Part 1. An Ran Chen MY Camp 2 nd Round 2012

Transcription

1 Part 1 In the lecture, Professor Tom Körner focuses on the various mathematical ideas used to understand the spread of infectious diseases, focussing especially on Small pox. These are as follow: 1. Using probability in the determining the cost/gain of inoculation by evaluation of the risks of inoculation. By analysing using annuity life tables, Bernoulli was able to hypothesize that inoculation increase the average life expectancy thus profiting the population as a whole. However, there are limitations of the idea such as the reduced of happiness initially after inoculation which is subject to value judgement and cannot be analysed in a mathematical way. 2. Modelling from a macro point of view, by assigning as the infection rate, and depending on the size of, and the size of the population, it can be determined whether the disease will develop into an epidemic and the duration of the epidemic. 3. The Professor also outlines the limitations of mathematical modelling: it makes many assumptions. Smallpox differs from other disease as it satisfies many of these assumptions. As per the lecture, Professor Körner suggested that if an infectious disease has an infection rate of (the number of people an infected patient is capable of infecting), an epidemic will occur if. Thus, we can see that the initial number of infected, I, will infect I number of people and these I people would pass the disease onto another people each, overtime, the number infected would increase exponentially. However, we know that the actual rate of infection would decrease overtime. For example, a person at the latter stages of an epidemic will being meeting many that have already had the disease and acquired immunity for it. So we can deduce that must be correlated with the number of people at risk of being infected, we thus code this as S, susceptible to the disease. However we must consider those that are not susceptible to the disease because they have already had it. So we can categorize the population into three categories, Susceptible to infection (S), Infected by the disease (I) and Removed from S and I by death/immunity (R). To simplify this model, let s assume that we can define the time length of all individual s infection to be of length 1, during this interval of time, the individual would infect at the rate of, and by the end of this interval, the individual would have recovered from the infection. Thus: For t(time) = x, So, when t=0, I x is the number infected at time x, similarly with R x and S x. P is the total population. I 0 = I 0, R 0 = 0 S 0 = P I 0 When t= 1, I 1 = I 0 = (P I 0 )(I 0 )/P R 1 = I 0 S 1 =P ( I 1 +I 0 )

2 At t = 2, I 2 = I 1 = (P I 0 - I 1 )(I 1 )/P R 2 = I 0 + I 1 S 2 = P- ( I 2 + I 1 +I 0 ) So, we can generalize that, at t=i I i = (P- )( /P R i = S i = P - We can see that, for all 1, I is always decreasing, thus it develop become an epidemic, however, when if 1, then I would initially increase then decrease, indicating the occurrence of an epidemic. This is demonstrated by a plot of I vs t that I have generated on excel. For this demonstration, I defined P to be 50 and I 0 to be Relationship between I and t with respect to a = 2 a = 0.7 a = However, this model may differ from a practical situation as many assumptions were made in order to construct this model: Firstly, we have assumed that the length of infection to be homogenous in the population. This cannot occur in reality, some people will spend longer time recovering from the epidemic and other will recover sooner. We have also assumed that immunity is acquired upon recovery. Though this is the case with diseases such as small pox, chicken pox, it is not the case of many; in fact, virus and other antigens can undergo mutations thus preventing the formation of immunity. We have also restricted the factors that can influence I to just and S. In reality, however, other factors exists, e.g increase awareness of the emerging epidemic can lead to preventative procedures such as quarantine that reduces the growth of I.

3 This model also doesn t account for the differentiated effects an infection has on individuals. Different age/racial groups of the population would not have the same chance of contacting the pathogen, nor is everyone equally vulnerable to the disease. Part 2: The purpose of vaccination is to induce a miniature infection, so that immunological memory can be created, preventing from catching the real and more serious form of the disease. It is known that when a big proportion of a population is vaccinated, herd immunity occurs, this is when the non-immune are all separated from each other due to the being surrounded by those immune and so even if an infected person enters into a population, the chances of him infecting the non-immune is negligible, so an epidemic will not occur. This phenomenon can be investigated from an epidemiological point of view. Similar to the modelling presented in the previous section, is the infection rate of the infectious disease. However, because many of population are already immune due to vaccination, many of them can be considered as our Removed population. Thus the actual infection rate of the disease, which we will denote as is, (1-r) where r is the proportion of population vaccinated. And we know that there will only be an epidemic when the actual infection rate is bigger than one, thus to contain and eradicate the infection, must be smaller than one. So, For herd immunity to occur, we must achieve a vaccination rate of at least. However, there is one problem with this, being vaccinated does not mean one will definitely gain immunity; there is a small chance that the vaccine will not work on a person. So we need to replace r with ev, where e is the success rate of the vaccine, and V is the proportion of population vaccinated. Thus the minimum proportion of population needed to be vaccinated for herd immunity to occur is: In the long run, we can predict that the disease would be eradicated as the disease cannot spread. However, one draw-back with mass vaccination program is that immunological memory cannot be passed down onto our off springs, so proportion immunised will fall below the minimum threshold for herd immunity over time, thus if some reservoir of the pathogen has been rediscovered, then our offsprings will not be able to defend against the disease causing a relapse of the epidemic. Also the immunity induced from vaccine might wear out overtime, calling for the need for booster vaccines, thus one must research into how our response to the pathogen weakens overtime. The efficiency of the vaccine can also vary between geographical areas, race and age. For example it has been speculated that

4 there is a correlation between the effectiveness of a vaccine and the living standards of the population. A malnourished child is less likely to respond successfully to the vaccine than a well-fed child. A vaccination programme that fails to eradicate the disease occur because the percentage immune is less than the critical proportion needed for herd immunity to occur. From, we can see that as more people are successfully vaccinated (r increases), the spread of disease,, decreases. However as the threshold needed for herd immunity is not achieved, would still exceed 1, thus an epidemic would still occur. However, since the rate of infection is lower, it would take longer for the infection to spread. This could be advantageous in the public health point of view as it allows time for appropriate advertising/response from the health department. For example, if the onset of an epidemic is detected early then people can be educated and warmed, to try and minimize the effects of the epidemic. One of the disadvantages of an unsuccessful vaccination programme is that it slows the rate of infection which in turns delays the mean age of contacting the disease. Bad if the disease is more severe at older ages. Thus to tackle this problem, a vigorous vaccination programme must be administered, to try and achieve herd immunity and increase the rate of immunisation. There are several aspects which must be considered when planning the vaccination programme: 1) Attitude towards the vaccine: Most people decided against taking a vaccine as they perceive the risks of abnormalities arising from the vaccine to be high, though it is true that no vaccine is completely risk-free, but we tend to be selectively informed of the dangers from a vaccine, making us perceive the risk to be higher than what it actually is. To encourage people to follow the vaccination programme we need to correct the bias in our risk perception. This can achieved through surveying samples of the population and identify any common misconceptions about the vaccine. Also, many are under the free-riding psychology, relying on others to get vaccinated in the hope that herd vaccination is obtained. We need to convert these people to get vaccinated through ways of encouragement such as monetary rewards. We can conduct experiments on random samples of these freerider to determine the amount of monetary encouragement we might need to offer to convert enough of them to boost the proportion vaccinated up and above the threshold for herd immunity. 2) Double injection: Assuming that the outcome (immunity induced vs. no immunity induced) of each injection of vaccine is independent of each other, then administering the vaccine multiple times would greatly increase the number of successful immunisations. For example, administering the vaccine once would only have a success rate of e, whereas injected twice was have a success rate of e + (1-e)e. However, we need further investigations to find out whether or not the outcomes of each successive injections on one individual are indeed independent or not, and if not how are they linked. 3) Narrowing down the high-risk groups: Finding out the exact age groups with the highest risks, e.g. from a death table and targeting promotion of vaccine more on the higher risk groups.

5 Bibliography: Webb, P., Bain, C. (2010), Essential Epidemiology : An Introduction for Students and Health Professionals (2nd Edition). Cambridge University Press, London. Xia, Y. Ma, S. (2008), Mathematical Understanding of Infectious Disease Dynamics. World Scientific, Singapore Ma, Z. (2009) Dynamical Modelling and Analysis of Epidemics.World Scientific, Singapore Evans, G. Bostrom, A., et al. (1997). Risk communication and Vaccination: Workshop Summary, National Academies Press, Washington DC, USA. Moussa Tessa, O. (2006) Mathematical model for control of measles by vaccination. Retrieved 20/02/2012 Johnson, T. (2009) Mathematical Modelling of Diseases: Susceptible-Infected-Recovered (SIR) Model. retrieved 20/02/2012 Wikipedia. Mathematical Modelling of Infectious Disease n retrieved 15/02/2012