THE MATHEMATICAL MODELING OF EPIDEMICS. by Mimmo Iannelli Mathematics Department University of Trento. Lecture 4: Epidemics and demography.

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1 THE MATHEMATICAL MODELING OF EPIDEMICS by Mimmo Iannelli Mathematics Department University of Trento Lecture 4: Epidemics and demography.

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3 THE MATHEMATICAL MODELING OF EPIDEMICS Lecture 4: Epidemics and demography. Praeterea iam pastor et armentarius omnis et robustus item curvi moderator aratri languebat, penitusque casa contrusa iacebant corpora paupertate et morbo dedita morti. Exanimis pueris super examinata parentum corpora nonnumquam posses retroque videre matribus et patribus natos super edere vitam. 1 Lucretius De Rerum Natura, Liber VI, In this lecture we will model the spreading of an epidemic, taking care of the age structure of the population, i. e. taking into account the chronological age of the individuals. The importance of such a step arises from the fact that for many diseases the rate of infection varies significantly with age. In fact, if we consider exanthematic diseases we see that the transmission mainly involves early ages, while for sexually transmitted diseases the principal mechanism of infection involves mature individuals. Moreover some diseases are to some extent transmitted from parents to newborns (vertical transmission) and also immunity is vertically transmitted and lasts up to some age. Thus we expect that the vital dynamics of the population and the infection mechanism, interact to produce non-trivial behaviours and, in any case, a more realistic description arises when considering the demographic structure. In the following sections we will first introduce a few concepts from demography and the we will extend the general models discussed in Lecture 1, also focusing on some special cases that can be mathematically treated by the methods of the previous chapters. Actually we will see that though chronological age is conceptually different from the age of the infection and the models arising within this setting present different features, nevertheless they can be treated by the same procedures and methods. 1 Moreover, by now the shepherd and every herdsman, and likewise the sturdy steersman of the curving plough, would fall drooping, and their bodies would lie thrust together in the recess of a hut, given over to death by poverty and disease. On lifeless children you might often have seen the lifeless bodies of parents, and again, children breathing out their life upon mothers and fathers. 1

4 1 An excursus into Demography Among all population models, the simplest one is entitled from T. R. Malthus who wrote a famous treatise ([11]) on the growth of the human population, predicting that it would be exponential in time with all the catastrophic consequence that one can imagine. To introduce this model we consider a single homogeneous population; that is, we assume that all individuals of the population are identical so that the only variable that we have to deal with is the number of the individuals as a function of time P (t) (total population size). In addition we suppose that the population lives isolated, in an invariant habitat with no limit to resources. Thus the population is subject to constant fertility and mortality rates that we respectively call β and µ (their difference α = β µ is usually called the Malthusian parameter of the population) and the growth is governed by the following equation Thus d P (t) = βp (t) µp (t) = αp (t). dt P (t) = P ()e αt. Though the Malthus model is often used in simple discussion on population growth, all demographic thinking is based on age structure. In fact, age is one of the most natural and important parameters structuring a population, since many internal variables, at the level of the single individual, are strictly depending on it, then different ages mean different reproduction and survival capacities and, also, different behaviors. Within such a context, the evolution of the population is described by its age density function at time t: p(a, t) a [, a ], t, where a denotes the maximum age which we assume to be finite. Thus the integral: a2 a 1 p(a, t)da gives the number of individuals that, at time t, have age in the interval [a 1, a 2 ]; and P (t) = p(a, t)da is the total population at time t. Concerning fertility and mortality we first introduce: β(a) age specific fertility, which can be defined as the number of newborn, in one time unit, coming from a single individual whose age is in the infinitesimal age interval [a, a+da]. Thus a2 a 1 β(a)p(a, t)da 2

5 gives the number of newborn in one time unit, coming from individuals with age in [a 1, a 2 ]. We also consider the total birth rate B(t) = β(a)p(a, t)da which gives the total number of newborn in one time unit. We also introduce µ(a) age specific mortality. It is the death rate of people having age in [a, a + da]; then the total death rate is: D(t) = µ(a)p(a, t)da and gives the total number of deaths occurring in one time unit. Figure 1: A typical curve for fertility The functions β( ) and µ( ) are, of course, non-negative: they are also called vital rates and are viewed as deterministic rates; in practice they are determined on a statistical basis. In Figures 1 and 2 we show some classical examples of these functions, drawn from demography. Other meaningful quantities are derived from β( ) and µ( ); namely: Π(a) = e R a µ(σ)dσ, a [, a ] denotes the survival probability, i.e. the probability for an individual to survive to age a; thus it must be Π(a ) = ; moreover the function K(a) = β(a)π(a), a [, a ] is called the maternity function and it synthesizes the dynamics of the population; it is related to the parameter R = β(a)π(a) da, (1) 3

6 which is called the net reproduction rate and gives the number of newborn that an individual is expected to produce during his reproductive life. We will see Figure 2: A typical curve for mortality that this parameter will play a role in the discussion of the asymptotic behavior of the population; in fact we expect the population to show an increasing trend when R > 1, decreasing if R < 1, stable when R = 1. The basic system describing the evolution of an age structured population is the so called Lotka-McKendrick problem i) p t (a, t) + p a (a, t) + µ(a)p(a, t) = a ii) p(, t) = β(σ)p(σ, t)dσ (2) iii) p(a, ) = p (a). The first of these equations describes the aging of the population and the output due to deaths, the second provides the way newborns are produced and enter the population at age a =. The problem above is equivalent to the linear integral convolution equation of Volterra type on the birth rate B(t) with: F (t) = t B(t) = F (t) + t Π(a) β(a) Π(a t) p (a t)da = K(t s)b(s)ds (3) K(t) = β(t)π(t), β(a + t) Π(a + t) p (a)da, Π(a) where t, and the functions β, Π, p are extended by zero outside the interval [, a ]. In fact, the solution of (2) is related to that of (3) by the formula Π(a) p p(a, t) = (a t) if a t, Π(a t) (4) B(t a) Π(a) if a < t. 4

7 The analysis of the asymptotic behaviour of (2) which is usually performed through the equivalent renewal equation (3) leads to the so called theory of the stable distribution. Namely it can be proved that p(a, t) b e α t e α a Π(a) (5) as t, where α is the (unique) real and leading solution of the characteristic equation K(λ) = 1. Note that the net reproduction rate defined in (1) is equal to K(), so that α is positive if and only if R > 1. In fact R is the number of newborns produced by one individual during his lifespan. The meaning of (5) is that, as time goes on, the population as a whole may increase or decrease according to the sign of α, but the age profile attains a well defined form (see Figure 3) e α a Π(a). Figure 3: Age profile Italy 22 2 Epidemics through an age structured population We consider a population that, in the absence of the epidemic that we are going to consider, can be described by the linear model discussed in the previous section, i.e. we consider a population which is isolated, in an invariant habitat, structured by age, with vital rates β(a) and µ(a). Because of the epidemics, the population is partitioned into the three classes of susceptibles, infectives and removed which are described by their respective age-densities s(a, t), i(a, t), r(a, t), at time t. Thus the age-density p(a, t) of the whole population must satisfy p(a, t) = s(a, t) + i(a, t) + r(a, t). 5

8 Denoting by γ(a), δ(a), λ(a, t) the age specific removal rate, cure rate and infection rate respectively, we have the following equations describing the transmission dynamics of the disease: s t (a, t) + s a (a, t) + µ(a)s(a, t) = λ(a, t)s(a, t) + δ(a)i(a, t) i t (a, t) + i a (a, t) + µ(a)i(a, t) = λ(a, t)s(a, t) (γ(a) + δ(a))i(a, t) r t (a, t) + r a (a, t) + µ(a)r(a, t) = γ(a)i(a, t). with the following renewal conditions s(, t) = b 1 (t), i(, t) = b 2 (t), r(, t) = b 3 (t). (7) Actually, each class undergoes the same demographic evolution determined by the vital rates β(a) and µ(a), while the passage from a class to another is ruled by the rates γ(a), δ(a), λ(a, t) (see Figure 4). (6) Figure 4: A sketch of the general age-structured epidemic model Together with system (6), we must consider the initial conditions s(a, ) = s (a), i(a, ) = i (a), r(a, ) = r (a) and constitutive equations for the birth rates b 1 (t), b 2 (t), b 3 (t). Concerning the latter we assume b 1 (t) = β(a) [s(a, t) + (1 q)i(a, t) + (1 w)r(a, t)] da b 2 (t) = q β(a)i(a, t)da (8) a b 3 (t) = w β(a)r(a, t)da, where q [, 1], w [, 1], are the vertical transmission parameters of infectiveness and immunity, respectively. These parameters indicate the fraction of 6

9 newborns who is born in the class of their parents; thus, if q = w =, all newborn are susceptible. In this model we assume that the intrinsic fertility β(a) and mortality µ(a) are not (significantly) affected by the disease, so we expect that the total population undergoes the same demographic process of the model of the previous section. In fact, if we add the equations in (6) and (8), we obtain the following problem for p(a, t) p t (a, t) + p a (a, t) + µ(a)p(a, t) = a p(, t) = β(a)p(a, t)da p(a, ) = p (a) = s (a) + i (a) + r (a), that is, we obtain problem (2). In this respect we make the following hypothesis on the demography of the population R = β(a)π(a)da = 1, i.e. we assume that the population is at zero growth (α = ) and, consequently, there exists a unique stationary solution Moreover, we suppose that p (a) = P ω (a) = b Π(a). p(a, t) = p (a) = p (a); i.e. we suppose that the population has reached the steady-state distribution p (a) and the total population is constant N = p (a)da Finally, we must give a constitutive form to the infection rate λ(a, t); a general linear form for it is given by λ(a, t) = K (a)i(a, t) + K(a, a )i(a, t)da, where the two terms on the right hand side are called the intracohort term and intercohort term, respectively. The following special cases λ(a, t) = K (a)i(a, t), (9) λ(a, t) = K(a) i(a, t)da, (1) correspond to two extreme mechanisms of contagion; in fact (9) represents the situation in which individuals can be infected only by individuals of their own age, while in (1) they can be infected by those of any age. A more structured form of the intercohort term, related to the mechanism of encounters, is given by the so called proportionate mixing form K(a, a ) = 7 c(a)c(a )χ c(a)p (a)da (11)

10 where c(a) is the rate of contacts of an individual of age a and χ is the infectiveness of a contact. If moreover the contact rate is independent of a then we have the homogeneous mixing form K(a, a ) = cχ N (12) and λ(a, t) = cχ N i(a, t)da A substantial reduction of the problem occurs when we consider the S-I-S epidemic. In fact, when we specialize problem (6)-(8) assuming γ(a), r (a), we have s t (a, t) + s a (a, t) + µ(a)s(a, t) = λ(a, t)s(a, t) + δ(a)i(a, t) i t (a, t) + i a (a, t) + µ(a)i(a, t) = λ(a, t)s(a, t) δ(a)i(a, t) a s(, t) = β(a)[s(a, t) + (1 q)i(a, t)]da, s(a, ) = s (a) (13) a i(, t) = q β(a)i(a, t)da, i(a, ) = i (a). and, since s(a, t) + i(a, t) = p (a) (14) we can set s(a, t) = p (a) i(a, t) in the second equation of (13) getting the following problem in the single variable i(a, t) i t (a, t) + i a (a, t) + µ(a)i(a, t) = λ(a, t)[p (a) i(a, t)] δ(a)i(a, t) a i(, t) = q β(a)i(a, t)da (15) i(a, ) = i (a), and we can limit ourselves to the study of this system. Another reduction concerns the S-I-R case, which corresponds to the assumptions δ(a) and w = 1; in this case we have the following system s t (a, t) + s a (a, t) + µ(a)s(a, t) = λ(a, t)s(a, t) i t (a, t) + i a (a, t) + µ(a)i(a, t) = λ(a, t)s(a, t) γ(a)i(a, t) a s(, t) = β(a) [s(a, t) + (1 q)i(a, t)] da i(, t) = q β(a)i(a, t)da s(a, ) = s (a) i(a, ) = i (a). (16) Actually, we can disregard the third equation in (6) because the first two are enough to determine the evolution of the two classes of susceptibles and infectives; however, because of the presence of the class of removed individuals, (14) is not true and we cannot further reduce the system. 3 The S-I-S intracohort model Here we consider the S-I-S case (15) and discuss existence of endemic states, i.e. non-trivial stationary states of the problem. 8

11 First we investigate (15) assuming the purely intracohort form (IV6) for the infection rate. Under this assumption (15) becomes i t (a, t) + i a (a, t) + µ(a)i(a, t) = i(, t) = q i(a, ) = i (a), = K (a)[p (a) i(a, t)]i(a, t) δ(a)i(a, t) β(a)i(a, t)da (17) and a stationary state i (a) must satisfy d da i (a) + µ(a)i (a) = K (a)[p (a) i (a)]i (a) δ(a)i (a) i () = q β(a)i (a)da. (18) We first note that (18) admits the trivial solution i (a) and that, if q = (i.e. when the disease is not vertically transmitted), this is the only solution. Then, letting q > and setting i () = v >, we see that the first equation in (18) yields: i v E(a) (a) = a, (19) 1 + v K (σ)e(σ)dσ where we have set: E(a) = e R a [µ(σ)+δ(σ) K(σ)p (σ)]dσ. (2) Plugging (19) into the second equation in (18), we get the following equation for v : β(a)e(a) 1 = q 1 + v a K da. (21) (σ)e(σ)dσ Of course, solving this equation is equivalent to solving (18), via formula (19). We note that the right hand side of (21) is a decreasing function of v, unless the following condition is satisfied: β(a) a K (σ)dσ = a.e. for a [, a ]. (22) This condition means that the fertility window lies below the infectiveness one and since the disease is sustained intracohort and by vertical transmission this is a critical case that we exclude. Then we state the following theorem which gives a threshold condition for the existence of endemic states. Theorem 1 Let q > and assume that (22) is not true, then (18) has one non-trivial solution if and only if q and, if such a solution exists, it is unique. β(a)e(a)da > 1 (23) 9

12 Proof: If (22) is not true, then the function: Φ(x) = q is strictly decreasing and: where: lim Φ(x) x + = q a a β(a)e(a) 1 + x a K da, x [, + ], (σ)e(σ)dσ a β(a)e(a)da = q β(a)e R a µ(σ)dσ < a a = sup{a K = a.e. in [, a]}. β(a)e R a [µ(σ)+δ(σ)]dσ da β(a)e R a µ(σ)dσ da = 1 Then there exists v > satisfying (22) if and only if Φ() > 1, and this solution is unique. The threshold condition Φ() > 1 is exactly (23). We are now ready to investigate the asymptotic behaviour of the problem. The specific case allow to show a global result that is typical of all the S-I-S models with a more general force of infection. We start integrating the first equation in (17) along the characteristics t a = const. We obtain the following formula i (a t)e(a) E(a t) + i i(a, t) = (a t) t K if a t (a τ)e(a τ)dτ (24) i(, t a)e(a) 1 + i(, t a) a K if a < t, (τ)e(τ)dτ where E(a) is defined in (2). Formula (24) is the starting point for the analysis of the model. First we rule out the case q =, that is the case with no vertical transmission of the disease. In fact, with this condition we have i(, t) and, consequently, i(a, t) = for t > a ; thus the disease dies out. Next we consider q >. To treat this case we must transform the problem into a Volterra integral equation on the infectives birth rate v(t) = i(, t). In fact, putting (24) into the second equation of (17), we get a non-linear integral equation of the form v(t) = F (t) + t G(a, v(t a))da, (25) where (extending all the functions by zero outside of [, a ]), F (t) = q β(a + t)e(a + t)i (a) E(a) + i (a) a+t da, t, K a (τ)e(τ)dτ 1

13 q β(a)e(a)z G(a, z) = 1 + z a K (τ)e(τ)dτ, a z. We are now able to give a complete description of the asymptotic behaviour of v(t). We start with the following preliminary result: Proposition 1 Suppose and let i be such that β(a) > a.e. in [a 1, a 2 ] (26) β(a + t)i (a)da > for some t. (27) Then the solution of (25) is eventually positive. Proof: If (27) is fulfilled, then F (t) and, consequently, v(t) are not identically zero on [, a ]. Suppose that v(t) > for t [α, β] [, a ]. Then for t [α+a 1, β+a 2 ], v(t) = F (t) + t min t [α,β] v(t) G(t s, v(s))ds t α (t β) q β(a)e(a) 1 + v(t a) a K da >, (τ)e(τ)dτ because (a 1, a 2 ) ( (t β), t α). Thus, iterating this argument we get v(t) > for t [α + na 1, β + na 2 ] for any positive integer n and, since [α + na 1, β + na 2 ] [t, + ), for some t >, we have v(t) > for t > t. n Note that condition (27) means that the initial datum i has a support which, if translated to the right, hits the fertility window: if this condition is not satisfied then F (t) is identically zero and consequently also v(t) vanishes for t. Now we analyze the behaviour of v(t) under conditions (26), (27). This behaviour depends on the threshold condition (23). First we have Theorem 2 Let (26), (27) be satisfied and assume q β(a)e(a)da 1. Then lim v(t) =. t + 11

14 Proof: Let I n = [na, (n + 1)a ] for any integer n ; then define: M n = max t I n v(t), Mn = max{m n, M n 1 }. Note that, by the proof of Proposition 1, we have M n > for any n ; then, if t I n with n >, we have v(t) = G(s, v(t s))ds G(s, M n )ds = M n Φ( M n ), (28) where Φ(z) is the function defined in the proof of Theorem 1. In fact, since s [, a ], we have t s I n I n 1 and, for a [, a ], G(a, z) is a nondecreasing function of z. From (28) we get M n M n Φ( M n ) n > (29) and, since Φ(z) is strictly decreasing and Φ() 1, we have M n < M n Φ() M n ; that is, M n < M n 1. Thus the sequence {M n } is decreasing and, setting M = lim n M n, we have, passing to the limit in (29): M M Φ(M ). If M >, we would now have the contradiction 1 < Φ (), which is absurd because Φ () 1. So, necessarily M = and the proof is complete. Besides, we have Theorem 3 Let (26), (27) be satisfied and assume Then, q β(a)e(a)da > 1. lim v(t) = t + v. Proof: Let I n, M n, Mn be defined as before. We first prove the following statement: if M n v, then M n+1 v. (3) In fact, we recall the following inequality already stated in (29): M n+1 M n+1 Φ( M n+1 ), n. (31) 12

15 Then, if M n+1 > v, we have M n+1 = M n+1 and, consequently, which is absurd. Next we prove that, and M n+1 M n+1 Φ(M n+1 ) < M n+1 Φ(v ) = M n+1, if M n > v for n > N, then M n+1 < M n for n > N, (32) lim M n = v. n + In fact, if M n > v and consequently M n+1 > v, we have from (31) M n+1 M n+1 Φ( M n+1 ) < M n+1 Φ(v ) = M n+1. That is, M n+1 < M n. Then, letting M = lim n M n v, we pass to the limit in (3): M M Φ(M ), so that, if M > v, we have M < M, which is absurd and then necessarily M = v. In addition, we define m n = min t I n v(t), m n = min{m n, m n 1 }, and, noticing that the sequence {m n } must eventually be positive by Proposition 1, we can also prove (the proof is similar to that of (3) and (31)) that, if m n v, then m n+1 v ; if m n < v for n > N, then m n+1 > m n for n > N and lim n + = v. Finally, putting together all the previous statements we get the proof of the theorem. References [1] Anderson, R. and May, R.Vaccination against rubella and measles: quantitative investigations of different policiesj. Hyg. Camb. 9, (1983) [2] Busenberg, S. and Cooke, K. Vertically transmitted diseases, Models and DynamicsSpringer, Biomathematics V. 23 (1993) [3] Busenberg, S., Cooke, K. and Iannelli, M. Endemic thresholds and stability in a class of age-structured epidemicssiam J. Appl. Math. 48, (1988) [4] Busenberg, S., Iannelli, M. and Thieme, H. Global behaviour of an age-structured S-I-S epidemic modelsiam J. Appl. Math. 22, (1991) 13

16 [5] Coale, A.J. The growth and structure of human populations. A mathematical investigationprinceton University Press, Princeton, New Jersey, (1972) [6] Dietz, K. and Schenzle, D. Proportionate mixing models for age-dependent infection transmission J. Math. Biol. 22, (1985) [7] Greenhalgh, D. Threshold and stability results for an epidemic model with an age-structured meeting rate IMA J. Math. Appl. Med. Biol. 5, 81-1 (1988) [8] Iannelli, M., Milner, F.A., Pugliese, A.Analytical and numerical results for the age structured SIS epidemic model with mixed inter-intra-cohort transmission SIAM J. Math. Anal. 23, (1992) [9] Inaba, H.Threshold and stability results for an age-structured epidemic modelj. Math. Biol. 28, (199) [1] Lotka, A.J. The stability of the normal age distribution Proceedings of the National Academy of Sciences 8, (1922) [11] Malthus, T.R.An Essy on the Principle of Population. (First edition), London 1798 [12] Thieme, H. Stability change of the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases In Differential Equations Models in Biology, Epidemiology and Ecology, S. Busenberg and M. Martelli, Eds., Springer Lecture Notes in Biomathematics 92, (1991) [13] Tudor, D. An age dependent epidemic model with application to measles Math. Biosci. 73, (1985) 14