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For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.

Forecasting when an epidemic wave is likely to end is an important component of disease management, allowing deployment of limited control resources to be planned efficiently. Here, we report an analysis that we conducted in real-time during the first COVID-19 epidemic wave in mainland China. We developed a mathematical model to construct bounds on the end date of the first epidemic wave there, assuming that strong quarantine and testing measures remained in place until the epidemic wave was confirmed over. We used reported data on case numbers in China from January 20 to April 9, 2020. We first developed an analytic approach, obtaining a formula describing the probability distribution of the epidemic wave end date using a combination of deterministic modelling and the theory of continuous-time Markov processes. Then, we ran simulations of an individual-based model to demonstrate that our analytic predictions were accurate. We found that the predicted end date of the first epidemic wave in China depended on the proportion of infected individuals that are symptomatic and appear in case notification data, as opposed to remaining asymptomatic throughout their courses of infection. We therefore provide an easy-to-use approach for predicting the ends of epidemic waves, as well as a clear demonstration that predicted end-of-epidemic times depend on the extent of asymptomatic infection. Our framework can be applied to predict the ends of epidemic waves during future outbreaks of a wide range of pathogens.