Epidemic plateau in critical susceptible-infected-removed dynamics with nontrivial initial conditions

Thermally activated intermittent flow in amorphous solids

Using mean field theory and a mesoscale elastoplastic model, we analyze the steady state shear rheology of thermally activated amorphous solids. At sufficiently high temperature and driving rates, flow is continuous and described by well-established rheological flow laws such as Herschel-Bulkley and logarithmic rate dependence. However, we find that these flow laws change in the regime of intermittent flow, where collective events no longer overlap and serrated flow becomes pronounced. In this regime, we identify a thermal activation stress scale, xa(T,), that wholly captures the effect of driving rate  and temperature T on average flow stress, stress drop (avalanche) size and correlation lengths. Different rheological regimes are summarized in a dynamic phase diagram for the amorphous yielding transition. Theoretical predictions call for a need to re-examine the rheology of very slowly sheared amorphous matter much below the glass transition.

Epidemic control in networks with cliques

Social units, such as households and schools, can play an important role in controlling epidemic outbreaks. In this work, we study an epidemic model with a prompt quarantine measure on networks with cliques (a clique is a fully connected subgraph representing a social unit). According to this strategy, newly infected individuals are detected and quarantined (along with their close contacts) with probability f. Numerical simulations reveal that epidemic outbreaks in networks with cliques are abruptly suppressed at a transition point f_{c}. However, small outbreaks show features of a second-order phase transition around f_{c}. Therefore, our model can exhibit properties of both discontinuous and continuous phase transitions. Next, we show analytically that the probability of small outbreaks goes continuously to 1 at f_{c} in the thermodynamic limit. Finally, we find that our model exhibits a backward bifurcation phenomenon.

Sars-Cov2 world pandemic recurrent waves controlled by variants evolution and vaccination campaign

While understanding the time evolution of Covid-19 pandemic is needed to plan economics and tune sanitary policies, a quantitative information of the recurrent epidemic waves is elusive. This work describes a statistical physics study of the subsequent waves in the epidemic spreading of Covid-19 and disclose the frequency components of the epidemic waves pattern over two years in United States, United Kingdom and Japan. These countries have been taken as representative cases of different containment policies such as "Mitigation" (USA and UK) and "Zero Covid" (Japan) policies. The supercritical phases in spreading have been identified by intervals with RIC-index > 0. We have used the wavelet transform of infection and fatality waves to get the spectral analysis showing a dominant component around 130 days. Data of the world dynamic clearly indicates also the crossover to a different phase due to the enforcement of vaccination campaign. In Japan and United Kingdom, we observed the emergence in the infection waves of a long period component (~ 170 days) during vaccination campaign. These results indicate slowing down of the epidemic spreading dynamics due to the vaccination campaign. Finally, we find an intrinsic difference between infection and fatality waves pointing to a non-trivial variation of the lethality due to different gene variants.

Periodic recurrent waves of Covid-19 epidemics and vaccination campaign

While understanding of periodic recurrent waves of Covid-19 epidemics would aid to combat the pandemics, quantitative analysis of data over a two years period from the outbreak, is lacking. The complexity of Covid-19 recurrent waves is related with the concurrent role of i) the containment measures enforced to mitigate the epidemics spreading ii) the rate of viral gene mutations, and iii) the variable immune response of the host implemented by vaccination. This work focuses on the effect of massive vaccination and gene variants on the recurrent waves in a representative case of countries enforcing mitigation and vaccination strategy. The spreading rate is measured by the ratio between the reproductive number Rt(t) and the doubling time Td(t) called RIC-index and the daily fatalities number. The dynamics of the Covid-19 epidemics have been studied by wavelet analysis and represented by a non-linear helicoid vortex in a 3D space where both RIC-index and fatalities change with time. The onset of periodic recurrent waves has been identified by the transition from convergent to divergent trajectories on the helicoid vortex. We report a main period of recurrent waves of 120 days and the elongation of this period after the vaccination campaign.

Effect of initial infection size on a network susceptible-infected-recovered model

We consider the effect of a nonvanishing fraction of initially infected nodes (seeds) on the susceptible-infected-recovered epidemic model on random networks. This is relevant when the number of arriving infected individuals is large, or to the spread of ideas with publicity campaigns. This model is frequently studied by mapping to a bond percolation problem, in which edges are occupied with the probability p of eventual infection along an edge. This gives accurate measures of the final size of the infection and epidemic threshold in the limit of a vanishingly small seed fraction. We show, however, that when the initial infection occupies a nonvanishing fraction, f, of the network, this method yields ambiguous results, as the correspondence between edge occupation and contagion transmission no longer holds. We propose instead to measure the giant component of recovered individuals within the original contact network. We derive exact equations for the size of the epidemic and the epidemic threshold in the infinite size limit in heterogeneous sparse random networks, and we confirm them with numerical results. We observe that the epidemic threshold correctly depends on f, decreasing as f increases. When the seed fraction tends to zero, we recover the standard results.

Plateaus, rebounds and the effects of individual behaviours in epidemics

Plateaus and rebounds of various epidemiological indicators are widely reported in Covid-19 pandemics studies but have not been explained so far. Here, we address this problem and explain the appearance of these patterns. We start with an empirical study of an original dataset obtained from highly precise measurements of SARS-CoV-2 concentration in wastewater over nine months in several treatment plants around the Thau lagoon in France. Among various features, we observe that the concentration displays plateaus at different dates in various locations but at the same level. In order to understand these facts, we introduce a new mathematical model that takes into account the heterogeneity and the natural variability of individual behaviours. Our model shows that the distribution of risky behaviours appears as the key ingredient for understanding the observed temporal patterns of epidemics.

Epidemic spreading in an expanded parameter space: the supercritical scaling laws and subcritical metastable phases

While the mathematical laws of uncontrolled epidemic spreading are well known, the statistical physics of coronavirus epidemics with containment measures is currently lacking. The modelling of available data of the first wave of the Covid-19 pandemic in 2020 over 230 days, in different countries representative of different containment policies is relevant to quantify the efficiency of these policies to face the containment of any successive wave. At this aim we have built a 3D phase diagram tracking the simultaneous evolution and the interplay of the doubling time,T_d, and the reproductive number,R_tmeasured using the methodological definition used by the Robert Koch Institute. In this expanded parameter space three different main phases,supercritical,criticalandsubcriticalare identified. Moreover, we have found that in thesupercriticalregime withR_t> 1 the doubling time is smaller than 40 days. In this phase we have established the power law relation betweenT_dand (R_t- 1)^-νwith the exponentνdepending on the definition of reproductive number. In thesubcriticalregime whereR_t< 1 andT_d> 100 days, we have identified arrested metastable phases whereT_dis nearly constant.

Metastable states in plateaus and multi-wave epidemic dynamics of Covid-19 spreading in Italy

The control of Covid 19 epidemics by public health policy in Italy during the first and the second epidemic waves has been driven by using reproductive number Rt(t) to identify the supercritical (percolative), the subcritical (arrested), separated by the critical regime. Here we show that to quantify the Covid-19 spreading rate with containment measures there is a need of a 3D expanded parameter space phase diagram built by the combination of Rt(t) and doubling time Td(t). In this space we identify the Covid-19 dynamics in Italy and its administrative Regions. The supercritical regime is mathematically characterized by (i) the power law of Td vs. [Rt(t) - 1] and (ii) the exponential behaviour of Td vs. time, either in the first and in the second wave. The novel 3D phase diagram shows clearly metastable states appearing before and after the second wave critical regime. for loosening quarantine and tracing of actives cases. The metastable states are precursors of the abrupt onset of a next nascent wave supercritical regime. This dynamic description allows epidemics predictions needed by policymakers interested to point to the target "zero infections" with the elimination of SARS-CoV-2, using the Finding mobile Tracing policy joint with vaccination-campaign, in order to avoid the emergence of recurrent new variants of SARS-CoV-2 virus, accompined by recurrent long lockdowns, with large economical losses, and large number of fatalities.

Nonuniversal power-law dynamics of susceptible infected recovered models on hierarchical modular networks

Power-law (PL) time-dependent infection growth has been reported in many COVID-19 statistics. In simple susceptible infected recovered (SIR) models, the number of infections grows at the outbreak as I(t)∝t^{d-1} on d-dimensional Euclidean lattices in the endemic phase, or it follows a slower universal PL at the critical point, until finite sizes cause immunity and a crossover to an exponential decay. Heterogeneity may alter the dynamics of spreading models, and spatially inhomogeneous infection rates can cause slower decays, posing a threat of a long recovery from a pandemic. COVID-19 statistics have also provided epidemic size distributions with PL tails in several countries. Here I investigate SIR-like models on hierarchical modular networks, embedded in 2d lattices with the addition of long-range links. I show that if the topological dimension of the network is finite, average degree-dependent PL growth of prevalence emerges. Supercritically, the same exponents as those of regular graphs occur, but the topological disorder alters the critical behavior. This is also true for the epidemic size distributions. Mobility of individuals does not affect the form of the scaling behavior, except for the d=2 lattice, but it increases the magnitude of the epidemic. The addition of a superspreader hot spot also does not change the growth exponent and the exponential decay in the herd immunity regime.

Self-driven criticality in a stochastic epidemic model

We present a generic epidemic model with stochastic parameters in which the dynamics self-organize to a critical state with suppressed exponential growth. More precisely, the dynamics evolve into a quasi-steady state, where the effective reproduction rate fluctuates close to the critical value 1 for a long period, as indeed observed for different epidemics. The main assumptions underlying the model are that the rate at which each individual becomes infected changes stochastically in time with a heavy-tailed steady state. The critical regime is characterized by an extremely long duration of the epidemic. Its stability is analyzed both numerically and analytically in different models.