We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain C k lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only C 0 and neither star-shaped nor Lipschitz.

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The COVID-19 pandemic has imposed many strenuous effects on the global economy, community, and medical infrastructure. Since the out- break, researchers and policymakers have scrambled to develop ways to identify how COVID-19 will affect specific sub-populations so that good public health decisions can be made. To this end, we adapt the work of Evensen et al [1] which introduces a SEIR model that incorporates an age-stratified contact matrix, a time dependent effective reproduction number R, and uses ensemble data assimilation to estimate model parameters. The adaptation is an extension of Evensen’s modeling framework, in which we model sub-populations with varying risks of contracting SARS-CoV-2 (the virus that causes COVID-19) in a particular state, each with a characteristic age-stratified contact matrix. In this work, we will focus on 9 U.S. states as well as the District of Columbia. We estimate the effective reproductive number as a function of time for our different sub-populations and then divide them into two groups: frontline communities (FLCs) and the complement (NFLCs). Our model will account for mixing both within populations (intra-population mixing) and between populations (inter-population mixing). Our data is conditioned on the daily numbers of accumulated deaths for each sub-population. We aim to test and demonstrate methodologies that can be used to assess critical metrics of the pandemic’s evolution which are difficult to directly measure. The output may ultimately be of use to measure the success or failures of the pandemic response and provide experts and policymakers a tool to create better plans for a future outbreak or pandemic. We consider the results of this work to be a reanalysis of pandemic evolution across differently affected sub-populations which may also be used to improve modeling and forecasts.