A cluster-based model of COVID-19 transmission dynamics.

Many countries have manifested COVID-19 trajectories where extended periods of constant and low daily case rate suddenly transition to epidemic waves of considerable severity with no correspondingly drastic relaxation in preventive measures. Such solutions are outside the scope of classical epidemiological models. Here, we construct a deterministic, discrete-time, discrete-population mathematical model called cluster seeding and transmission model, which can explain these non-classical phenomena. Our key hypothesis is that with partial preventive measures in place, viral transmission occurs primarily within small, closed groups of family members and friends, which we label as clusters. Inter-cluster transmission is infrequent compared with intra-cluster transmission but it is the key to determining the course of the epidemic. If inter-cluster transmission is low enough, we see stable plateau solutions. Above a cutoff level, however, such transmission can destabilize a plateau into a huge wave even though its contribution to the population-averaged spreading rate still remains small. We call this the cryptogenic instability. We also find that stochastic effects when case counts are very low may result in a temporary and artificial suppression of an instability; we call this the critical mass effect. Both these phenomena are absent from conventional infectious disease models and militate against the successful management of the epidemic.

Limit cycle oscillations of a violin string.

In this work, we write and solve a first principles model for the motion of a bowed string. We find limit cycle oscillations driven by stick-slip friction. The shape of these oscillations is in accordance with the Helmholtz-Rayleigh motion. We observe that when bow force, bow speed, and other parameters are varied, the stable limit cycle occurs in a narrow region of parameter space. This explains why it is difficult for amateurs to produce musically acceptable sounds from the instrument.

Impact of reproduction number on the multiwave spreading dynamics of COVID-19 with temporary immunity: A mathematical model.

OBJECTIVES: The recent discoveries of phylogenetically confirmed COVID-19 reinfection cases worldwide, together with studies suggesting that antibody titres decrease over time, raise the question of what course the epidemic trajectories may take if immunity were really to be temporary in a significant fraction of the population. The objective of this study is to obtain an answer for this important question. METHODS: We construct a ground-up delay differential equation model tailored to incorporate different types of immune response. We considered two immune responses: (a) short-lived immunity of all types, and (b) short-lived sterilizing immunity with durable severity-reducing immunity. RESULTS: Multiple wave solutions to the model are manifest for intermediate values of the reproduction number R; interestingly, for sufficiently low as well as sufficiently high R, we find conventional single-wave solutions despite temporary immunity. CONCLUSIONS: The versatility of our model, and its very modest demands on computational resources, ensure that a set of disease trajectories can be computed virtually on the same day that a new and relevant immune response study is released. Our work can also be used to analyse the disease dynamics after a vaccine is certified for use and information regarding its immune response becomes available.