Section 8.1 Def. and Examp. Systems Key Terms: SIR Model of an epidemic o Nonlinear o Autonomous Vector functions o Derivative of vector functions Order of a DE system Planar systems Dimension of a system Reduction of higher order DEs to systems of first order DEs
We investigate systems of differential equations. Initially we discuss definitions and examples. Naturally we will be able to model many more applications using systems. Then we examine ways to visualize solutions and make geometrical interpretations of a system. A system of differential equations is a set of one or more equations, involving one or more unknown functions. In many applications there are several interrelated quantities changing with respect to an independent variable, often time. Modeling such an application leads to a system of differential equations. We start with a model. The SIR model of an epidemic We start by modeling an epidemic. Suppose we have a population of N individuals that is subject to a communicable disease. We will assume the following facts about the disease: The disease is of short duration and rarely fatal. The disease spreads through contact between individuals. Individuals who have recovered from the disease are immune. These features are present in measles, the mumps, and in the common cold.
We divide the population into three groups. The susceptible, S(t), are those individuals who have never had the disease. The infected, I (t), are those who are currently ill with the disease. The recovered, R(t), are those who have had the disease and are now immune. The total population is the sum of these three, N = S + I + R. We must compute the rate of change for each of these subpopulations. Since the disease is of short duration and rarely fatal, we may ignore deaths and births. This means that the total population N is a constant. It also implies that S(t), the number of susceptibles, changes only because some of them catch the disease and pass into the infected population. Since the disease spreads through contacts between susceptible and infected individuals, the rate of change is proportional to the number of contacts. Assuming that the two populations are randomly distributed over area, the number of contacts is proportional to the product SI of the two populations. Thus there is a positive constant a such that ds = -asi
The number of infected individuals, I(t), changes in two ways. First, it increases as susceptible individuals get sick. Assuming that there is a fairly standard time in which a recovery takes place, the rate of recovery is proportional to the number of infected. So there is a positive constant b such that the rate of recoveries is bi. Putting these together, we see that di = asi - bi The number of recovered individuals, R(t), increases as those who are infected are cured so dr = bi ds = -asi Thus we have a system of 3 DEs: di = asi - bi This called the SIR model. It is nonlinear (note the products of unknown functions S and I). It is autonomous. dr = bi A solution to the SIR model is a triple of functions S(t), I (t), and R(t), which satisfy the equations the three DES.
We cannot solve the system explicitly, but we can compute numerical solutions. To do this we need to select initial conditions and values for the rate constants a and b. Let S(0) = 4, I (0) = 0.1, and R(0) = 0 and a = b = 1. So we have IVP ds = -SI di = SI - I dr =I S(0) = 4 I (0) = 0.1 R(0) = 0 4 3.5 3 2.5 2 SIR EXAMPLE from text S(t) I(t) R(t) 1.5 A 4 th order Runge-Kutta Method was used with h = 0.5. 1 0.5 0 0 1 2 3 4 5 6 The number of infected individuals starts small, rises quickly, and then falls off. Interestingly, almost the entire population eventually gets the disease.
Although the SIR model as originally set up involves three equations and three unknowns, the last equation is not really needed. Remember that part of our model requires that the total population N = S + I + R is constant. If we add the equations we get dn/ = 0, which serves as verification. In addition, the first two equations form a planar, autonomous system. If we were to solve only these equations for S and I, we could compute the number of recovered individuals from the equation R = N S I. In this way, we get the system Sꞌ = asi, I ꞌ = asi bi, which is also referred to as the SIR model.
Vector Notation We will use vector notation when dealing with first-order systems of DEs. Often it easiest to relabel dependent variables using subscripted names. For example the SIR system can put into vector notation as follows. ds Let u 1 (t) = S(t), u 2 (t) = I(t), and u 3 (t) = R(t). Then system = -asi is equivalent to uꞌ 1 = au 1 u 2 uꞌ 2 = au 1 u 2 bu 2 uꞌ 3 = bu 2. We now introduce the vector-valued function and then the derivative of u is If we define vector function u' 1(t) -au1u 2 u'(t) = = u' 2(t) au1u 2 - bu2 u' 3(t) bu2 -au1u 2 f(u) = au u - bu bu2 1 2 2 di = asi - bi dr = bi u 1(t) St () u(t) = = It () u 2(t) u 3(t) Rt () Most software for approximating systems of DEs using this format of input. then the system is given by uꞌ = f(u).
IVP in vector formulation System u' 1(t) -au1u 2 u'(t) = u' 2(t) = f(u) = au1u 2 - bu2 u' 3(t) bu2 requires 3 initial condition at initial time t 0, S(t 0 ) = S 0, I(t 0 ) = I 0, and R(t 0 ) = R 0 then the IVP is expressed as uꞌ = f(u) u(t 0 ) = u 0 = (S 0 I 0, R 0 ) T Terminology The SIR model involves only first-order derivatives of the unknown functions S, I, and R. For that reason it is called a first-order system. The order of a system of differential equations is the highest derivative that occurs in the system The general first-order system of two equations has the form xꞌ = f (t, x, y), yꞌ = g(t, x, y), where f and g are functions of the three variables t, x, and y. A solution to the system is a pair of functions x(t) and y(t) that satisfies xꞌ(t) = f (t, x(t), y(t)), yꞌ(t) = g(t, x(t), y(t)), for t in some interval.
Example: Show that the pair x 1 (t) = -t and x 2 (t) = -1 form a solution of the system x' = 1, x = 1 ==> x' = x 1 2 1 2 x' 1 = x2 x' = -x x - x 2 1 2 1 Just substitute in each side and show the results are equal. x' = 0, -x x -x = ( t)( 1) ( t) = t + t = 0, ==> x' =-x x -x 2 1 2 1 2 1 2 1 General systems of DEs; Systems of n equations with n unknowns. If the unknown functions are x 1 (t), x 2 (t),..., and x n (t), then the system has the form xꞌ 1 = f 1 (t, x 1, x 2,..., x n ), xꞌ 2 = f 2 (t, x 1, x 2,..., x n ),... xꞌ n = f n (t, x 1, x 2,..., x n ), We will always require that the number of equations is equal to the number of unknowns, and this number is called the dimension of the system. A system of dimension 2 is called a planar system. The SIR model in the form Sꞌ = asi, I ꞌ = asi bi is a planar system and so is x' 1 = x2 x' = -x x - x 2 1 2 1
Reduction of higher-order equations and systems to first-order systems An application involving a higher-order equation (that is, a DE with second derivative or higher) has an equivalent model using a first-order system. Most numerical solvers are written to solve first-order systems. To solve a higher-order equation, it is necessary to use the equivalent first-order system. Example: Find a first-order system equivalent to the third-order, nonlinear equation xꞌꞌꞌ + xxꞌꞌ = cos t. The idea is to introduce new dependent variables for the unknown function and each derivative up to one less than the order of the equation. Hence, we introduce u 1 = x, u 2 = xꞌ, and u 3 = xꞌꞌ. Then differentiate each of the new variables. Finally solve the original DE for the highest derivative, and use the definitions of the new variables. We get the first-order system uꞌ 1 = u 2 uꞌ 2 = u 3 uꞌ 3 =xꞌꞌꞌ = u 1 u 3 + cos t.