Chapter 9 Continuous-Time Age-Structured Models …

1 Chapter 9. Continuous-Time Age-Structured Models in population dynamics and epidemiology Jia Li and Fred Brauer Abstract We present Continuous-Time Models for Age-Structured populations and disease transmission. We show how to use the method of character- istic lines to analyze the model dynamics and to write an Age-Structured population model as an integral equation model . We then extend to an age- structured SIR epidemic model . As an example we describe an Age-Structured model for AIDS, derive a formula for the reproductive number of infection, and show how important a role pair-formation plays in the modeling process. In particular, we outline the semi-group method used in an Age-Structured AIDS model with non-random mixing. We also discuss Models for populations and disease spread with discrete age structure. Why Age-Structured Models ? In the simplest Models for a single population all members are assumed to be interchangeable. However, even the simplest Models for disease transmission include structuring the population by disease state (susceptible, exposed, infective, or removed).

2 More advanced population Models add some structure to the population such as speci cation of spatial location or age. Age is one of the most impor- tant characteristics in the modeling of populations and infectious diseases. Individuals with di erent ages may have di erent reproduction and survival capacities. Diseases may have di erent infection rates and mortality rates for di erent age groups [1]. Department of Mathematical Sciences, University of Alabama in Huntsville, 301, Sparkman Dr., Huntsville, AL 35899, USA Department of Mathematics, University of British Columbia, 1984, Mathematics Road, Vancouver BC, Canada V6T 1Z2 205. 206 J. Li and F. Brauer Individuals of di erent ages may also have di erent behaviours, and be- havioural changes are crucial in control and prevention of many infectious diseases. Young individuals tend to be more active in interactions with or between populations, and in disease transmissions. Sexually-transmitted diseases (STDs) are spread through partner interac- tions with pair-formations, and the pair-formations are clearly age-dependent in most cases.

3 For example, most AIDS cases occur in the group of young adults. Childhood diseases, such as measles, chicken pox, and rubella, are spread mainly by contacts between children of similar ages. More than half of the deaths attributed to malaria are in children under ve years of age due to their weaker immune systems. This suggests that in Models for disease trans- mission in an age structured population it is necessary to allow the contact rates between two members of the population to depend on the ages of both members. In order to describe Age-Structured Models for disease transmission we must rst develop the theory of Age-Structured populations. In fact, the rst Models for Age-Structured populations [34] were designed for the study of disease transmission in such populations. Modeling Populations with Age Structure Let (t, a) be the age-density function at time t with a [0, a+ ], where a+ < is the maximum age of individuals, or with a [0, ) for convenience a2. of mathematical description.]

4 Then (t, a)da is the number of individuals a1.. having ages in the interval [a1 , a2 ] at time t, and (t, a)da = P (t) is the 0. total population size at t. Let be the age speci c fertility rate, or birth a2. rate, so that (t, a)da is the number of o spring produced by individuals a1.. with ages in [a1 , a2 ] in unit time at time t. Then (t, a)da = B(t) is the 0. total number of newborns, at time t. The age speci c fertility may depend on the population density so that = (a, (t, a)), or may depend on the total population so that = (a, P ). The reader should note that here . is not related to the contact rate for disease transmission in compartmental Models introduced in earlier chapters. Here we assume the fertility to be time-independent. Let be the age speci c mortality, or death rate, so that . (t, a)da is the total number of deaths at time t, occurring in one unit 0. time. Similarly, the age speci c mortality may depend on the population 9 Age-Structured Models 207. density so that = (a, (t, a)), or may depend on the total population size so that = (a, P ).

5 Again we assume the mortality to be time-independent. In this Chapter we consider the case in which both the fertility and mortality depend on the total population size rather than on the age-speci c population density. Suppose that the population changes from time t to t + h, with h > 0. t+h The number of newborns in the time interval [t, t + h] is t B(s)ds =. t+h . ( , P ) (s, )d ds. Note that the number of individuals who die t 0 a+s at time t + s, having age less than or equal to a + s, is 0 ( , P ) (t +. s, )d . Then the total number of deaths in the time interval [t, t + h] is h a+s ( , P ) (t + s, )d ds. 0 0 a Let N (t, a) = 0 (t, )d be the number of individuals having ages less than or equal to a at time t, and assume that there is no migration. Then the change in the population size from time t to t + h is the total number of births minus the total number of deaths during the time interval [t, t + h], that is, t+h h a+s N (t + h, a + h) N (t, a) = B(s)ds ( , P ) (t + s, )d ds.

6 T 0 0. ( ). The instantaneous rate of change of the population size is a N (t + h, a + h) N (t, a). lim = Nt (t, a) + Na (t, a) = t (t, )d + (t, a). h 0 h 0. Dividing ( ) by h and then letting h 0 yields a a t (t, )d + (t, a) = B(t) ( , P ) (t, )d . ( ). 0 0. Setting a = 0 in ( ), we have (t, 0) = B(t). Di erentiating equation ( ) with respect to a, we have t (t, a) + a (t, a) = (a, P ) (t, a). ( ). Then we arrive at the following system of a rst order partial di erential equation with corresponding initial and boundary conditions: t (t, a) + a (t, a) = (a, P ) (t, a), (t, 0) = 0 (a, P ) (t, a)da = B(t), ( ). (0, a) = (a), where (a) is the initial age distribution. For continuity at (0, 0) it would be necessary to require that 208 J. Li and F. Brauer . (0) = (a, P ) (a)da, 0. but because it is possible to allow discontinuous solutions of ( ) this re- quirement is usually ignored. The partial di erential equation in ( ) is commonly called the Lotka . McKendrick equation [26, 42]. Solutions along Characteristic Lines Fix t0 and a0 and consider the functions (h) := (t0 + h, a0 + h) and.

7 (h) :=. (a0 + h, P (t0 + h)). This amounts to following the age cohort of members of the population with age a0 at time t0 . Then equation ( ) is equivalent to d .. + . (h) . = 0. ( ). dh Solving ( ) yields h (h) = (0)e 0.. ( )d . , ( ). that is, h (t0 + h, a0 + h) = (t0 , a0 )e 0.. ( )d .. ( ). For a > t, setting (t0 , a0 ) = (0, a t) and h = t, we have t t (t, a) = (0, a t)e 0.. ( )d . = (a t)e 0. (a t+ ,P ( ))d . , ( ). and for t > a, setting (t0 , a0 ) = (t a, 0) and h = a, we have a a (t, a) = (t a, 0)e 0.. ( )d . = B(t a)e 0. ( ,P (t a+ ))d . , ( ). [17, 38, 42]. Then, we obtain the following expressions for solutions along the lines of characteristics for system ( ): " t (a t)e 0 (a t+ ,P ( ))d , a > t, (t, a) = a ( ). B(t a)e 0 ( ,P (t a+ ))d , t > a. Thus we have obtained an expression for the population density function for all (t, a) by following each age cohort along a characteristic line. Notice, however, that the solutions in ( ) involve the total population size P which depends on (t, a).

8 9 Age-Structured Models 209. Equilibria and the Characteristic Equation One of the important properties in the study of population dynamics is the asymptotic behavior of the steady states or equilibria of the populations. For system ( ), a steady state, or an equilibrium distribution, (a), satis es the equations d (a). = (a, P ) (a), da ( ). (0) = 0 (a, P ) (a)da, .. P = 0 (a)da. Suppose that system ( ) has a solution (a). Then we can investigate the local stability of this steady state or equilibrium by linearization of system ( ) about (a) as follows.. Let y(t, a) = (t, a) (a), and write Y (t) = 0 y(t, a)da. Then substi- tution into ( ) yields yt + ya = t + a a = (a, Y + P ) (y + ) a , and . y(t, 0) = (t, 0) (0) = (a, Y + P ) (y + ) da (0), 0.. where P = 0. (a)da. For (t, a) near , we have, using ( ), yt + ya (a, P )y (a, P ) P (a, P )Y a ( ). = (a, P )y P (a, P )Y, and . y(t, 0) 0 ( (a, P )y + (a) P (a, P )Y ) da . = 0 (a, P )y(t, a)da + 0 (a) P (a, P )daY (t) ( ).. = 0 K(a, , P )y(t, a)da, where.

9 K(a, , P ) = (a, P ) + ( ) P ( , P )d . ( ). 0. Hence, for (t, a) near , we arrive at the linearized equation . yt + ya = (a, P )y P (a, P ) y(t, a)da, ( ). 0. with the linearized integral boundary condition . y(t, 0) = K(a, , P )y(t, a)da. ( ). 0. 210 J. Li and F. Brauer . Suppose further that y(t, a) = u(a)e (t a) , and write w = 0 u(a)e a da. By substituting them into ( ) and ( ), respectively, we have du(a) . = (a, P )u(a) P (a, P )e a 0 u(a)e a da ( ). da = (a, P )u(a) P (a, P )e a w, and . u(0) = K(a, , P )u(a)e a da. ( ). 0. Solving ( ), we have a ( ,P )d . u(a) = e 0 (u(0) E( , a) w) , ( ). where a s ( ,P )d + s) . E( , a) = e( 0 P (s, P )ds. 0. Then substituting ( ) into ( ) and w, we obtain the following linear system a ( ,P )d + a).. u(0) = 0 Ke (. a 0 da u(0) . 0. Ke . a E( , a)da w, ( ( a ( ,P )d + a). w= 0 e 0 ( ,P )d + a). da u(0) 0 e 0 E( , a)da w, or equivalently, the linear system a .. 1 0 Ke ( 0 ( ,P d + a) da u(0) + 0 Ke a E( , a)da w = 0, ( a ( ,P )d + a) . ( a ( ,P )d + a).)

10 0. e 0 da u(0) 1+ 0. e 0 E( , a)da w = 0, ( ). in the unknowns u(0) and w. Hence, there exists a non-zero solution (u(0), w). to system ( ) if and only if a . a .. 1 0 Ke ( 0 ( ,P )d + a) da 1 + 0 e ( 0 ( ,P )d + a) E( , a)da . + 0 Ke a E( , a)da 0 e ( + )a da = 0. ( ). Equation ( ) is an equation in . There exists a solution of the form y(t, a) = u(a)e (t a) of the linearization ( ) and ( ) if and only if there exists a solution to equation ( ). Equation ( ) is called the charac- teristic equation of system ( ) as in [9 11]. 9 Age-Structured Models 211. Age-Structured Integral Equations Models Integral equations have also been used for modeling of Age-Structured popu- lations. These integral equations can be derived from system ( ), or more speci cally from ( ).. a Write (a, P ) = e 0 ( ,P (t a+ ))d . Then it follows from ( ) that t . B(t) = 0. (a, P ) (t, a)da + (a, P ) (t, a)da t t (a, P ). = (a, P ) (a, P )B(t a)da + t (a, P ) (a t)da, 0 (a t, P ). ( ). and t . P (t) = 0. (t, a)da + t (t, a)da t (a, P ) ( ).