Guest post by British doctor DeeTee. See more by DeeTee here.

As the H1N1 swine flu pandemic gathers pace, with 29 deaths to date, the attitude of the public seems to vary from rank indifference to blind panic. This view is reflected within my own hospital, where news of a possible flu admission is greeted by some staff saying "so what?" while others demand the equivalent of a biological "hazmat" suit before they get within a mile of a case.

One of the intriguing aspects has been the poor grasp people have on how quickly epidemics can evolve and what factors can influence this. Having spent some time recently reminding myself about the mathematics behind the spread of epidemics, I thought it would be a good idea to highlight some of the underlying principles.

Reproductive Numbers.

The extent of spread of any infection such as flu depends on how "contagious" the virus is. For an infection to spread within a population, an infected person has to pass it on to at least one other person. Depending on its degree of infectiousness, each infection may be allocated a number indicating its likelihood of spreading rapidly, representing the number of people on average who will be infected by one other case. This is known as the Reproductive Number, signified mathematically as R0. If an infection has an R0 of greater than 1, it will spread exponentially and become an outbreak or epidemic, but if it is less than 1, it will soon die out (hence the old adage that 1 is the most important number in epidemiology). But Reproductive Numbers can be variable, and are also situation specific (e.g. it will be higher in a children’s playgroup in Birmingham than in a rural Scottish village). The R0 for influenza is estimated to be between 1.5 and 3, and around 2 on average (by comparison the R0 for measles is around 15, and polio around 12, indicating their higher potential for spread).

But don’t think flu having an R0 of 2 is good news. A useful analogy for the transmission of an infection like flu through a population is the spread of gossip. Let’s imagine you have a unique bit of gossip; perhaps you know that a well-known local celebrity has got secretly engaged and you just cannot resist telling a couple of acquaintances. If they each tell 2 other people, you can see the starting of an epidemic of gossip (with R0 = 2). If the gossip is passed on every hour (although with texting and twitter, it would probably travel further and faster), how many people might have heard the gossip within a day? This question formed the basis of a scenario I ran by some non-clinical colleagues at work recently. Their answers ranged from a paltry "150" to an impressive-sounding "Ten thousand".

Or what if your gossip was not quite as juicy, and it only infects only one other person on average (R0 = 1)? Perhaps to get your news onto the grapevine you wish to give it a boost, by emailing it to 200 people in the company. After 24 hours, which method/infection type will have resulted in the most infections?

Most of those I asked indicated that this second situation would result in the most infections. In fact the gossip with R0 of 1, despite getting a big kick-start, can only infect a possible maximum of 4800 people. (200+200+200 and so on for 24 generations). However, the gossip with an R0 of 2 will have infected a gob-smacking 33,554,430 people, or over half the population of Britain (2+4+8+16+32+64 and so on for 24 generations). Any potential bioterrorists reading this please take note: You can efficiently generate a globally-devastating epidemic by infecting just a couple of people with a pathogen with a high R0. There is no need to contrive a massive release of biological agents by detonating germ-filled explosives in the skies over New York or in the Paris Metro. (Movie producers, feel free to ignore this tip and continue doing what you always do).

In practice however, epidemics of infection often quickly run out of steam, even if they have a high R0. Clearly you are unlikely to infect over 33 million people with your tittle-tattle of celebrity gossip. The obvious flaw lies in the fact that after a few generations it will be hard to find someone in the localised population who has not already heard the gossip, so they cannot be "infected" anew, and the further spread of gossip will cease, and the epidemic collapses. (Some of you will also realise this is the reason why moneymaking scams like pyramid selling and ponzi schemes will all fail within a short time of being created). The gossip’s R0 starts out as 2, but then drops as there are fewer and fewer people to potentially infect anew, and once the R0 drops below one, the epidemic burns itself out unless new vulnerable populations continue to be infected.

In 1927 a mathematical model of epidemics was devised by Kermack and McKendrick. This supposes there are 3 classes of individuals, namely those who are susceptible (S), those who are infected (I), and those who are recovered (i.e immune)(R). This is the SIR "compartment" model (and unfortunately further exploration of it requires differential equations beyond my understanding, so my foray into it must come to a grinding halt). It is sometimes adapted to SEIR, for Susceptible, Exposed, Infectious, and Recovered, for infections where there is a significant latent period between infection taking place and the ability to infect. For those of you interested in the background to compartmental modelling, there is a good overview on wikipedia.

As unfeasible as the gossip scenario for flu might seem in practice, it does illustrate the potential infections have to spread explosively. Various estimates are that half the UK population could still get swine flu, so is the gossip analogy valid and what does it tell us about real infections? Could they eventually infect millions of people in the same way? The answer is yes, but there are many other factors at work.

Let’s use the analogy to look at some of these. You tell two work colleagues your celebrity gossip. One of your friends tells her manager, and later on the other friend tells some of the staff in accounts. The epidemic is under way. However, within a day or two, you find that most of the staff in your company have heard the news. They are no longer susceptible, having become "immune" through already having been exposed to the gossip previously. These individuals are like the real-world examples of those who have recovered from an infection. According to the SEIR model, they are "recovered" and are now immune to catching it again, and will not pass it on even if re-exposed, and the pool of "susceptibles" has declined rapidly. Some of the people hearing the gossip could also be regarded to be "immune" because although exposed to your gossip, they couldn’t care less about it, don’t give it a second thought and wouldn’t dream of telling someone else. These are the equivalent of people who have been immunised by vaccination. An important point to remember is that not everyone in the company will hear your news. Perhaps the cleaners who only come in at night or the staff who work in the post room remain blissfully unaware of it because they fortuitously avoided coming into contact with anyone who was infectious.

Herd Immunity.

The gossip analogy is useful to explain the dynamics of other aspects of an outbreak, like the protection afforded by herd immunity. Let’s say someone visits your company from head office a few days later. A bit behind the times, he mentions the celebrity engagement by way of conversation. But by now this is old news in your workplace, the 3 people he tells it to are already immune, having already heard the gossip the previous week and they don’t bother mentioning it to anyone else. This is herd immunity at work. A sufficiently large proportion of the people in your company have been immunised to such a degree that new infections will not be able to spread easily. They make up a protective "herd", and the still susceptible cleaners (even now they have switched to the day shift) will never get to hear the gossip.

Generation Time.

Another crucial factor that determines the rapidity of spread of an infection is the “Generation Time” (Tg), or the interval it takes between one individual becoming infected and passing it on to the next. Some infections will spread more rapidly than others, even though they have a similar R0, just because they have a shorter generation time. For example, the 1918 pandemic influenza strain had an R0 of around 2, which is about the same as does HIV. The difference is that the generation time for flu is only about 2-3 days, whereas for HIV it is many months, and often several years. This explains why flu pandemics seem to zip around the globe, while the HIV pandemic has chugged along at a relatively sedate snail pace. (Bioterrorists take note once more: Choose an agent with a short generation time as well as a high R0).

Calculation of susceptibles versus protected/immunes

There is a simple and useful formula for calculating the number of people in a population who need to be “susceptible” before an outbreak or epidemic is likely. This is most often used to estimate the proportion of the susceptible population that needs to be immunised in order to avert outbreaks. If the proportion of immune people in the population drops below 1-1/ R0, then outbreaks are likely.

For measles, 1-1/ R0 is 1-1/15 (or 93.3%), hence the advice that we need measles vaccination rates of 90%-95% to avoid outbreaks. For influenza, 1-1/R0 is 1-1/2, or 50%. So if half the population is immune from having had the swine flu before or because they are vaccinated, a major epidemic is unlikely. Obviously with the current swine flu epidemic, there are very few who are immune to H1N1 (which last touched our shores in 1957) and as yet there is no vaccine, so the epidemic would appear to be unstoppable.

Strategies for Management.

The dynamics of epidemics tells us something about how they are best managed. Initially in the UK, the plan was one of containment, where any suspected cases who had travelled to the UK from Mexico or the USA would be isolated, treated and all close contacts given antivirals to prevent them being infected. This it was hoped would contain the epidemic by rendering the R0 <1 and reducing spread. Unfortunately, once widespread onward transmission took place in the general population, this ring-fencing became impracticable, and the shift is now towards treatment and isolation of cases only as they occur. Some estimates indicate that if the R0 for influenza were less than 1.6, then a system of isolation and targeted antivirals would be successful. The fact that this has not succeeded with swine flu indicates its R0 is greater than this, and more likely to be around 2, as for the 1918 H1N1 epidemic.

The mathematics here involves not just R0, which is the most relevant factor, but also latent periods, infectious periods and generation times, which are again is too complex to delve into here. A good overview of the influence of these can be found here.

Infections are fascinating. One of their most intriguing and unique qualities is their ability to spread, often explosively, causing major epidemics of disease or pandemics. Infections have helped shape evolution, history and civilisation, and have ranged from the biblical plagues of antiquity to the more modern global devastations wrought by malaria and AIDS. People usually don’t give more than a passing thought to the historical relevance or consequenses of infection, but occasionally this apathy is punctured by episodes of sheer panic, as is currently happening with swine flu. If nothing else, I hope this blogpost has given food for thought, and hopefully not resulted in even greater concern about the consequenses of swine flu, which fortunately does not seem to be any more lethal than the usual outbreaks of seasonal influenza.