Time-dependent SI model for epidemiology and applications to Covid-19

A generalisation of the Susceptible-Infectious model is made to include a time-dependent transmission rate, which leads to a close analytical expression in terms of a logistic function. The solution can be applied to any continuous function chosen to describe the evolution of the transmission rate with time. Taking inspiration from real data of the Covid-19, for the case of cumulative confirmed positives and deaths, we propose an exponentially decaying transmission rate with two free parameters, one for its initial amplitude and another one for its decaying rate. The resultant time-dependent SI model, which under extra conditions recovers the standard Gompertz functional form, is then compared with data from selected countries and its parameters fit using Bayesian inference. We make predictions about the asymptotic number of confirmed positives and deaths, and discuss the possible evolution of the disease in each country in terms of our parametrisation of the transmission rate.

The asymptotic number of weighted partitions with a given number of parts

For a given sequence $$b_k$$ b k of non-negative real numbers, the number of weighted partitions of a positive integer n having m parts $$c_{n,m}$$ c n , m has bivariate generating function equal to $$\prod _{k=1}^\infty (1-yz^k)^{-b_k}$$ ∏ k = 1 ∞ ( 1 - y z k ) - b k . Under the assumption that $$b_k\sim Ck^{r-1}$$ b k ∼ C k r - 1 , $$r>0$$ r > 0 , and related conditions on the Dirichlet generating function of the weights $$b_k$$ b k , we find asymptotics for $$c_{n,m}$$ c n , m when $$m=m(n)$$ m = m ( n ) satisfies $$m=o\left( n^\frac{r}{r+1}\right) $$ m = o n r r + 1 and $$\lim _{n\rightarrow \infty }m/\log ^{3+\epsilon }n=\infty $$ lim n → ∞ m / log 3 + ϵ n = ∞ , $$\epsilon >0$$ ϵ > 0 .

Counting Triangles in Power-Law Uniform Random Graphs

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the probabilities that edges are present are moderated by asymptotically linear functions of the products of these vertex weights.

Estimating the size of COVID-19 epidemic outbreak

In this work, we analyze the epidemic data of cumulative infected cases collected from many countries as reported by WHO starting from January 21st 2020 and up till March 21st 2020. Our inspection is motivated by the renormalization group (RG) framework. Here we propose the RG-inspired logistic function of the form as an epidemic strength function with n being asymmetry in the modified logistic function. We perform the non-linear least-squares analysis with data from various countries. The uncertainty for model parameters is computed using the squared root of the corresponding diagonal components of the covariance matrix. We carefully divide countries under consideration into 2 categories based on the estimation of the inflection point: the maturing phase and the growth-dominated phase. We observe that long-term estimations of cumulative infected cases of countries in the maturing phase for both n = 1 and n ≠ 1 are close to each other. We find from the value of root mean squared error (RMSE) that the RG-inspired logistic model with n ≠ 1 is slightly preferable in this category. We also argue that n determines the characteristic of the epidemic at an early stage. However, in the second category, the estimated asymptotic number of cumulative infected cases contain rather large uncertainty. Therefore, in the growth-dominated phase, we focus on using n = 1 for countries in this phase. Some of them are in an early stage of an epidemic with an insufficient amount of data leading to a large uncertainty on parameter fits. In terms of the accuracy of the size estimation, the results do strongly depend on limitations on data collection and the epidemic phase for each country.

PRIMES REPRESENTED BY INCOMPLETE NORM FORMS

Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$ . We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$ . In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$ , we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$ . Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^{2}+Y^{4}$ and of Heath-Brown on $X^{3}+2Y^{3}$ . Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.

Quantitative recurrence results for random walks

First, we prove an almost sure local central limit theorem for lattice random walks in the plane. The corresponding version for random walks in the line has been considered previously by the author. This gives us an extension of Pólya’s Recurrence Theorem, namely we consider an appropriate subsequence of the random walk and give the asymptotic number of returns to the origin and other states. Secondly, we prove an almost sure local central limit theorem for (not necessarily lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove a version of the almost sure central limit theorem for multidimensional random walks. This is done by exploiting a technique developed by the author.

Magic state distillation with low space overhead and optimal asymptotic input count

We present an infinite family of protocols to distill magic states for T-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log⁡log⁡1/δ). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.