key: cord-0964797-6x37ihv5 authors: Khatri, B. S. title: Stochastic extinction of epidemics: how long does would it take for Sars-Cov-2 to die out? date: 2020-08-12 journal: nan DOI: 10.1101/2020.08.10.20171454 sha: 636ef7b30e398faf49ce1f7d3e318889e351cd4d doc_id: 964797 cord_uid: 6x37ihv5 Worldwide, we are currently in an unprecedented situation with regard to the SARS-Cov-2 epidemic, where countries are using isolation and lock-down measures to control the spread of infection. This is a scenario generally not much anticipated by previous theory, and in particular, there has been little attention paid to the question of extinction as a means to eradicate the virus; the prevailing view appears to be that this is unfeasible without a vaccine. We use a simple well-mixed stochastic SIR model as a basis for our considerations, and calculate a new result, using branching process theory, for the distribution of times to extinction. Surprisingly, the distribution is an extreme value distribution of the Gumbel type, and we show that the key parameter determining its mean and standard deviation is the expected rate of decline Re = {gamma}(1-Re) of infections, where {gamma} is the rate of recovery from infection and Re is the usual effective reproductive number. The result also reveals a critical threshold number of infected I† = 1/(1-Re), below which stochastic forces dominate and need be considered for accurate predictions. As this theory ignores migration between populations, we compare against a realistic spatial epidemic simulator and simple stochastic simulations of sub-divided populations with global migration, to find very comparable results to our simple predictions; in particular, we find global migration has the effect of a simple upwards rescaling of Re with the same Gumbel extinction time distribution we derive from our non-spatial model. Within the UK, using recent estimates of I0{approx}37000 infected and Re= 0.9, this model predicts a mean extinction time of 616{+/-}90 days or approximately ~2 years, but could be as short as 123{+/-}15 days, or roughly 4 months for Re = 0.4. Globally, the theory predicts extinction in less than 200 days, if the reproductive number is restricted to Re < 0.5. Overall, these results highlight the extreme sensitivity of extinction times when Re approaches 1 and the necessity of reducing the effective reproductive number significantly (Re<{approx}0.5) for relatively rapid extinction of an epidemic or pandemic. The SIR model has remained a popular paradigm to understand the dynamics of epidemics 1,2 , despite its simplifications compared to real world epidemics, which have spatial structure 3, 4 and heterogeneity 5, 6 in connection between regions. But it is this simplicity that distills the course of an epidemic down to a few key parameters that gives it its power. One such parameter is the reproductive number, which is a dimensionless number that describes whether an epidemic is growing (R e > 1) or shrinking (R e < 1) in a population, and encapsulates the number of secondary infections within the period of infectivity (1/γ). Simple models can therefore be very instructive on qualitative behaviour, particularly when the parameters of such models are interpreted as effective and coarse-grained representations of more detailed models. These detailed models are necessary for quantitative accurate predictions, but such highly parameterised models can suffer from a lack of transparency in understanding the effect on the dynamics of changing different parameters 6, 7 . For this reason, simplified coarsegrained models, such as the SIR model, have the major advantage of being highly transparent, particularly when a) Electronic mail: bkhatri@imperial.ac.uk analytical solutions are available. Both approaches used together provide a powerful approach to prediction in epidemics. This paper examines the SIR model, but fully accounting for the discreteness of individuals, that leads to stochasticity in the progress of an epidemic. Why do we need to worry about stochastic effects in an epidemic, and in particular, to understand extinction? A key danger of deterministic models, whether we treat as continuous the number of infected individuals or the density of infected individuals, is the infinite indivisibility of a population. Particularly, in the latter case, if we consider a continuous model of isolation or lockdown and potential rebound (second wave), then there is danger of the model giving densities that actually correspond to much less than 1 individual in the population -a physical impossibility; unchecked this model will incorrectly predict regrowth of infections once restrictions are relaxed 7 . In a continuous model, the problem is what to do when the number of infected corresponds to less than 1 individual, and what is really required is a full stochastic description, as we do in this paper; in a stochastic model there can be exactly 0 individuals in a population, so unless there is migration of cases, the epidemic will be extinct in that population once there are 0 infected, and there can be no rebound or second wave. There has been a considerable amount of work done to understand stochastic aspects of epidemics 8 from un-derstanding critical community sizes in diseases such as measles 9, 10 , stochastic phases in the establishment of epidemics 11 , to stochastic extinction. With regard stochastic extinction most results have been focussed on understanding the time to extinction either through the whole course of an epidemic 12, 13 or assuming a quasiequilibrium has been reached through herd immunity in the population 14 . However, the situation currently faced in many countries has not been considered in previous modelling, that is the situation where significant herd-immunity has not been achieved in the population, and isolation measures have led to the reproductive number being reduced to less than 1, the critical threshold for growth. It is with this scenario in mind, where a population still has many more susceptible (& recovered), compared to infected, where the main results of the paper are focussed, and we find the stochastic dynamics tractable within a simple birth-death branching process framework. Our main theoretical result is the distribution of the times to extinction of an epidemic, which surprisingly, we find is a Gumbel-type extreme value distribution. Key to this result is a new threshold I † = 1/(1 − R e ), where † indicates extinction, below which stochastic changes dominate; when R e > 0.6 and approaches one, we show this gives significantly shorter mean extinction times compared to deterministic predictions. As this result ignores spatial structure and heterogeneity of an epidemic, we then compare to simple and more complex spatial epidemic simulations, and find our theory captures the extinction time distribution very well, as long as R e is appropriately rescaled to account for migration. We then use this theory to make broad predictions of extinction times within the UK, and globally to serve as a guide to more complex and detailed models. Our key message is that for reproductive numbers R e > 0.6 and approaching one, extinctions times increase very rapidly, and are of order many years, but can very quickly drop to times much less than a year or a few months if restricted to R e < 0.5. The SIR model divides the population of N individuals in a region into 3 classes of individuals: susceptible S (not infected and not immune to virus), I infected and R recovered (and immune, so cannot be re-infected). If we assume a rate β of an infected individual infecting a susceptible individual (S + I → I), and a rate γ that an infected person recovers from illness (S → R), the ordinary differential equations describing the dynamics of this process are: The most important thing about this model is that it allows a simple characterisation of when number of infections will grow or decline: whatever the previous history of the epidemic, for growth we need dI dt > 0 and this happens for the following condition on RHS of the 2nd equation above: βS(t)/N − γ > 0, or equivalently, R e = βS(t)/N γ > 1, where we have defined the dimensionless number R e as the combination shown, and will in general be time-dependent, as the number of susceptible individuals in a population change. R e represents the average number of individuals an infected person infects through the duration of the infection τ = 1/γ. It is important to understand that this interpretation of R e is within the context of a well-mixed model. In reality, locally there may be deviations from the global density of susceptible individuals (S(t)/N ) and also differences in connectivity between different regions causing differing rates of infection locally. The parameter β is the rate per infectious individual of causing a successful secondary infection. It is useful to decompose this parameter as β = να, where ν is the rate of making contacts per individual and α is the probability of each contact causing an infection; the probability of infection α is largely controlled by the biology of the virus/pathogen, whilst the rate of making contacts ν is mainly controlled by human behaviour, and so interventions like social distancing, lock-downs and track, trace and isolate will have an aggregate effect on ν, by reducing the rate of contacts between individuals. In an idealised SIR epidemic, if β does not change due to behaviourial changes, the reproductive number is in general a constantly decreasing number, due to the susceptible pool of individuals diminishing -this is what would eventually lead to herd immunity as the decreasing susceptible fraction brings R e < 1. However, changes in social behaviour can also bring R e < 1 by controlling β, before any significant herd immunity is established, which is currently the case in many countries. In this case, if we assume that the fraction of the population that have been infected is small, we can assume that the effects of herd immunity are negligible. Estimates by the UK Office for National Statistics 15 (June 12th 2020), estimate from serological testing in England that 6.8% of the population have been infected (≈ 4.6 × 10 6 ) and from random PCR testing a current incidence of 0.055% (≈ 3.7 × 10 4 actively infected in UK) -in which case, extrapolating to a UK population size of ≈ 67 × 10 6 , the total number of susceptibles (≈ 62 × 10 6 ) is much greater than the number currently infected (as of 9th Aug 2020 there are more recent estimates but these numbers have not changed significantly). In this case, as long as R e isn't very close to 1, it is reasonable to assume that the population of susceptible individuals S(t) = S 0 is roughly constant and the reproductive number unchanging R e = βS(t)/N γ ≈ βS0/N γ ; although each time an individual is infected, we loose exactly one susceptible the relative change of the susceptible pool is negligible, since the total number of susceptible individuals is very large. It can be shown that this approximation is good as long as 1 − R e (I 0 /S 0 ), which corresponds to when the decline is sufficiently rapid, for the initial value of R e , that the error due to ignoring the change in the susceptible pool is negligible; for I 0 = 3.7 × 10 4 and S 0 = 62 × 10 6 , we expect this approximation to be good for 1 − R e 0.024. The last differential equation involving R is really only there for book-keeping, as there is no direct effect of R on the dynamics of S and I -even with a more complicated model where immunity only lasts a finite time (SIRS), thereby increasing the susceptible pool again, if we assume the total number of susceptibles is very large in comparison, these effects will be negligible. This means for the case where only a small fraction of the population are ever infected, the SIR dynamics results in a single differential equation for I(t): The solution to this is of course an exponential function: where ρ e = γ(R e −1) is the effective growth rate for R e > 1, and decay rate when R e < 1, and I 0 = I(0) the initial number of infected individuals. Note that R e is not a rate, it does not, in absolute terms, tell you anything about the time scales of change; however, ρ e is a rate, and if it could be measured empirically, it would give information in the speed of spread of the infection, as well as having the same sign information for the direction of change (ρ e > 0 the epidemic spreads, while ρ e < 0 means the epidemic cannot spread). We are interested in understanding extinction of an epidemic and so from here on we define the rate ρ e = γ(1−R e ) to be a positive quantity, making the assumption that R e < 1. In this case we can make a simple deterministic prediction for the time to extinction, by calculating the time for the infected population to reach I(t) = 1: Of course, we want to know the time to complete elimination I(t) = 0, but we cannot answer this question with a deterministic continuous approximation, since the answer would be ∞; the time it takes to go from 1 infected individual to 0 cannot be handled in a deterministic approach, since it ignores the discreteness of individuals and the stochasticity that lies therein. In fact, without understanding the stochasticity of the extinction process, it is difficult a priori to say anything about the goodness of this deterministic calculation, since in general we would expect stochasticity to be important far before there remains only a single infected individual. The above analysis assumes deterministic dynamics with no discreteness -it ignores any randomness in the events that lead to changes in number of infected individuals; an infected person might typically take the tube to work, potentially infecting many people, whilst on another day decide to walk or take the car, reducing the chances of infecting others. When the epidemic is in full flow with large numbers of individuals infected, all the randomness of individual actions, effectively average out to give smooth almost deterministic behaviour. However, at the beginning of the epidemic, or towards the end, there are very small numbers of individuals that are infected, so these random events can have a large relative effect in how the virus spreads and need a stochastic treatment to analyse. We are interested in analysing the stochasticity of how the number of infected decreases when R e < 1 and eventually gives rise to extinction, i.e. when there is exactly I = 0 individuals; in particular, we are primarily interested in calculating the distribution of the times to extinction. We can initially confirm that the assumptions of a constant R e due to a negligibly changing susceptible population of the previous section are accurate, by running multiple replicate stochastic continuous time simulations with Poisson distributed events (known as Gillespie or kinetic Monte Carlo simulations) 16 of the SIR model with R e = 0.7, γ = 1/7 days −1 , I 0 = 3.7 × 10 4 and an initial recovered population of R(0) = 6 × 10 6 (for simplicity we take 10% infected and recovered as a more conservative estimate than from the serological surveillance data in the UK 15 ). Fig.1 plots the decline in number of infected over time I(t). Each of the trajectories from the Gillespie simulations is a grey curve, whilst the deterministic prediction (Eqn.5) is shown as the solid black line. We see that for I(t) 1 the stochastic trajectories are bisected by the deterministic prediction, indicating that the assumption of a constant R e is a good one. We can see from Fig.1 that as I(t) approaches extinction, as expected the trajectories become more and more varied as the number of infected becomes small. A simple heuristic treatment inspired from population genetics 19 would define a stochastic threshold I † , below which stochastic forces are more important than deterministic; the time to extinction is then a sum of the time it takes to go deterministically from I 0 to I † ( 1 ρe ln(I 0 /I † )) and the time it takes to go from I † to I = 0 by random chance. Assuming such a threshold I † exists, this latter stochastic time can be approximated as follows: if there are I † individuals and changes are mainly random, then we are randomly drawing individuals from a pool of I † infected individuals and N − I † non-infected individuals -a binomial random walk -which when I † N has standard deviation ≈ √ I † per random draw, which means we need k = I † random draws, such that the standard deviation over those k draws is √ kI † ≈ I † ; a single random draw corresponds to one infection cycle of the virus, which is τ = 1/γ days, so the time to extinction starting with I † individuals is approximately I † /γ. How do we estimate I † ? It is given by the size at which random stochastic changes, change the number of infected by the same amount as the deterministic decline. In one cycle or generation of infection, if there was no stochasticity, the number of infected would decline by ≈ ρ e I † /γ, so equating this to the expected standard deviation of purely random changes, As discussed below, and in more detail in the Appendix, a more exact calculation of these considerations, using branching processes, gives exactly the same expression for I † . This means the typical stochastic phase lasts I † /γ = 1 ρe days and so adding the deterministic and stochastic phases, the mean time to extinction ≈ 1 ρe (1 + ln(I 0 /I † )) (see Eqn.8 below for a more exact expression of the mean). The branching process framework used to calculate the distribution of extinction times is standard, but detailed, and so we will sketch the derivation here and leave details for Appendix 1. The first step is to recognise that there are two independent stochastic events that give rise the net change in the numbers of infected individuals, as depicted in Eqn. 4 for the continuum deterministic limit: 1) a susceptible individual is infected by an interaction with a infected individual, such that I → I +1 and 2) an infected individual recovers spontaneously such that I → I − 1. This is a simple birth death branching process for which it possible to write down differential equations (dp I (t)/dt) for how the probability of I infected individuals changes with time in terms of the birth and death events just defined. It is possible to find after some calculation the probability generating function G(z, t) of the birth-death process, from which the probability of having exactly I = 0 individuals as a function of time is given by: If p † (t) is the distribution of times to extinction (i.e. the probability of an extinction occuring between time t and t + dt is p † (t)dt), then clearly the integral of this distribution, between time 0 and t is exactly Eqn.6, and hence the distribution of times to extinction is simply the derivative of p 0 (t) with respect to time. Doing this and also taking the limit that I 0 I † , we find: where τ † = 1 ρe ln(I 0 /I † ), which is the time it takes for number infected to change from the initial number I 0 to the critical infection size, which this calculation shows is given by I † = 1 1−R0 , which is the same result as arrived by the heuristic analysis above. Fig.2 shows a histogram (grey bars) from Gillespie simulations of the SIR model with 5000 replicates of the number of infected individuals for R e = 0.7, γ = 1/7 days, I 0 = 3.7 × 10 4 and an initial recovered population of R(0) = 6 × 10 6 and the corresponding prediction from Eqn.7 (solid black line) -we see that there is an excellent correspondence. Surprisingly, the extinction time distribution is a Gumbeltype extreme value distribution 20,21 ; it is surprising as an extreme value distribution normally arises from the distribution of the maximum (or minimum) of some quantity, although here it is not clear how this relates to the extinction time. There are number of standard results for the Gumbel distribution Eqn.7, so we can write down (or directly calculate) the mean and standard deviation of the extinction time: where Υ ≈ 0.577 is the Euler-Mascheroni constant. We see that the heuristic calculation overpredicts the stochastic part of the extinction time by a factor of ≈ 2. Note that the standard deviation or dispersion of the distribution only depends on the inverse of the rate of decline ρ e and as expected not on the initial number of infected individuals I 0 ; hence as ρ e decreases (R e gets closer to 1), we see that the distribution of extinction times broadens (as we see below in Figs.4&6). We can also calculate the cumulative distribution function from which the inverse cumulative distribution function T † = (P † ) −1 can be calculated: which enables direct generation of random numbers drawn from the extinction time distribution, by drawing uniform random u on the unit interval and calculating T † (u). It also allows calculation of arbitrary confidence intervals, for example, the 95% confidence intervals, by calculating T † (0.025) and T † (0.975), as well as the median T † (1/2) = τ † − 1 ρe ln (ln (2)). Finally, it is important to stress that the distribution of extinction times Eqn.7 and the following results all assume that I 0 I † , so that there is a clear separation of the deterministic and stochastic phases of the decline in infections. A more general and exact result for the distribution of extinction times is given in Appendix 1. A potentially valid criticism is that real populations have spatial structure and heterogeneity of contacts between regions. To make comparison to our simple predictions, we used a complex epidemic simulator GleamViz (v7.0) 17,18 , which includes a gravity model of migration, where rates of migrations between sub-populations are proportional to their population sizes (see Fig.1 inset map of UK), and each sub-population based on accurate census data within a grid of 25 km. We ran 50 replicate simulations for an SIR epidemic within the United Kingdom and with zero air travel to other countries, with the same parameters as the stochastic SIR simulations in the previous section (R e = 0.7, γ = 1/7 days, initial recovered population R(0) = 6 × 10 6 -in addition, each definable sub-population in the UK was given a current infection incidence of 0.06% giving a total I 0 ≈ 3.7 × 10 4 ). We see the trajectories ( Fig.1 -yellow lines) and histogram of extinction times (Fig.2 -yellow bars) compare very favourably to the predictions of the stochastic SIR model (black solid line and grey histogram bars); the mean and standard deviation including the gravity migration model is 211 ± 16 days, which is slightly smaller than the prediction of the stochastic SIR model which has no migration or spatial structure (231 ± 30 days). This suggests that heterogeneity and migration might together have the net effect of reducing extinction times, as below we see increasing migration uniformly, has the opposite effect; nonetheless within the UK it would seem the overall effect of heterogeneity and migration is of second order to predictions of a well mixed model. Overall, at a national level, we find the results of our simple model are accurate to within the width of the distributions of the extinction times. It was not possible to repeat these simulations on a global scale as GleamViz does not record individual level changes FIG. 2. Probability density of extinction times for the same parameters as in Fig.1 . Grey bars are a histogram of 5000 replicate simulations of Gillespie simulations normalised to give an estimate of the probability density, and the black curve is the prediction of the analytical calculation given in Eqn.7, which we see matches the simulations extremely well. The yellow bars are histograms from the GleamViz spatial epidemic simulator with 50 replicates, which we see gives similar results to the predictions of the stochastic SIR model. in infections and deaths in its global output. Here instead we first consider the total extinction time distribution for a number of isolated regions (countries) with no migration between, but each with the same R e . As we show in Appendix 2, in fact, the extinction time distribution of the whole region (i.e. the distribution of the maximum time of all the groups) is exactly the same distribution as assuming a single unstructured/undivided population for the region. We verify this by Gillespie simulation of a simple birth-death model with growth rate γR e and death rate γ for n isolated populations; the grey histogram in Fig.3 is the estimate of the extinction time distribution for isolated sub-populations and this matches the grey solid line, which is exactly the solid black line in Fig.2 . We now look at the effect of migration, where we examine the same Gillespie simulations of birth and death, but with a probability of global migration per individual of φ. As we increase φ we see that the extinction time distribution shifts to longer times, yet still maintains the same form as given by Eqn.7 -fitting to this equation using only R e as a free parameter, we find for φ = {0.01, 0.05, 0.1}, R e = {0.709 ± 0.001, 0.732 ± 0.001, 0.760 ± 0.001}, respectively (minimum R-sqd statistic of 0.975). These fits are shown as the blue and red solid lines in Fig.3 for φ = 0.05 and φ = 0.1, respectively, and we see that the fits follow the data very closely (the histogram and fits for φ = 0.01 are not shown in Fig.3 for clarity, as they overlap closely with φ = 0). We can see that can predict these estimated reproductive numbersR e by simply rescaling the base R e to R e → (1 + φ)R e = {0.707, 0.735, 0.77} for φ = {0.01, 0.05, 0.1}, respectively. This finding is closely related to the literature on the group level reproductive number R * 3-5 , except here we are studying the decline and extinction of an epidemic/pandemic as opposed to its establishment, which has not been previously studied in this context. Overall, these results suggest that under the assumption that each national region has the same R e , that the extinction time distribution is given by the stochastic SIR model (Eqn.7) but with a rescaled R e to account for air traffic or migration between regions/countries. We first consider what this model predicts for the extinction of the SARS-Cov-2 epidemic within the United King- dom, given an incidence of 0.055% (12th June 2020 -a more recent estimate show this incidence has not changed significantly) or I 0 ≈ 37 × 10 4 were infected at the beginning of June 15 . In Fig.4 we have plotted the estimates of mean (solid lines), median (dashed lines) and deterministic (Eqn.5 -dots) extinction times, together with 95% confidence intervals (shaded region), given an initial number of infected I 0 for various reproductive numbers R e between 0 < R e < 1, and for various values of γ. We see three broad trends: 1) for R >≈ 0.6 the extinction times are very long of order years, whilst at the same time the 95% confidence intervals becomes increasingly broad (of order years themselves); 2) also for R e >≈ 0.6, the deterministic prediction increasingly and significantly overestimates the mean extinction time; 3) for R e < 0.5 the extinction times plateau with diminishing returns for further decreases in the reproductive number; and 4) as the mean infection duration 1/γ increases the extinction times increase and the 95% confidence intervals go up in proportion. Regarding point 3), we see that there is minimum time to extinction, given by R e = 0; from Eqn.8 in the limit that R e → 0, the mean time to extinction t → 1/γ(ln(I 0 )+Υ), which shows the extinction process is ultimately limited by the rate of recovery γ, imposing a maximum speed limit on the rate of decline of infections. To make specific predictions, we need to estimate the recovery rate γ. Current direct estimates for the mean or median infection durations are quite broad, and also in need of careful interpretation; different individuals will have very different disease progressions and also very different transmission potentials, from those that are asymptomatic to individuals that are hospitalised. A recent comprehensive survey of the literature 22 looked at three different types of infection durations: 1) asymptomatic (T 2 ); 2) pre-symptomatic (T 3 ) and 3) symptomatic (T 5 ). Their survey suggest a median T 2 of ≈ 7 days (4−9.5), a mean T 3 of 2 days, and a median T 5 of 13.4 days. On the other hand an early study of time to recovery of patients within and outside of China, suggests 1/γ ∼ 20 days 23 . Overall, it is arguable that the T 2 time is likely to be most relevant, since infection events are likely to be dominated by asymptomatic carriers, and symptomatic times T 5 in hospital are likely to be an overestimate of the duration for which an individual has high transmission potential. In addition, as suggested by this study the T 5 time is typically estimated by time to a negative RT-PCR test, but the time when the individual had viral loads sufficient Re with an initial infected population of I0 = 3.7 × 10 4 for various values of the recovery rate: 1/γ = 7 days (yellow); 1/γ = 10 days (red); 1/γ = 20 days (green). Solid lines are the predictions of the mean (Eqn.8), dashed lines the median, the dots correspond to the deterministic prediction (Eqn.5), whilst each shaded region corresponds to the 95% confidence interval; the median and confidence intervals are calculated directly from Eqn.11. to infect is likely to much shorter. However, the scope of this paper is not to make very precise predictions, but to demonstrate broad trends, and so as well as plotting predictions for what seems to be the most likely infection duration 1/γ = 7 days (yellow), we have plotted predictions for 1/γ = 10 days (red) and 1/γ = 20 days (green). In general, we see that longer infection durations give rise to longer extinction times, since as mentioned above the recovery rate sets the overall tempo for the decline in infections. Assuming a duration of 1/γ = 7 days, we see that the simple stochastic SIR model predicts that extinction or elimination of the epidemic can occur within the United Kingdom of order about 100 days or a few months, if R e can be kept to below about 0.5. More precisely, if R e = 0.4, the mean time is 123 ± 15 days with 95% confidence intervals:(102, 160) days. However, as of the end of June, both the UK government's own estimate (SPI-M modelling group) 24 and the MRC Biostatistics Unit, Cambridge 25 estimated a reproductive number put R e ≈ 0.9 for the region of England, which gives a mean time of 616 ± 90 days (484, 832), or roughly two years; as R e = 0.9 is close to a value of the reproductive number where would expect our underlying assumptions of an unchanging susceptible population to be less good (R e = 1 − I 0 /S 0 = 1 − 0.024 = 0.976), we checked this prediction by Gillespie simulation of the SIR model, and find a mean extinction time of 592 ± 85 days, which is slightly less than our prediction, but with a difference which is still within the spread of times we would expect in reality. Estimating extinction times from direct estimate of ρ e Precise predictions of extinction times based on Eqn.7, as shown in Fig.4 , require knowledge of both R e and γ, which are generally quite difficult to estimate. However, Eqn.7 is mainly dependent on the rate of decline of the epidemic ρ e , with weak dependence separately on R e and γ. ρ e can be determined more straightforwardly if we assume current daily deaths are proportional to the number infected; if infections are declining exponentially at a rate ρ e then so will the number of daily deaths, so a curve fit will give an accurate measure of ρ e even if we cannot determine the proportionality constant to translate deaths to infections. An alternative, could be to look at time-series of number of daily infections per number of tests performed, to remove biases due to testing, however, here the aim is to illustrate why a direct estimate of ρ e is useful, rather than to estimate this number very accurately. As shown in Fig.5 , fitting decaying exponentials to daily number of deaths (date of death, not date of reporting) from the NHS UK 26 , from the moment of decline to 13th July 2020, shows a very good fit (showing deaths are declining as a simple exponential and giving further weight to our simple model), giving an England wide estimate of ρ e = 0.043 ± 0.001 days −1 , with a range of ρ e = 0.036±0.001 days −1 (slowest decline) for North Yorkshire and ρ e = 0.069 ± 0.001 days −1 (most rapid decline) in London. Using the rate of decline of England, we estimate a mean extinction time in the UK as 231±30 days (95% CI: (187, 303) days), for R e = 0.7, 1/γ = 7 days and 238 ± 30 days (95% CI:(195, 310) days), for R e = 0.57, 1/γ = 10 days; as we can see changing R e and γ for a fixed ρ e does not change the predictions significantly, and we suggest in general, determining ρ e could be a more robust way to estimate extinction times. These estimates are much shorter than the approximately 2 years estimated above using separate estimates of R e and γ to calculate ρ e . It is clear that in some of the regions, there is relatively poor fit at later times, with an indication that overall in England the rate of decline ρ e is decreasing, which could account for the difference in estimates. However, a more careful probabilistic analysis 6 is required to support such a conclusion, which is outside the scope of this paper; as the above stochastic calculation and simulations show, as the number of infections -or deaths in this case -is small, trajectories become more stochastic and varied, and importantly, can't be simply treated as a deterministic trend plus (Gaussian) noise, which is reasonable when numbers are large. Trying to fit a more complicated model could result in fitting noise; in other words we might be wrongly concluding a decrease in ρ e due to a trajectory in a region which just happens to randomly decline slightly less quickly. It is likely that the UK government's advisory SPI-M modelling group 24 does exactly this in its direct estimates of ρ e (as far as the author is aware details are not publicly available), which do indicate an overall slower rate of decline with a range quoted of ρ e = 0.01 → 0.04; assuming 1/γ = 7 days, we would arrive at the same estimate above for the extinction time, for R e with ρ e ≈ 0.014, which is consistent with the range quoted. We can also use our calculation to make approximate predictions of global extinction times of SARS-Cov-2 , with all the same broad caveats, given the simplifications of the model. However, as argued above at a global level, we can have confidence that the predictions of the extinction time distribution Eqn.7 can be quantitatively correct, when the R e value is effectively rescaled to account for migration/air-traffic between nations. The current global death rate is approximately 5000 deaths per day (9th August), and so with an approximate infection fatality rate η ≈ 0.01, 23,27 the rate of change of deaths would be roughly d(deaths)/dt = ηγI(t); inverting this gives a crude estimate of the current number of actively infected globally as I 0 ≈ 3.5 × 10 6 , or roughly 3.5 million. More precise estimates would require understanding the age structure of the infection fatality rate, which is highly biased to older populations. Given a global population of 7.8 billion this corresponds to a global incidence of ≈ 0.05%, which is very similar to that measured more directly within the UK. In Fig.6 we plot how the predicted extinction time changes with effective reproductive numbers between 0 < R e < 1 for 1/γ = 7 days (yellow), given I 0 = 3.5 × 10 6 , and also for 1/γ = 10 days (red) and 1/γ = 20 days, where we correct our initial estimate of I 0 for different values of γ, giving I 0 = 5 × 10 6 , and I 0 = 10 7 , respectively. As would be expected, the results mirror the predictions for within the United Kingdom; for sufficiently small values of R e (R e < 0.5), the theory predicts global extinction on a timescale of approximately 200 days or 6 to 7 months, whilst R e > 0.6 lead to very long extinction times (∼years). More precisely, for R e = 0.4 and assuming 1/γ = 7 days, we find a mean extinction time of 177 ± 15 days (95% CI: (155, 213) days). We have presented a new analysis of extinction in the stochastic SIR model in the context of populations with very little herd immunity and yet a significant number of infected individuals. Using heuristic analysis we calculate the mean time to extinction and show that there is a critical threshold I † = 1/(1 − R e ), below which random stochastic changes are more important, which suggests that for R e < 1, but approaching 1, a simple deterministic prediction will be poor. With a more exact branching process analysis, we then calculate in closed form the extinction time distribution of an epidemic, which, surprisingly, is an extreme value distribution of the Gumbel type, and is mainly dependent on the rate of decline of the epidemic ρ e = γ(1 − R e ), with a weak logarithmic dependence on R e explicitly. A key ad-vantage of a closed form solution for the distribution is that we can discern broad trends very quickly without doing a large number of complex simulations over a large parameter space. Given the simplicity of SIR well-mixed model, we compared our results to a complex spatial epidemic simulator, GleamViz 17,18 with explicit heterogeneity in connections between sub-populations, as well as simulations of uniform migration between sub-populations and find good overall agreement for the distribution of extinction times with our simple predictions. We use these results to produce predictions of extinction times within the UK and globally of the SARS-Cov-2 epidemic/pandemic, as a function of R e as a guide to the expected trends in more complex models; the results suggest that if the reproductive number can be constrained to R e = 0.4 (assuming 1/γ = 7 days), within the UK we would expect extinction times of approximately 123±15 days, or 4 months, given an incidence of infection ≈ 0.06% and globally, roughly 177 ± 15 days, assuming a similar level of incidence. However, given current estimates of R e ≈ 0.9 within the UK, the extinction times become very long, of order a couple of years; this clearly demonstrates the sensitivity of extinction times for R e > 0.6, as shown in Figs.4&6. On the other hand the same figures show that decreasing R e much below R e = 0.4 produces diminishing gains in reductions of extinction times; hence, given social consequences of lockdown measures this suggests an optimal R e ≈ 0.4 → 0.5, which in the United Kingdom was achieved just after lockdown in regions such as London 25 . However, these predictions need to be treated with care when 1 − R e becomes small, as relative changes in the susceptible pool have a large relative effect on R e , so we can no longer assume it is constant; in particular, we estimate this occurs for 1 − R e ∼ I 0 /S 0 , and will in general lead our extinction time estimates being an over-prediction. This does not affect our broad conclusions that for all practical purposes when R e > 0.6 the extinction times become very long and of order years. However, it is an interesting open question to calculate the extinction time when R e = 1, in the same limit that I 0 S 0 , since infections reducing the susceptible pool now have a significant effect to reduce R e ; intuitively, we might expect that R e will continue to decrease until again changes in S become negligible, such that R e stabilises and then the rest of the analysis in this paper would be valid. Note that a naive analogy to evolutionary theory where R e = 1 would correspond to neutrality, and so the epidemic could potentially increase, would be incorrect for the reasons outlined -even for R e = 1 we would expect the epidemic to decline. Note that within the UK, throughout we have taken an incidence level of 0.055% measured by random population sampling in England between 25 May to 7 June 2020, corresponding to I 0 ≈ 3.7 × 10 4 . A more recent estimate (20 to 26 July 2020) by the ONS shows a small increase in the incidence to ≈ 0.07%, which means our estimates are still relevant as of end of July. In fact, the month of June saw a decrease to an incidence of 0.03% by random population sampling, indicating that infections levels are currently on an upward trend, which of course would be inconsistent with current government central estimates that R e < 1. We have also used England and United Kingdom, somewhat interchangeably, purely as a matter of convenience, as the main scope is to provide approximate and qualitative predictions. The Scottish government 28 estimates the current (7th Aug 2020) number of infected as I 0 ≈ 200 and a weekly reduction in the infectious pool of 24%, which translates to ρ e = 0.038 (R e ≈ 0.73, assuming 1/γ = 7days) giving a mean extinction time of 119 ± 33 days (95% CI: (70, 199) days); this reduces to 63 ± 15 days if ρ e = 0.086, corresponding to a reproductive number of R e = 0.4. In Wales there is no direct report of currently infected, but a very crude estimate of I 0 ≈ 1400 can be obtained given approximately an average of 2 deaths per day (2nd Aug → 9th Aug), an infection fatality rate of η ≈ 0.01, 23,27 and 1/γ = 7 days. The rate of decline of infections is estimated to have a wide range ρ e = 0.01 → 0.08 29 , so taking a central estimate of ρ e = 0.045 (R e = 0.685, 1/γ = 7 days), we arrive at mean extinction time prediction of 148 ± 29 days (95% CI: (106, 217) days); if the rate of decline is increased to ρ e = 0.086 (R e = 0.4) then this reduces to 85 ± 15 days (95% CI: (63, 121) days). In Northern Ireland, it is currently estimated R e = 0.8 → 1.8 30 , and so is likely to be above 1, which means an extinction time estimate cannot be made; however, we can make an estimate assuming R e as for the other nations. As for Wales there is no estimate of the size of the currently infected pool, but the same crude approximation gives I 0 ≈ 47, given 2 deaths in period 10th July to 9th Aug (30 days), and an extinction time estimate of 46 ± 15 days (95% CI: (24, 82) days). Again, these predictions should be viewed as a guide to expected qualitative changes in extinction times given changes in R e , and in particular, to demonstrate the sensitivity of extinction times on R e in specific local contexts. Reducing the reproductive number to R e = 0.4 is ambitious, but was achieved in London not long after the lockdown on 23rd March 2020 25 ; in other regions of England, the lockdown was less successful with numbers in the range R e = 0.6 → 0.7, which if repeated again, from Fig.4 would give an extinction time of roughly half a year for the United Kingdom as a whole. The basic calculation in this paper ignores spatial structure 3, 4 , heterogeneity of contacts between individuals 5 , and also dependence of transmission on age structure of the population. To assess the realism of our simple model, we performed simulations using a realistic spatial epidemic simulator, GleamViz, 17,18 which gave a distribution of extinction times matching reasonably closely the Gumbel distribution in Eqn.7, with a slightly smaller mean time; this suggests that despite the heterogeneity of contacts between different regions, the overall decay of infection is exponential and the stochastic variation follows closely the well-mixed predictions of the stochastic SIR model presented here. We have also made a very crude extrapolation of our results to extinction at the global level. The most problematic assumption, of course, is a single value of R e globally, and so the predictions in Fig.6 should be viewed as a guide to what could be achieved globally if all countries acted roughly in the same way on average. As we show, sub-divided regions with uniform migration, simply lead to a upwards rescaling of R e by factor (1 + φ) where φ is the migration probability, and so we can have confidence in these predictions, as long as R e at a global level is suitably interpreted. Previous work on generalising the concept of the reproductive number to include spread between different regions, uses a different approach, by defining an analogous reproductive number R * for regions, 3-5 , i.e. how many sub-populations or regions have at least one infection, where migration plays an analogous role to individual contacts for the spread of infection in a single population. This gives rise to a condition for spread or decline of infections to multiple regions and eventually globally, if R * > 1 or R * < 1, respectively. In the context of a local reproductive number R e < 1, as has been discussed previously 3 , it is still theoretically possible that R * > 1, particularly when infection is highly prevalent in a region (due to a previous time when R 0 > 1, as has occurred across in many countries with SARS-Cov-2 before lockdowns were imposed); this can happen if the global contact probability is sufficiently large, meaning that the infection can continue to spread between countries and regions. However, these long distance seedings of infection will not in themselves lead to regional or national outbreaks, as long as locally R e < 1 7 ; the simple upward rescaling of R e → (1 + φ)R e , which we observe for small φ captures this phenomenon. These results also suggest that at the national level, a simple rescaling of R e should describe the reduction in the rate of decline of infections due to importation of infected cases, and should be accounted in more accurate estimates of extinction times. Finally an aspect, which we have not considered, is the possibility of a non-human reservoir of SARS-Cov-2 , which could allow re-infections of human populations, such that actually extinction is not a permanent (stable) state, as we have assumed in this paper. This could be accounted in a similar way as we account for migration in this theory, except, unlike human populations (and arguably even in human populations), we have much less control, or even understanding, of a potential non-human reservoir. However, if such reservoirs, whether bat or pangolin 31,32 , can be identified and surveyed 33 , measures can be taken to control contact with human populations, such that effective global elimination in human populations is possible, even if the virus cannot be eliminated from non-human populations. To conclude, the theory presented and closed-form analytical equation for the distribution of extinction times of an epidemic, despite the many simplifications of the SIR model, provides a useful and quick guide to estimate, with confidence intervals, the time to extinction of an epidemic; different intervention, and test and trace, strategies are encapsulated in an overall effective reproductive number R e to give extinction time predictions as shown in Figs.4&6. In particular, we have shown that these results have broad applicability, beyond the specific assumptions of the calculation, when the reproductive number is suitably rescaled to account for migration. The broad conclusion when applied to SARS-Cov-2 is that to achieve rapid extinction, on times of order or less than half a year, then the goal should be to restrict R e to numbers much less than 1 and optimally in the region R e ≈ 0.4 → 0.5. Code to plot extinction time predictions and distributions, as well as for performing the Gillespie simulations can be found at https://github.com/BhavKhatri/Stochastic-Extinction-Epidemic. A birth-death process with birth rate b = γR e and death rate d = γ, corresponds to a pure exponential growth (R e > 1) or decay (R e < 1) phase of an epidemic, when the number of susceptible individuals S(t) are far in excess of the total number infected I(t). In this appendix, for presentational clarity, we will use n = I to represent the number of infected individuals. To write down the rate of change of the probability of n infected individuals at time t, we need only consider the probability of having n − 1, n, n + 1 individuals and rates of transitions between them, since in the limit of infinitesimal (continous) changes in time, we consider only changes of single individuals. The rate of transition from n − 1 → n happens with rate γR e (n − 1) and the rate of transition from n + 1 → n happens with rate γ(n + 1), which both lead to an increase in the probability of n, while the rate of transition from n → n + 1 happens with rate γR e n and the rate of transition from n → n − 1 happens with rate γn, which both decrease the probability of n. Using these facts we can write down the rate of change of the probability of n at time t: dp n (t) dt = γ R e (n − 1)p(n − 1, t) − (R e + 1)np(n, t) + (n + 1)p(n + 1, t) . However, this description isn't complete, and we need to consider how the probability of the n = 0 state changes, since the above equation won't work for n − 1, since we cannot have a negative number of individuals: dp n=0 (t) dt = −γ (R e np(n, t) + (n + 1)p(n + 1, t)) . (13) We can encompass both equations together in one by using the unit step function U n = 1 for n ≥ 0, while U n = 0 for n < 0: dp(n, t) dt = γ U n−1 R e (n − 1)p(n − 1, t) − U n (R e + 1)np(n, t) + (n + 1)p(n + 1, t) . For n < 0, as long as we have an initial condition, p n<0 (t = 0) = 0, the ODEs above guarantee that p n<0 (t)∀t. Considering each value of n : 0 ≤ n < ∞, we have an infinite set of coupled differential equations. The standard way to solve this is to use probability generating functions: which is in general a complex function of a complex variable z. Using the fact that z∂G(z, t)/∂t = np n (t)z n , it is straightforward to show that the set of ODEs give the following first order partial differential equation for G(z, t): where α(z) = γ(R e z 2 − (R e + 1)z + 1). This PDE can be solved by using the method of characteristics, which finds a parametric path z(s), t(s) along which our original PDE is obeyed. The rate of change of G(s) along this path in terms of our parameterisation is: and so with reference to the original PDE (Eqn.16), we can identify that dt ds = 1 dz ds = −α(z). population. Extinction will occur when all sub-populations have zero infected individuals. We can record the extinction times in each sub-population: t 1 , t 2 , ..., t k , ..., t n and the extinction time of the whole population will be the maximum of this set:t = max{t 1 , t 2 , ..., t k , ..., t n }. The cumulative distribution function of the maximum timet will be the probability of the joint event that each sub-population k has an extinction time less thant: P n (t) = P (t 1