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# Mathematical modelling of epidemics

N L. O araigh
School of Mathematical Sciences, University College Dublin, Beleld, Dublin 4, Ireland (Dated: May 28, 2012)

First of all, a confession: I know very little about mathematical modelling of epidemics. By the end of these two sessions, you will know as much as I do! On Friday, Professor Gleeson will give you much more information than I possibly could about dierent and more thorough approaches to this modelling problem. However, I will endeavour to make up for this ignorance with enthusiasm, as I nd some of the ideas in this modelling problem to be both devastating simple and elegant. The rst talk is based on Chapter 4 of the textbook of Stephen Strogatz Nonlinear Dynamics and Chaos [1], which in turn, is based on an earlier paper by Kermack and McKendrick [2]. Epidemics are complicated things. Even the simple model of Kermack and McKendrick involves three ODEs that aect one another (or are coupled together). However, we will show how, in this particular case, the ODEs can be simplied down to a single equation. We will then analyse that equation using the techniques we have learned (xed points, stability analysis).
I. SOME BACKGROUND INFORMATION

The epidemic modelling of Kermack and McKendrick is based on ordinary dierential equations. An ordinary dierential equation is a relation of the kind du = F (u, t), u(t = 0) = u0 , (1) dt where u(t) is some trajectory in Rn and u0 is some constant vector in Rn . Of course, a simple special case is when n = 1, in which case, you should recover a familiar kind of equation, which you have hopefully solved in your classes: du = f (u, t), u(t = 0) = u0 . dt In general, local existence theory guarantees that Eq. (1) has a solution for nite times t [0, T ] (with T to be determined) provided F is suciently well behaved. More precisely, if F is Lipschitz, i.e. if there exists a t-independent positive number K R such that, for all u1 , u2 Rn , F (u2 , t) F (u1 , t) 2 K u2 u1 2 , then Eq. (1) has a unique solution in [0, T ], for each T < 1/K . Here 2 is the usual L2 norm on Rn ; the choice of norm does not matter here. Dont pay too much attention to this condition for now [3]! In loose, everyday language, provided the F function is not too badly-behaved, the ODE (1) is guaranteed to have a solution. A special kind of ODE is called autonomous, which means that the F -function contains no explicit dependence on t: du = F (u), dt u(t = 0) = u0 , (2)

2 is an autonomous ODE problem. For such ODEs, it is meaningful to talk about xed points. Roughly speaking, this is an equlibrium point, where there is no motion. A vector u is a xed point of Eq. (2) if F (u ) = 0. Obviously, a xed point does not evolve over time, since d u = F (u ) = 0. dt For such xed points, it is meaningful to talk about their stability. More precisely, it is useful to know if a trajectory that starts near a xed point, stays near the xed point as time goes by. Unfortunately, there are many types of stability! The most elementary kind is linear stability: we take an initial value u0 = u + u0 , where u0 is small in an appropriate norm, and we evolve this condition over time, letting u0 u(t), u + u0 u + u(t).

An explicit evolution equation for u(t) is obtained by substituting the proposed time evolution into Eq. (2) to obtain d F u(t) = F (u + u) = dt u u.
u

The perturbation u(t) will stay small (stability) or blow up (instability) depending on the eigenvalues of the matrix F J := . u u
II. THE MODEL ASSUMPTIONS

We consider a human population suering from an epidemic. The population is divided into three classes: healthy people, x(t), sick people, y (t), and dead people, z (t). The population is subject to the following assumptions: The total population x + y + z is constant in size. That is, births, deaths (from nonepidemic-related causes), and migration, are unimportant. This will certainly be the case in a rapidly-evolving epidemic. Sick people do not recover and die at a constant rate . These assumptions, together with one further assumption (below) suce to develop our model. The model is now based on conservation of people. First, imagine that no people die from the epidemic, = 0. Then, the number of people who get sick is proportional to the probability that a sick person and a healthy person encounter one another. This in turn is proportional to the product of the population of sick and healthy people: Probability of a healthy person getting sick xy.

3 When a healthy person gets sick, this diminishes the sub-population of healthy people. Thus, we formulate an equation like dx = kxy, dt where the negative sign indicates that the healthy population decreases as more and more people get sick. Remember, we have imagined for a minute that the death rate is zero. That means that the number of people, x + y , is constant, hence dy dx d (x + y ) = 0 = = = +kxy. dt dt dt We now have two equations: dx = kxy, dt dy = +kxy. dt

Now, let us restore the non-zero death rate. We have assumed that people die at a constant rate, . The number of people dying per unit time is thus y . This will serve to diminish the y -population, without aecting the x population directly. We therefore modify the y -equation: dy = kxy y. dt Finally, conservation of people now requires that dx dz dx dy (x + y + z ) = 0 = = = (kxy ) (kxy y ) = + y. dt dt dt dt Let us gather up our model equations:

dx = kxy, dt dy = kxy y, dt dz = y. dt

## (3a) (3b) (3c)

Note that this system of equations can be connected with the formalism in Section I (where we had the general equation du/dt = F (u) by identifying u = (x, y, z )T , F (u) = (kxy, kxy y, y )T .

## Clearly, this is an autonomous system of equations.

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III. REDUCTION TO A SINGLE EQUATION

## Observation 1: Consider the x- and z -equations alone: dx = kxy, dt dz = y. dt

In the z -equation, observe that y = (dz/dt)/ . Substitute this into the rst equation to obtain dx 1 dz = kx . dt dt This looks a little bit like a separation-of-variables problem! First, divide both sides by x: 1 dx k dz = . x dt dt Now formally cancel the dts on both sides to obtain dx k = dz. x This identity can now be integrated to yield k log x log x0 = (z z0 ), x0 = x(0), z0 = z (0).

However, we assume no dead people at the onset of the epidemic, so z0 = 0. Exponentiation gives x = x0 ekz/ . Here, however, we need to remind ourselves that x and z are in fact functions of time, so the correct form is x(t) = x0 ekz(t)/ . Observation 2: By assumption, the population is constant: d (x + y + z ) = 0 = x + y + z = N = Const. dt Thus, y = N x z. However, by Eq. (4), x = x0 ekz/ . Thus, y = N x0 ekz/ z. We substitute this into the z -equation dz/dt = y : (4)

dz = dt

N x0 ekz/ z ,

(5)

And this is a single equation in a single dependent variable. This equation can therefore be solved on a computer, or studied using the qualitative methods.
IV. RESCALING

In this section, we rescale the equation (5) to reduce the number of parameters and to make it more amenable to analysis. First, we introduce a variable u= k z.

Because k and both have the same physical units (rate constants), and because z has the dimensions of a pure number (population), u also has dimensions of a pure number. Next, multiply Eq. (5) by k/ : dk k z= N x0 eu z . dt In other words, du = kN kx0 eu kz. dt Divide both sides by : 1 du kN kx0 u k = e z, dt or 1 du kN kx0 u = e u. dt We are going to do one more division that may seem unnecessary but will help in the xed-point analysis: we divide both sides by kx0 / and obtain
kx0

1 1 du 1 kN 1 = kx0 eu kx0 u. dt

## Lets tidy up:

1 du N = eu u. kx0 dt x0 kx0 Now, we use the old trick and introduce a new time variable d = kx0 dt. Hence, the nal, rescaled equation is du = a bu eu , d where a= N , x0 b= kx0 . (6a) (6b)

## Note that a = N/x0 = (x0 + y0 )/x0 1, while b = /kx0 > 0.

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V. FIXED-POINT ANALYSIS

In this section we analyse the xed points of Eq. (6). Basically, we must plot the graph of f (u) = a bu eu , a 1, b > 0, u 0.

To sketch such a curve, it suces to know the behaviour as u 0, u , and to know if the curve has maxima and minima. Behaviour at zero: We have, f (0) = a 1. Since a 1, f (0) 0. Let us consider the case where a > 1. Thus, y0 > 0, and there are some sick people present initially. Then f (0) > 0. Behaviour for large u-values: As u , the exponential term dies away, and we are left with f (u) bu, u . In other words, f (u) becomes large and negative and behaves like a straight line, as u . Maxima and minima: We compute f (u0 ) = 0 and nd b + eu0 = 0. In other words, u0 = log b. For b < 1, this point exists in the range u > 0. The second derivative is f (u) = eu , and this is always negative. Hence, the point u0 = log b, if it exists, is a maximum. Crossing the u-axis: We have: limu0 f (u) > 0, limu f (u) < 0, f (u) at most has one local maximum on u > 0. Therefore, f (u) has precisely one root on u > 0 one xed point. Examination of the one-dimensional vector eld shows that this is a stable xed point Fig. 1.
VI. THE PEAK OF THE EPIDEMIC

The peak of the epidemic occurs when the number of people getting sick is at a maximum: dy = 0 = The number of sick people is at a maximum. dt Not that dz/dt = y = d2 z/dt2 = (dy/dt), so that, at peak, d2 z = 0 at peak. dt2

1
b<1

0 f(u)

1
b>1

3 0

2 u

## Figure 1: The function f (u) for the two representative b-values.

In other words, at peak, the number of deaths stops accelerating. Since z u, d2 u = 0 at peak. dt2 d2 u d du d du = = f (u) = f (u) = f (u)f (u). 2 dt dt dt dt dt Thus, peak corresponds to f (u) = 0 of f (u) = 0. We do not call f (u) = 0 the peak of the epidemic, because this is a xed point. Therefore, f (u) = 0 at peak. However, by Fig. 1, this condition is realised only for b < 1. Therefore, the epidemic peaks only when b < 1. In other words, Starting with an initial condition u0 = 0 (not dead people at t = 0), the number of dead people accelerates, with d2 u/dt2 = f (u)f (u) > 0. At peak, the acceleration stops, f (u) = 0. After peak, the number of dead people decelerates, with d2 u/dt2 < 0. When b > 1, d2 u/dt2 = f (u)f (u) < 0 always. That means that the number of dead people decelerates from the start. The parameter-value b = 1 is therefore called the threshold value: If b < 1 the epidemic accelerates before slowing down, while if b > 1, the epidemic decelerates from the start. Consider again b = /(kx0 ). For this discussion, take x0 to be xed. Physically, b 1 means that k . In other words, the rate at which the disease spreads exceeds the death rate. Sick people, before they die, manage to infect lots of healthy people. On the other hand, b 1 means that k . The sick people die before they have time to infect people. The deadlier the disease, the slower it spreads! Note:

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VII. VALIDITY OF MODEL

Kermack and McKendrick applied their model to data from the Bombay Plague of 1906. More complicated models (albeit ones that share similarities with this one) have been used to model the spread of AIDS.

[1] Nonlinear Dynamics and Chaos, S. Strogatz, Westview Press (1994), Chapter 4, pp. 91-92. [2] Kermack, W. O. and McKendrick, A. G. Contributions to the mathematical theory of epidemics I. Proc. Roy. Soc. 115A, 700 (1927) [3] But do worry about it later, e.g. in the Maths contest.
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