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

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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 >/