition C 0 via the relation: C 0 = K/ [1 + α exp ( αrt c )] 1 /α or, alternatively, One can also obtain explicit expressions for the points of zero jerk for the RM [25]: The expression for the point t 4 of maximum jerk for the RM can be obtained from the corresponding result for the BLM, given in Appendix C, after setting q = p = 1.…”

“…It will be argued that the mathematical analysis provided by our application can be useful to public health authorities, not only because it indicates the current dynamical stage of the epidemic in a given place but also because it can help to predict its likely evolution in the near future. This type of information, combined with other analyses, can in turn help the authorities in their decision making process regarding, say, the adoption or relaxation of non-pharmacological interventions [25] and other containment measures. It should also be emphasized from the outset that the Modinterv app, by directly implementing mathematical models, provides information about the epidemic dynamics which cannot be obtained neither from a mere visual inspection of the raw empirical curve nor by using only its moving-average smoothed version.…”