|UBC Calculus Online Course Notes|
There's good reason for the despair that hangs over Vancouver's Downtown Eastside...Researchers have found that the 6,000 to 10,000 heroin addicts who inhabit this pocket of poverty have the developed world's highest rate of HIV transmission - 18.6 per cent. Translation: If 1,000 addicts are free of HIV, 186 of them will contract the AIDS- related virus within 12 months."The depressing news contained in a series of articles in the Globe and Mail outlines a serious epidemic raging in the Lower Eastside of Vancouver, and threatening to spread beyond. The main victims of HIV and its consequent AIDS are the intravenous drug users that live under poverty conditions in tenement and block houses. The mode of transmission is dominantly the sharing of needles, though unprotected sex probably accounts for part of the problem and the main mechanism of transmission of this epidemic to the population at large.
M. Cernetig, Globe and Mail, Oct 8, 97.
On this page, we will formulate a case study based on the HIV epidemic, and illustrate how some of the mathematical ideas discussed in the calculus course can enter into the discussion of the basis of this epidemic, and possible measures to control it. The purpose is to use simple mathematics and illustrate what predictions can be based on mathematical arguments, not to produce a definitive model for a complex socio-medical phenomenon. There are many research papers filling thousands of journal pages which deal with the many complex issues that have to be considered. Here we can only get a brief glimpse at some of the most basic steps. Nevertheless, even this gives a certain insight into the disease progression.
(a) To understand the dynamics of the HIV epidemic.
(b) To suggest health measures for controlling its spread.
Here are some facts gathered from newspaper articles (see citations at the bottom of this page). At the time of writing of this module:
- There were 6,000-10,000 intravenous (i.v.) drug users living in the Vancouver Lower Eastside (VLE).
- Among these, about 1,500 were already infected with HIV, and more were acquiring the infection.
- An injected- cocaine addict gets a "fix" 15-20 times per day.
- 40 per cent of the i. v. drug users share their needles.
- Poverty leads to significantly increased mortality from complications of AIDS, the disease associated with HIV.
- In V L E out of 1000 uninfected individuals, it was estimated that 186 would acquire the disease as a result of contact with the infected population.
Modelling the disease dynamics
We will start with some definitions of variables that will help discuss the progression of this epidemic.
= number of i.v. drug users infected with HIV at time t.
= number of i.v. drug users susceptible (NOT infected) at time t.
= total number of i.v. drug users at time t.
In order to simplify the situation so that it is easy to understand, we will make the following assumption:
Assumption 1:We will keep track of the number of infected individuals in the population. To do so we notice that new infectives arise as the disease is spread, and those already infected are removed by mortality and other causes. Thus, the equation of the model will look like:
The housing situation in VLE is such that the population of i v drug-users is roughly constant. When one user dies or leaves, another moves into the vacancy left in the rooming house or tenement. Thus,
This equation shows how changes over time as infectives leave and enter the population. We now consider the detailed forms of the terms in this equation. To do so we need the following two assumptions:Assumption 2:
The population is homogeneous. This means that we are assuming that all its members have similar habits, live in similar conditions, and share similar traits. (This assumption is far from realistic, but it permits us to make a statement about the population on average.)
The population is "well mixed". This means that we assume that each infected person comes into contact with all other people equally often. (Again, this assumption is unrealistic, since physical and social connections would surely influence how needle sharing occurs. Again, it is a simplification we make cautiously at this stage.)
Transmission and mortality
A constant parameter that we here call will stand for the average rate of transmission of the disease through i v needle sharing. The definition of this parameter and consequences are:
= average probability that 1 infected transmits HIV to 1 susceptible in this population per year.
= average number of susceptibles that get HIV from contact with a single infected individual per year.
=average number of susceptibles that get HIV from contact with any of the infected individuals per year= total number of new infections created per year.
About the mortality, we assume that
= average probability that 1 infected dies per year.In actual fact, we know that HIV takes many years to develop into AIDS from which the risk of mortality is very high. There is a significant delay between the time of infection and the time at which the disease leads to death. Here is another place where a simplification is being made.
= average number of infected individuals that die per year.
With the terms collected above, we now arrive at the differential equation that describes the spread of the epidemic:
This equation has two unknown functions that are changing with time, , but we can use assumption 1 to eliminate one of these. Plugging in we obtain:
We can expand terms, regroup them, and rewrite this equation in the more revealing form which is shown at the end of the following questions.
For your consideration:
- (1) In the above equation, identify which terms are constants and which signify unknown time-varying quantities. What are we looking for when we are trying to find a solution to this equation ?
- (2) Expand the equation shown above and demonstrate that it can be written in the form:
- (3) By grouping together the constant terms inside the braces, show that the equation can be written in the simpler form
where the new constant stands for .
- (4) Comment on the similarities/differences between this equation and the Logistic equation discussed in Limited and Unlimited growth
- (5) Is this constant positive or negative ? Could it be either ? Under what circumstances is positive? Explain how the sign of this constant depends on the size of the population.
- (6) For what values of is there no change in the number of infections? (Such values are called steady states.) Interpret each of the two possible cases in which this occurs.
Analysis of the Equation
In the parts above, you will have discovered that the equation for spread of infection can be written in the form
with . You may also have noted that the levels for which are just , so that either the disease is absent entirely, or else it has reached a level of . We can use the qualitative method of the direction field to understand how the solutions to this differential equation behave. As before, we are aided by a table which connects values of and the sign of the slope and we find that:
Value of 0 K Sign of 0 + 0 -
This leads to the direction field shown below. The red lines represent the tangents to any solution curve . The blue curve is one such solution. Its initial value can be changed by moving its red dot up and down along the y axis. Since the constant depends on other parameters, it is possible, as you may have found above that its value would be either positive (if is very large) or negative (if ). The constant is represented in the Applet below by the pink line with its red control node. By moving this node, you can experiment with the behaviour of the solution when the constant is either positive or negative.
For your consideration:
- (1) Explain why only values of are of interest in this model.
- (2) Set in the interactive diagram above, and describe how the solution to the differential equation changes as you vary the initial condition. For which initial values of does the solution decrease? For which does it increase? For which values is it exactly constant?
- (3) Set the pink line at a level that corresponds to . Describe the blue solution curve.
- (4) Set the pink line at a level that corresponds to . Describe the blue solution curve.
- (5) Interpret each of these results. For which of these cases does the infection die out ? For which does it reach a constant endemic level ?
You should have discovered a result which is known as the threshold effect of epidemic spread, namely that
There is a threshold level of the population, . When the population is larger than this level, i.e. , the epidemic will become endemic, and the number of infections will stabilize at . If the population is small, i.e. , the infection will die out.
"Real" parameter values and the HIV infection
We now use the parameters estimated from newspaper articles to estimate the values of the constants in the problem.
Transmission: We use the fact that (at the time of writing) the estimated rate of infection in the Vancouver Lower Eastside is 18.6 per cent given the then current number of infections in this community. Thus, using the fact that
we find that
Population: We further estimate that the total population of drug users is
Mortality:Finally, from Murray Campbell's article (Globe and Mail, Nov 15, 1997) we can estimate that the mortality associated with AIDS/HIV is roughly 15 percent. This means that out of 100 infected individuals, 15 would die within one year. Thus,
We find that in this case, the parameters satisfy so that the disease will become endemic unless some measures are taken to control it. Further, the constant defined above has the value
From our discussion above, we remember that a positive value of represents the value of that is eventually reached. Thus, our analysis predicts that the number of HIV positive individuals would eventually reach a total as high as 4,800 given that the epidemic continues unabated. This is of course an approximation in which many simplifying assumptions entered, and which neglects any efforts made to curb the progress of infection.
We can use Euler's method to solve the differential equation with its parameter values in place. The figure below displays the results:
For your consideration:
- (1) Now that you know something about the dynamics of the disease, describe what might be the effect of each of the following changes in the community:
- (a) Increasing the extent of crowding - i.e. having a larger population of drug users in the same community.
- (b) Supplying many sterile needles constantly, so that addicts need to share needles less often. Which parameter would this affect ? How would it affect the parameter? What would be the result ?
- (c) Reducing mortality with the new generation of medications that can keep AIDS patients alive longer. Discuss the mathematical and ethical implications of this effect.
- (2) What strategies would you suggest might help to combat the epidemic?
- (3) The model we have discussed is quite simplified. How might that affect our conclusions? What should we consider to formulate more accurate models which would be more accurate and have more robust predictions?
Other sources of information:
Miro Cernetig "Where death gets a double shot", The Globe and Mail, October 8, 1997.
Robert Matas "Vancouver HIV actually down", The Globe and Mail, Oct 25, 1997.
Murray Campbell "AIDS: step forward, slide back", cover story, The Globe and Mail, November 15, 1997.
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