Spring 2006
(Revised 04/25/06)
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EEB/MCB 182H:
Spring 2006 |
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Modelling Malaria.
(Revised 04/25/06) |
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Ross-MacDonald Equations. The simplest models [r11, r16, m52, m57] of malaria, the so-called Ross-MacDonald (R-M) equations, date to the early 20th century. These equations compress the manifestly complex interactions among man, mosquito and plasmodium into a pair of coupled differential equations that specify the time evolution of two variables: the numbers, X(t) and Y(t), of parasite-laden humans and female mosquitos. In particular, R-M assumes that transmission of the plasmodium - from mosquito to man and from man to mosquito - depends jointly on the numbers of susceptible and infected individuals of the appropriate species, which is to say, upon their product. Let N and M be the total numbers of humans and female mosquitos. If parasite transfer is from mosquito to man, the product in question is Y(t) × [N - X(t)], which is the number of plasmodium-bearing mosquitos, Y(t), times the number of malaria-free (and by assumption susceptible) humans, [N - X(t)]. Conversely, the transfer rate from man to mosquito is presumed proportional to the number of infected humans, X(t), times the number of plasmodium-free mosquitos, [M - Y(t)]. This leads us to
dX/dt = A B Y (N - X) - r X
where the parameters, A, B, C, m and r are as defined in Table I. In particlar, A is the per mosquito bite rate. Equations (1) assume constant total numbers, N and M, of humans and mosquitos. They further presume that infected humans recover (and become immediately susceptible) at rate, r, and that sporozite-bearing mosquitos die off at rate m — i.e., both loss processes are assumed to be exponential.1 As to the constants, A, B and C, these relate transmission to biting by female mosquitos — the event by which transfer occurs — and the probabilities of parasite transfer from vector to man and man to vector. Note that the products, A × B and A × C, are respectively the per mosquito production rate of newly infected humans and the per human production rate of newly infected mosquitos.
Exercise 1. Verify that Equations (1) "balance," i.e.,
that the units on both sides of the equality sign are humans/year in the case of the first
equation and mosquitos/year in the case of the second.
Replacing Absolute Abundance with Proportion. Inasmuch as Equations (1) presume constant populations, they are customarily simplified by replacing absolute abundances of men and mosquitos with proportions. In this spirit, we introduce the following new variables and parameters:
Substituting the new quantities into Equations (1) then yields
dx/dt = a b q y (1 - x) - r x
which is the form most frequently encountered in the literature - see, for example, Aron and May [am82].2
Exercise 2. Verify that the units of Equations (3) -
both equations, both sides - are y-1.
Isoclines and Equilibria.
One way of studying systems of differential equations is to integrate them numerically. Especially if nonlinearities are involved, this is often the only approach possible - i.e., most of the equations arising in mathematical biology do not admit to closed form solutions. In the present case, however, the equations' simplicity, allows us to take an alternative tack that yields geometric insight. Specifically, we study the zero-growth isoclines,
dx/dt = 0: y = (r/abq)[x/(1-x)]
that are obtained by setting the right-hand sides of Equations (3) equal to 0 and solving for y as a function of x. We now plot these functions in the x-y plane, as shown in Figure 1. Note the essential points:
Negative numbers of humans and mosquitos being biologically implausible, we restrict our
attention to the plane's positive quadrant, (x ≥ 0; y ≥ 0).
As shown in Figure 1, there are two
possibilities:
Top graph. The so-called no-disease state, (x, y) = (0, 0),
is the sole, non-negative equilibrium, and it is stable.
Bottom graph. There are two non-negative equilibria. One is
the origin, which is now unstable; the other, the so-called endemic state,
(x, y) = (x*, y*), is stable.
Which possibility obtains depends on the value of a quantity, R, called the basic reproductive rate of the disease. In the present case,3 R = a2 b q/m r . (5)
How one goes back and forth between these alternatives can be understood by observing the following:
It follows that as R is varied through R = 1, the equilibria collide, and there is an exchange of stability as shown in Figure 2. This qualitative change in dynamics is called a transcritical bifurcation.4
Exercises.
x* = (R-1)/[R + (a/m)],
4. For the parameter values in Figure 1, compute the value of a at which the transcritical bifurcation occurs. Compare with Figure 2. 5. Equations (3) are "partially non-dimensionalized," i.e., only time has units. Complete the non-dimensionalization by replacing t, the units of which is years, with the dimensionless quantity, t = r t and by introducing new, dimensionless parameters, a = a/m and b = bmq/r. Write the resulting equations in the form
(dx/dt) = ...
where e = r/m. Why might e be small? If it is, what are the consequences?
To incorporate latency into our model, we distinguish a third class of female mosquitos, z, which we will say are exposed, and replace our previous equation for (dy/dt) with two equations.5 We further note that the rate at mosquitos enter the infectious class is proportional to the product, x˜ (1- y˜ - z˜), of infected humans and susceptible mosquitos, t time units in the past, i.e., x˜ = x(t-t), etc. Moreover, this product must be multiplied by e-mt, which is the fraction of newly infected mosquitos that survive to become infectious. Accordingly, we write6
dx/dt = a b y (1 - x) - r x
From these equations, we calculate R˜ = [a2 b q /(r m)] e-mt (9) where the symbol, R˜, is used to indicate the inclusion of latency in the model, and
x* = (R˜-1) / [R˜ + (a/m)]
From these exprerssions, we conclude that latency diminishes both the basic reproductive rate, R˜ = Re-mt, and the equilibrium density of infectious mosquitos, y*, by a factor of e-mt (Figure 3). As pointed out by Aron and May [am82], the latter change brings the model into better accord with observation - i.e., empirically, the fraction of female mosquitos with parasites in the salivary glands is on the order of a few per cent.
The most important implication of the foregoing analyses, i.e., with and without the time delay, is that you don't have to kill all the mosquitos or immunize all of the human population to eradicate malaria. What is is necessary is to disrupt propagation to the point that R < 1. This having been accomplished, the disease will then die out of its own accord, as the system tends to the "no-disease" state at the now-stabilized origin.7 What does this tell us about possible control strategies? In terms of the parameters listed in Table I, R˜ = [(A B M) / m] × [(A C N) / r] × e-mt (11)
where the quantities in square brackets can be viewed respectively as the per mosquito
production
of newly infected humans (the term, [(A B M) / m]) and the per case
production of newly parasitized mosquitos (the term, [(A C N) / r]). Obviously,
one wants to reduce the quantities, A, B, C and M, while
increasing m and r. In addition, we note the following:
The mosquito mortality rate, m, enters twice, once in
proportion to (1/m), the other time, exponentially as
e-mt. As a result, reducing
mosquito longevity, as might be accomplished by spraying, provides the surest approach to
reducing R˜.
The bite rate, A, also enters twice - in proportion to
A2. This makes sense,
since completing the plasmodium's life cycle requires two bites. Accordingly, a 50%
reduction in the number of bites divides R by a factor of 4, etc. It follows
that, even
without killing mosquitos, control strategies (screens, pesticides, netting) that keep
insects out of peoples' houses, or at least out of their beds, yield favorable
results. Together with point # 1 above, this fact explains the historical success
[a00] of spraying programs (now largely curtailed) in which houses were
dusted with DDT once or twice a year. The remaining parameters enter once: Vaccines that reduce human susceptibility reduce B; drugs that reduce the infectious period by killing plasmodia within the human body increase the recovery rate, r; interupting the parasite's life cycle within mosquitos, e.g., via the creation of transgenic mosquitos, reduces C; reducing the overall mosquito population by spraying or draining marshes and other breeding sites, reduces M; reducing local human population density (relocation programs) reduces N. In sum, control strategies that focus on the death rate, m, of adult mosquitos or the bite rate, A, are inherently more efficient than "non-A/m" strategies. This is not to deny that complete interuption the plasmodium life cycle at any point would end the scourge. After all, the basic reproductive rate, R, depends multiplicatively on all the parameters, and multiplying anything by 0 gives you 0. At the same time, non-A/m control strategies must, of necessity, be more effective than m- or A- strategies to achieve the same results. Illustrative calculations are given in Table II. Here, R˜D is the value of R˜ that obtains upon multiplying or dividing the parameter in question by a factor of D. Caveat. Equations (3), ignoring as they do, enormous amounts of biological complexity, are highly simplified.8 Still, something like the partitioning of the basic reproductive rate into two components, and the multiplication of their product by e-mt, is likely to fall out of any reasonable mathematical description of the disease. In effect, R˜ = R˜hm × R˜mh × e-mt (12) where R˜hm is the per mosquito production of newly infected humans, and R˜mh, the per case production of newly parasitized mosquitos, both of which depend on the bite rate, A. This, we believe, is the biological essential, and it guarantees the distinction between "m-", "A-" and "non-A/m" control strategies noted above.
Exercise 6. Enumerate additional aspects of the biology
omitted from Equations (8) and discuss their probable conbsequence to the conclusions of the
preceding section. In particular, think about the selective consequences to mosquito and
plasmodium life histories of various control programs.
[am82] Aron, J. L. and R. M. May. The population dynamics of malaria.
Pp. 139-179. In, Anderson, R. 1982. The Population Dynamics of Infectious Diseases:
Theory and Applications. Chapman and Hall. London.
Notes.
1. According to this, the mean recovery time in humans is (1/r),
and the mean longevity of mosquitos, (1/m). Since (1/r) > (1/m),
we say that the time scale of Y is "fast" relative to that of X.
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