Monkeypox is a zoonotic disease caused by a virus that is a member of the orthopox genus, which has been causing an outbreak since May 2022 around the globe outside of its country of origin Democratic Republic of the Congo, Africa. Here we systematically analyze the data of cumulative infection per day adapting model-free analysis, in particular, statistically using the power law distribution, and then separately we use reservoir computing-based Echo state network (ESN) to predict and forecast the disease spread. We also use the power law to characterize the country-specific infection rate which will characterize the growth pattern of the disease spread such as whether the disease spread reached a saturation state or not. The results obtained from power law method were then compared with the outbreak of the smallpox virus in 1907 in Tokyo, Japan. The results from the machine learning-based method are also validated by the power law scaling exponent, and the correlation has been reported.
We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree Model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.
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