Modern concepts of epidemic theory

originated from the work of William Farr, the first epidemiologist to begin

discerning mathematical principles governing the behaviour of infectious

diseases. The basic reproduction number of such an infection, R0, is

defined as ‘the number of cases that would result directly from the

introduction of a single infectious individual into a susceptible population’ and

is therefore effectively synonymous with ‘transmissibility’:

R0 = C x

P x D

In which C represents the average rate of

contacts made between an infected individual and susceptible individuals in the

population, P is the probability of transmission from each contact and D is the

duration of infectiousness.

An R0 value of 1 therefore

implies that a single infectious case will cause, on average, one other, whereas

an R0 of less than 1 indicates the disease will eventually disappear,

and an R0 greater than 1 indicates continual spread of infection. Hence,

in order to eradicate an infection, we must attempt to alter the host pathogen

relationship in such a way that R is decreased below 1.

It is unlikely in any actual population

that every single individual will be susceptible to a particular disease, the

effective reproductive rate, R, therefore estimates the average number of

secondary cases in a given population consisting of both susceptible and

non-susceptible individuals.

R = R0x

In which X represents the fraction of the

population which is susceptible to the disease. X will be reduced in

populations which have previously encountered a disease, and hence a greater

proportion of individuals have acquired immunity. This is also seen in public

health measures which promote immunization; “When an individual is successfully immunised, not only is there

one less person who will ever be infected, there is also one less person who

will be infectious” (McLean, 1992). Vaccination therefore acts to decrease the

infectious pool in such a way that it provides protection even to those who

have not received the vaccine, known as herd immunity. In order to completely

eradicate a disease, the critical level of vaccination (also known as the threshold

for herd immunity) can be calculated from R0, giving the extent to

which the pool of succeptible individuals must be reduced;

Pc =

1-1/R0

The larger R0,

and hence the greater the transmissibility of a condition, the larger Pc

must be in order to eradicate the disease.

Measles

Measles has a high R0 and

therefore a very high proportion of the population must be vaccinated in order

to eradicate the disease. The viral disease causes serious complications,

including encephalitis and pneumonia as well as suppressing the immune system,

increasing the P value for other epidemics. As a result, measles remains a major causes of

death, however, the introduction of a successful vaccine is believed to have decreased

child deaths by 1/5th since 1990. The importance in this vaccination

program can be seen in the aftermath of the publication of Wakefield’s work in

a 1998 Lancet paper. This paper implicated the MMR vaccine in the development

of autism in young children, which, despite the paper stating that no causal

relationship had been proven, and further studies finding no relationship at

all, resulted in a huge decline in confidence in the vaccine, and hence a rise in

the number of parents refusing it for their children. This has been implicated

as the major cause of the rise seen in measles cases following the papers

publication, In 2008 for example, measles was declared endemic for the first

time in 14 years. This illustrates the dramatic effect that may be caused by a

seemingly small decline in vaccination rate – it is thought that a 5% fall in

MMR vaccine administration may result in a threefold increase in measles cases.

The 2008 endemic was particularly prevalent in festivals, thought to be a

result of the increased rate of contact, increasing R0;

R0 = ?/?

In which B is the no. of contacts per unit

time which will result in new infections and 1/y the mean infectious period.

Small

pox

Small pox is also a virus which, in

contrast to measles, has a reasonably low R0 and has therefore been one

of the only 2 diseases to be officially declared eradicated (alongside rinderpest

in 2011), despite over 15million cases occurring each year as recently as 1967.

Small pox is highly contagious, with a high P value, however the duration of

the infectious phase is short and occurs only following onset of the rash, and

with reasonably close contact (within 1.8m), and hence R0 is lower

than measles and many other viral diseases. The eradication of small pox has

been made far easier by the lack of alternative hosts which may provide a

reservoir for the disease (for example mosquitos in malaria) and has relied

primarily on widespread vaccination programs and careful surveillance and

isolation of outbreaks.

HIV

One of the most effective methods of reducing

the incidence of AIDS, for which there is no vaccine, is to reduce the risk of

transmission, P, by the use of anti-retroviral therapy. By 2020 the joint

United Nations program on HIV and AIDS has set a target to ensure 90% of all

people infected with HIV are aware of their status, 90% of them are on

anti-retroviral therapy and 90% of those on therapy will have full viral load

suppression. This will result in 73% of those who have HIV achieving full viral

load suppression, which, if maintained should enable elimination of HIV in 70%

of Sub-Saharan countries and reduce R0 to less than 2 in the remaining 12

countries, compared to the current median R0 of 4.3. in combination with other

high-impact preventative methods such as promotion of condom use and an increase

in the availability of Pre-exposure prophylaxis (PrEP) could potentially see

the eradication of AIDS.

PrEP enables a reduction in the number of susceptible

individuals who may come in contact with those which are infectious, and hence

a reduction in C similarly to vaccination. If taken consistently it has been

shown to reduce the risk of infection by up to 92% in high risk individuals. Without treatment, the risk of transmission of

HIV from sexual intercourse is considered to be approximately between 0.001 and

0.1 but the use of condoms has been found to reduce this by 80 – 85%. One study

found that in 123 discordant couples who consistently used condoms none of the

uninfected partners became infected, in comparison to 12 uninfected partners

who contracted the disease out of 122 couples using condoms inconsistently.

IMMUNITY

and mutation? FLU

Childhood

diseases

R0 can in fact be estimated from the

average age of infection. This is because there is a greater chance of

encountering a disease with a high R0 (i.e. a high transmission risk and many

infectious people) earlier in life. Such infections are commonly termed ‘childhood

diseases’ since the majority of people are infected at a young age, conferring acquired

immunity for the remainder of their lives. By this estimation:

R0 = 1 + L/A

In which L is the average lifespan and A

the average age at infection.

Chicken pox is one such disease in which

early infection is common since the infectious period begins 1-2 days prior to

a rash appearing, unlike small pox, preventing cases from being identified and

isolated. This is coupled with a high probability of transmission – studies of

transmission have found that over 90% of close, susceptible people in contact

with a diseased individual will be infected, resulting of an R0

value of approximately 11. Generally, contraction of the disease results in

immunity to future infection (although latent infection may be reactivated) so

outbreaks are rarely seen in adult populations. Pertussis, by comparison, has a

much lower R0, of approximately 5.5 and as a result is more commonly

seen in older populations, as supposed to just the very young. Since the R0

value of chicken pox is so high, but the symptoms of the disease, particularly in

children, are relatively mild in comparison to pertussis, it is not commonly vaccinated

against in the UK.

Limitations

of R0:

R0 is always an average value –

as not all infected individuals will transmit the disease to exactly the same

number of people, therefore if there is great variation in the rate of spread

among different subgroups if the population, the average R0 will be

largely meaningless. R0 is therefore most useful in explaining the dynamics of

a disease which is spread broadly among individuals who meet at random.

R0 is difficult to observe and

calculate in the field and hence mathematical models are frequently used to

estimated its value, however, the true value of R0 is rarely

actually derived from many mathematical models, which instead give a threshold

as supposed to the number of secondary infections. This does have some benefit

in determining the viability of an epidemic (with R0 >/< 1) however is of
relatively little use in comparing the dynamics of two different disease
epidemics.