The ecology of infectious diseases

In this assignment you will use your knowledge of the principles of disease ecology to explore the spread and management of a novel infectious disease.

General overview

You are a government advisor, with expertise on the management of infectious diseases. A new viral pathogen has emerged, which is rapidly spreading through the UK wild giraffe population, and your job is to (i) assess alternative control approaches and (ii) provide clear, evidence-based guidelines on how best to manage it.

You have available to you:

– Initial data on the number of infected individuals over the first 3 weeks of the pathogen’s spread through the population (see below for details).

– A mathematical model of host-microparasite dynamics (see below).

– Your knowledge, and some relevant literature (e.g., your class notes, and assigned readings), on the concepts of infectious disease spread, and the application of those concepts to the management of other, similar infectious diseases.

Your report will be structured around the questions in the ‘Report sheet’ below – these will guide your work in the following areas:

1. Estimate the basic reproductive number (R0) for the pathogen from the initial data available.

2. Use this value to parameterise your mathematical model, and explore alternative control/management strategies.

3. Comment on the confidence you have in your recommendations, the underlying assumptions, and priority areas for future work.

Remember, this report is intended for government employees who are non-experts in the field. So at each stage please describe clearly any assumptions you have made, and provide relevant background information to support your work.

Data and Model information

Data

The available data (‘Assignment 2 data+model.xlsx’, Sheet: Data) comprise information on the number of infected individuals recorded at regular intervals over the first 3 weeks of the epidemic.

The ‘Data’ sheet also gives current best-estimates of several key parameters (highlighted in orange) relating to the disease:

natural giraffe mortality rate (m = 0.1 week-1),

pathogen virulence (v= 0.5 week-1),

mean recovery rate (g= 0.4 week-1),

mean duration of infection (L = 1/(m+v+g)),

initial population size of susceptible giraffes (S = 99999 animals).

At the end of this document you will find guidance notes thatprovide information on carrying out various data manipulation tasks in Excel.

Model

The spreadsheet (‘Assignment 2 data+model.xlsx’, Sheet: Model) contains a standard SIR model (see lecture notes). The yellow-highlighted cells give the initial numbers of Susceptible, Infected and Recovered animals, and the total population size (there is no need to change these values for the assignment, but by all means feel free to explore what happens if you do).

The green-highlightedcells give the main parameters that control the epidemic. These are the ones you should focus on for your assignment. They should be self-explanatory, but note that it is assumed that the birth rate is the same as the natural death rate, which means the total population size would remain constant in the absence of the disease.

The blue-highlighted cells give ‘derived’ values that are calculated automatically from the parameters above – these are just for your information, and there is no need to change these values.

When you change any parameter values, the spreadsheet and graphs will automatically update to reflect those changes. Currently there are 2 graphs plotted:

i) The number of Susceptible individuals (using the scale on the left-hand axis) and the number of Infected individuals (using the scale on the right-hand axis) over time.

ii) The cumulative number of Infected individuals (red) and the cumulative number of individuals dying due to infection (blue) over time. The final values of these, at the last time point, will give the total number infected or killed by the infection throughout the epidemic.

Potential control alternatives

The model enables you to explore various alternative control methods by changing the relevant parameters in the green-highlighted cells on the ‘Model’ sheet. Your job is to run the model under various scenarios, and choose the best one(s) to recommend, with justification for that decision.

The main alternatives you could explore are:

a) Vaccination. The parameter p determines the proportion of the susceptible population vaccinated at birth (and therefore protected from future infection; it is assumed protection is lifelong). This parameter can be varied from 0 (no animals vaccinated) up to 1 (all animals vaccinated).

b) Quarantine. This isolates infected animals from contacting susceptible animals, thereby reducing transmission. You can model this scenario by varying the transmission rate (β) from the value you calculated in Q2a (which we assume to be the maximum contact rate in the absence of any quarantine). Reducing this value increases the proportion quarantined.

c) Medication. Here we assume the available medication only soothes the symptoms of infection, but does not directly affect the infection. As such infected individuals will remain infectious, but will not suffer from being infected. Model this by varying the virulence (v) from the value given (which we assume to be the maximum death rate due to infection in the absence of medication), down to 0 (the infection has no effect on host mortality).

d) Culling. Here we assume there’s an increased likelihood that infected individuals are detected and removed from the population. We can use the virulence (v) as a proxy for this process –but now we would increasev, to represent infected individuals being removed at an increased rate due to culling.

Report Sheet

• Insert your responses below.

• In all cases, clearly explain how you arrived at your answer, and add any figures/graphs as needed.

• Describe clearly any assumptions you have made, and provide relevant background information to support your work

• Please use outside reading to support your statements where appropriate – but remember to give full citations, and a reference list at the end, for any material taken from websites or the literature.

• Remember to use a sensible number of decimal places where relevant

Questions

1. Estimate the basic reproduction number (R0) from the data provided on the initial phase of the epidemic [hint: see Guidance notes at end for how to fit an exponential ‘trendline’ to data in Excel]. [4 marks]

2. Using the value estimated in (Q1), and the other parameter values given on the ‘Data’ spreadsheet,calculate the following values: [6 marks]

a) The transmission rate (β) [hint: remember R0 = βSL]:

b) The critical population size (ST; also known as the critical community size, CCS):

c) The threshold proportion to vaccinate to prevent disease spread (pT):

3. Use the values calculated above and the mathematical model to explore the consequences of the various control scenarios described on page 3. For each scenario, vary the relevant parameter to explore increasing levels of control by that method, and assess the effects on (i) the total number of individuals infected throughout the epidemic (the final value of the cumulative number infected), (ii) the total number of individuals dying due to disease (the final of the cumulative number dying) and (iii) the duration of the epidemic (the length of time until there is less than 1 infected case). The specific values you use, and overall design of this modelling ‘experiment’, are up to you – but do be systematic in your design – treat it like an experiment so you can clearly identify the effects of each control method.

Describe what you have done, and note the key findings (e.g., whether the control method brings the epidemic under control/eradicates the disease; whether the host population is maintained at a high level, or is driven very low/extinct; whether there are any interesting/unexpected effects of control). Includeany relevant plots to support your statements.

[40 marks]

4. Based on your analyses above, what is/are your recommendations for controlling this disease? Clearly justify your recommendation with reference to your above analysis, and with reference to any previous attempts to control other, similar diseases in humans, livestock or wildlife.

[30 marks]

5. Assess your confidence in coming to these recommendations. Describe key assumptions you have made, and identify potential sources of error or uncertainty. Where are more data or more information needed? Where would you recommend research efforts are prioritised? Finally, based on your knowledge of other similar diseases, outline any potential complicating factors that may need to be considered.

[20 marks]

Guidance notes for Excel

Here are some basic tips for manipulating data and calculating formulae in Excel, which may be useful for answering some of the questions.

Adding a trendline to a figure

If you want to add a trendline to an existing figure, right click on one of the datapoints in the figure, and click on ‘Add Trendline…’. Choose which type of trendline you would like (e.g., Exponential). To see the equation of the best-fit trendline, click ‘Display Equation on chart’.

Basic formulae

To enter a formula in a cell, start it with an equals ‘=’ sign. You can then either directly type the formula in (e.g. ‘=5+3’), or refer to values in other cells (e.g., ‘=A1+B1’).

Remember to use brackets to group terms together – for example if you wanted to calculate 3/(5 +2) you would have to remember those brackets: ‘=3/(5+2)’. Also remember that multiplication in Excel uses the ‘*’ symbol, so 3/(5×2) would be written as ‘=3/(5*2)’.

Reading scientific numbers

Sometimes you may see a value written in Excel like: 1.0E-05. This is Excel’s way of showing either very big or very small numbers; the ‘E’ means ‘x10 to the power of…’. So, 1.0E-05 means 1.0×10-5, or 0.00001. Similarly, 1.0xE05 means 1.0×105, or 100000.