2 F. A. MILNER 1 A. PUGLIESE 2 rate, μ > the mortality rate of susceptible and infected individuals; μ R the mortality rate, presumably larger than μ,

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1 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL OF EPIDEMICS F. A. Milner 1 A. Pugliese 2 We describe and analyze a numerical method for an S-I-R type epidemic model. We prove that it is unconditionally convergent and that solutions it produces share many qualitative and quantitative properties of the solution of the differential problem being approximated. Finally, we establish explicit sufficient conditions for the unique endemic steady state of the system to be unstable and we use our numerical algorithm to approximate the solution in such cases and discover that it can be periodic, just as suggested by the instability of the endemic steady state. I. Introduction. Simple epidemic models for diseases of short duration by comparison with the mean life span of the individuals of the population involved usually consist of a system of ordinary differential equations based on the classical Kermack McKendrick model [9]. When infected individuals may transmit the disease for a significant fraction of their lifetime, the role of variable infectiousness in the dynamics of transmission becomes essential [2,4-6,16] and partial differential equations become necessary because of the introduction of another independent variable, the age of infection. Such is the case, for example, for HIV infection, where the duration of the infectedinfectious phase before developing AIDS may be in the range of 6 14 years [3], that is, up to one fifth of a typical mean human lifespan of 7 years. A model frequently used [4,7,15,16] for HIV/AIDS is given by the following integro differential initial boundary value problem: (1.1) 8 >< >: S =Λ μs (t)s; S() = S @ = μi fl( )i; i( ; ) = i ( ) ; i(;t)= (t)s (t) = C(S + I) S + I R = μ R R + ffi( )i( ; t)d fl( )i( ; t)d ; R() = R : Here S(t), I(t) = R 1 i( ; t)d, and R(t) indicate, respectively, the total number at time t of susceptible, infected, and removed (AIDS) individuals; Λ indicates the recruitment 1 Department of Mathematics, Purdue University, West Lafayette, IN Department of Mathematics, University oftrento, 385 Povo (TN), Italy. The work of this author

2 2 F. A. MILNER 1 A. PUGLIESE 2 rate, μ > the mortality rate of susceptible and infected individuals; μ R the mortality rate, presumably larger than μ, of AIDS individuals; fl( ), ffi( ), and i( ; t) are, respectively, the removal rate from the class of infected, the infectiousness, and the density of infected individuals with respect to the time elapsed since infection. Finally, C(S + I) denotes the mean contact rate, that is, the average number of contacts an active individual has per unit of time (active means susceptible or infected); we will generally assume C to be a non-decreasing function (see [15] for more specific assumptions and examples). We assume i is integrable, which makes perfect sense biologically. This model has been analyzed by several authors [4,5] and used for AIDS modeling in several papers [7,16]. The occurrence of the same mortality rate in susceptible and infected individuals is for technical convenience in the theoretical analysis of the model. On the other hand, in the application to HIV/AIDS it seems likely that there is an increased mortality among HIV + individuals by comparison with HIV ones. If one were to assume the same mortality for all epidemic classes (μ R = μ), total population size would have a very simple learning curve" behavior: P + μp =Λ; P () = P S + ki k L 1 + R ; which implies that P (t) = Λ μ + P Λ e μt : μ This is a smooth monotone function with the following obvious properties: ρ ff (1.2)» P (t)» max P ; Λ ; μ (1.3) lim t!+1 P (t) = Λ μ : This will be a bound for solutions if μ R μ. Moreover, since clearly O» S(t);I(t);R(t)» P (t); the only interesting case for diseases of long duration is the one where Λ > since, otherwise, (1.4) lim t!+1 S(t) = lim I(t) = t!+1 lim R(t) =: t!+1 This just says that when the whole population dies out, so do all its subclasses. Analogously, toavoid trivial situations we assume i 6. The recruitment rate incorporates in this model both births and immigration and, for the latter, the questionable assumption is made that immigrants are all uninfected. For the numerical algorithm, it is not a problem to have three different recruitment rates into the three epidemic subclasses. We indicate only one for the sake of simplicity. However, the dynamical system defined by (1.1) becomes quite a bit more difficult to analyze in such a case and, in fact, this analysis has not been done. Moreover, a recruitment term in the partial differential equation for the infected would be impossible to model since it would require that we knew at each time the distribution of infected individuals by age of

3 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 3 The numerical computation of solutions of epidemic models has been addressed by several authors [8,11,14]. However, in none of these works was the driving idea a numerical method which would remain accurate for very long time simulations, or one that would preserve most of the qualitative or quantitative properties of the solution of the differential problem. This is precisely the driving force behind the algorithm we propose here. Many epidemics appear more or less cyclically in a population and, consequently, within the analysis of mathematical models of epidemics a top concern has been the existence of periodic solutions. For non autonomous systems with periodic coefficients and/or periodic source terms, periodic solutions have been proved to exist [1]. For the autonomous system (1.1) the closest result to a proof of existence of periodic solutions is found in [15,16], where it is shown that in certain cases a unique non trivial equilibrium can be unstable, when the threshold condition that guarantees its existence and the instability of the trivial equilibrium holds. This strongly suggests that periodic solutions would exist in such cases, and it can be seen numerically that this is indeed the case. In the existing numerical methods attempts were made to develop high order algorithms and some can perform superbly in test problems [1,14]. However, with real life data, the necessary compatibility for high order convergence is seldom valid and, given the sensitivity of such algorithms to the regularity of the solution, they usually yield an effective convergence of first order as soon as the compatibility conditions are violated. This fact suggests that, in most cases, a first order method should be the most adequate. In this paper we shall present a new first order algorithm, which is unconditionally convergent and preserves many useful qualitative and quantitative properties of the solution of the differential problem. The plan of the paper is as follows. In the next section we describe the numerical method we propose and prove that it converges unconditionally at first order rate. In x3 we compute the steady states of the discrete model and prove a threshold phenomenon very similar to the one known for the continuous model. In x4 we elaborate on the theorem of [15] about unstable non trivial equilibria and obtain explicit sufficient conditions and we present the results of simulations performed using such parameters that guarantee a unique non trivial and unstable steady state. The relevant parameters and functions are chosen to reflect realistic conditions for the HIV/AIDS epidemic in a small population of intra-venous drug users. The only exception is that the infectivity function ffi will be chosen to have just an initial peak and then vanish forever. Finally, in x5 we make some concluding remarks. II. The Numerical Algorithm and Its Convergence. We discretize system (1.1) using linearized implicit backward Euler finite differences for the ordinary equations, and the finite difference method of characteristics for the partial differential equation. Specifically, we let T be the final time of the simulation and N 2 N be the number of steps used to arrive to T. Then, the discretization parameter is defined as t = T=N. We shall denote by S n, i n j, In, R n, and n, j; n, respectively the approximations of S(n t), i(j t; n t), I(n t), R(n t), and (n t). For the coefficients

4 4 F. A. MILNER 1 A. PUGLIESE 2 our method is given by (2.1) 8 >< >: S n S n 1 =Λ (μ + n 1 )S n ; t n 1; i n j in 1 j 1 = (μ + fl j )i n j ; t j; n 1; i n n 1 S n 1 = (1 + μ t)[1+ t(μ + n 1 )] ; n 1; I n = t n = C(Sn + I n ) S n + I n R n R n 1 t initialized with the obvious choice i n j ; n ; t ffi j i n j ; n ; = μ R R n + t fl j i n j ; n 1; (2.2) S = S ; i j = i (j t) for j ; and R = R : The equations for S n, i n j and Rn come from backward finite differences; those for I n and n from simple quadrature formulae. On the other hand, the equation for i n does not come from a standard discretization of the differential equation, but from a conservation principle. Specifically, we require that when Λ = fl j =, the active population N n = S n + I n satisfy the implicit backward Euler finite difference equation N n N n 1 t = μn n : We should also point out that it is really unnecessary to compute R n because the equations for S n and i n j do not depend on it. However, we do use the numerical approximation of the number of removeds in order to have a way of comparing data from simulations with field data, since in HIV modeling only the AIDS cases, R(t), are known with some certainty. Note that (2.1) can be solved explicitly in the following form: (2.3) S n = for the susceptibles; Λ t + S n 1 1+ t(μ + n 1 ) ; n 1; (2.4) i n j = 1 1+ t(μ + fl j ) in 1 j 1 ; j; n 1; for the density of infecteds; (2.5) R n = 1» R n 1 +( t) 2 1 X flj i n j ; n 1;

5 » (Sn 1 + I n 1 )(1 + n 1 t)+λ t : PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 5 for the removeds. We shall start by showing that the discrete dynamical system (2.1) (2.2) shares several important qualitative properties with the continuous one (1.1). Let us introduce now for j, (2.6) ß j = jy k= t(μ + fl k ) ; the discrete probability for a newly infected to remain infected after j units of time. Lemma 2.1. For j; n we have (2.7) S n ;i n j ;I n ;R n ; n ; and (2.8) <S n + I n» max ρ ff S + I ; Λ : μ Also, for j n 1, (2.9) i n j = i j n ß j ß j n ; and, for n j 1, i n j = i n j ß j : Proof. First note that (2.7) follows trivially from (2.3) (2.5) together with the fifth equation in (2.1). Furthermore, it is trivial to show by induction on n that, S n > if S > or Λ > ; and I n > if i 6 : This implies that < S n + I n for all n. As for the second inequality in (2.8), note that for n 1, (2.11) S n + I n =» Λ t + S n 1 1+ t(μ + n 1 ) + tin + Λ t + S n 1 1+ t(μ + n 1 ) + I n 1 1+ t(μ + fl j ) = (Λ t + Sn 1 )(1 + μ t)+ n 1 S n 1 t (1 + μ t)[1+ t(μ + n 1 )] = Sn 1 1+μ t + In 1 1+μ t + n 1 S n 1 t (1 + μ t)[1 + t(μ + n 1 )] + In 1 1+μ t Λ t 1+ t(μ + n 1 ) + In 1 1+μ t

6 6 F. A. MILNER 1 A. PUGLIESE 2 Then, it follows from (2.11) and an iterative argument that (2.12) Λ Λ μ Sn I n μ (1 + μ t + μ n 1 ) Λ t S n 1 I n 1 n 1 t(sn 1 + I n 1 ) 1+ t(μ + n 1 ) Λ = n 1 1+ t n 1 μ Sn 1 I 1+ t(μ + n 1 ) n 1 Λ Y μ S I 1+ t j 1+ t(μ + j ) : Let now " = n 1 Y 1+ t j 1+ t(μ + j ) : Then, <"<1 and hence (2.12) yields (2.8): a) if S + I» Λ μ, then Sn + I n» Λ μ (1 ")+(S + I )"» Λ μ ; b) if S + I > Λ μ, then Sn + I n» Λ μ (1 ")+(S + I )"» S + I. Finally, (2.9) and (2.1) follow by iterating (2.4) in itself. Remark 2.1. Note that the product appearing in (2.9) is a first order approximation of Thus, (2.9) can be interpreted as saying e μn t P n 1 k= fl j k t ß e R j t (j n) t [μ+fl( )]d : i n j ß i ((j n) t)e R j t (j n) t [μ+fl( )]d ; that is, at time t n = n t, the number of infected for a length j = j t is approximately that of those who initially had been infected for a length j n multiplied by the probability that they did not die or recover during the ensuing time interval (;t n ). This is, of course, what any reasonable numerical method should be saying, since this is true of the differential problem, as is easily seen by integrating the second equation in (1.1) along the characteristics t =. A similar observation can be made about (2.1). Λ Next note that for the differential problem, we have from (1.1) (2.13) S(t) =S e R t (μ+ (fi))dfi +Λ Z t Λ e R t s [μ+ (fi)]dfi ds: We shall prove now a similar result for the approximation of a number of susceptibles. Lemma 2.2. The following relation holds for n 1: n 1 Y (2.14) S n = S k= X n t(μ + k ) +Λ `= n 1 Y t k=` 1 1+ t(μ + k ) :

7 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 7 Remark 2.2. Comparing (2.13) and (2.14) we see that S n does indeed have a similar decomposition as S(t n )interms of surviving initial number of susceptibles plus surviving immigrants" (recruited). Moreover, (2.13) and (2.14) prove that, if j (t k ) k j is of the order of t, then so is the absolute error in the number of susceptibles js(t k ) S k j. Λ We shall now demonstrate that the numerical method (2.1) converges at a first order rate. For convenience, let us introduce the following notation for evaluations of the solution of (1.1): (2.15) s n = S(t n ); y n j = i( j ;t n ); `n = (t n ); Y n = I(t n ); r n = R(t n ); and for the errors in their approximations: (2.16) ff n = s n S n ; n j = y n j i n j ; n = ο n = `n n ; ρ n = r n R n : n j t; k n k`1 = j n j j t; Theorem 2.1. Assume that all the coefficients of (1.1) are sufficiently smooth, that ffi and fl are bounded, and i is compactly supported. Then, for» n» N,» j, we have where the constant Q is independent of t. jff n j + j n j j + k n k + jο n j + jρ n j»q t; Proof. First note that (1.1) and (2.4) imply that, for i compactly supported, all sums in j,» j, really extend over a finite number of indexes. Also, (2.13) and (2.14) imply that, for» n» N, s n + Y n and S n + I n are bounded below by a positive constant, so that their reciprocals are bounded. Next, it is very easy to derive from (1.1) and (2.1), using Taylor expansions and simple quadratures, the following error equations for n 1: ff n ff n 1 = (μ + `n 1 )s n + O( t)+(μ + n 1 )S n ; t j n n 1 j 1 = (μ + fl j ) j n + O( t); j 1; t n = `n 1 ff n 1 + S n 1 ο n 1 + S n 1 n 1 [2μ t + n 1 t +( t) 2 (μ + n 1 )] + O( t); ρ n ρ n 1 t ο n = C(sn + Y n ) s n + Y n = μ R ρ n + t ffi j n j t + fl j n j + O( t);» C ( ) s n + Y n (ffn + n + O( t)) (ff n + n + O( t)) 1 X C(S n + I n ) (s n + Y n )(S n + I n ) ffi j i n j t + O( t):

8 n 8 F. A. MILNER 1 A. PUGLIESE 2 we can rewrite these as follows: ff n 1 = 1+ t(μ + `n 1 ) [ffn 1 ts n ο n 1 ]+O(( t) 2 ); j n 1 = 1+ t(μ + fl j ) n 1 j 1 + O(( t)2 ); j 1; n = `n 1 ff n 1 + S n 1 ο n 1 + O( t); ρ n 1 =»ρ n 1 +( t) 2 fl j j n 1+μ R t ο n = C(sn + Y n ) s n + Y n + O(( t) 2 ); ffi j n j t + O(jff n j + j n j + t): Taking absolute values on both sides of these relations we see that, for n 1, (2.17) (2.18) (2.19) (2.2) (2.21) jff n j»jff n 1 j + K tjο n 1 j + O(( t) 2 ); j n j j»j n 1 j 1 j + O(( t)2 ); j 1; j n j»k(jff n 1 j + jο n 1 j)+o( t); jρ n j»jρ n 1 j + K tk n k`1 + O(( t) 2 ); jο n j»k(k n k`1 + jff n j)+o( t): Here and in the sequel, K will denote a generic constant independent of t, and possibly dependent on lower bounds of S(t)+I(t) and S n + I n, as well as on Λ, μ, S + I, and upper bounds of ffi, fl, and C. Next, it follows from (2.18) and (2.19) by summing on j, that, for n 1, (2.22) k n k`1» (1 + K t)k n 1 k`1 + K tjff n 1 j + O(( t) 2 ): Also, combining (2.16) and (2.21) we obtain (2.23) jff n j»jff n 1 j + K t(jff n 1 j + k n k`1)+o(( t) 2 ): Adding (2.22) and (2.23) we derive the relation jff n j + k n k`1» (1 + K t)(jff n 1 j + k n 1 k`1)+o(( t) 2 ) which, iterated in itself yields, using ff =, jff n j + k n k`1» 1+ KT N k k`1 + N Using (2.2) we see that k k`1 = O( t) and, consequently, (2.24) jff n j + k n k`1» O( t); n 1;» 1+ KT N 1 O( t): N where the constant in the O term depends on T exponentially. Combining now (2.21) and (2.24) we obtain

9 Λ Λ Λ Λ Λ PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 9 and using this relation and (2.24) in (2.19) we see that (2.26) j n j»o( t); n 1: Finally, iterating (2.18) in itself and using (2.2) and (2.26) we arrive at the following relations: (2.27) (2.28) j n j j»j j nj + O( t) =O( t); j n 1; j n j j»j n j j + O( t) =O( t); n j 1: The theorem follows by combining (2.24) (2.28). Λ In order to see numerically that the method performs as described by the theorem, we chose a simple ordinary model of S I type with constant population size. This means that we set S + I = Λ=μ, R = fl, and ffi 1. In this case we can solve the resulting system explicitly and compute the errors of the numerical approximations exactly. We have ρ I(t) = cffi(1 e ρt )+ ρ I e ; ρ = Λ ρt μ cffi μ: We ran simulations for T = 1, using Λ = 5, S = 4, I = 1, μ = 1, C 1, and using t ranging from 1/4 to 1/128. We then postulated that the error is of the form E t = ji(1) I N j = Q( t) ff and by taking the logarithms of the ratios of the errors E t and E t=2 and dividing them by ln 2, we determined the effective rate of convergence ff. The results are presented below in Table 2.1. Table 2.1 Effective convergence rate t E t ff 1/ / / / / / III. Steady States of The Discrete Dynamical System. We shall study now the steady states of (2.1) and derive a threshold theorem for the existence of non trivial ones. Let us denote the equilibria of (2.1) by

10 1 F. A. MILNER 1 A. PUGLIESE 2 These must satisfy the following relations: (3.1) (3.2) (3.3) (3.4) (3.5) Λ (μ + Λ )S Λ =; i Λ 1 j = 1+ t(μ + fl j ) iλ j 1; j 1; i Λ = Λ S Λ (1 + μ t)[1 + t(μ + Λ )] ; Λ = C(SΛ + I Λ ) S Λ + I Λ μ R R Λ + ffi j i Λ j t; fl j i Λ j t =: First note that (3.2) and (2.6) imply that, for j 1, (3.6) i Λ j = i Λ from which jy k=1 (3.7) I Λ = i Λ 1 1+ t(μ + fl k ) = iλ ß j ß j t follows. Next, let us introduce the ultimate discrete force of infection per infected, (3.8) T = It then follows from (3.4), (3.6), and (3.8) that ffi j ß j t: (3.9) Λ = C(SΛ + I Λ ) S Λ + I Λ i Λ T : Also, (3.5) and (3.6) yield (3.1) R Λ = i Λ Now, (3.3) and (3.9) can be combined to obtain fl j μ R ß j t: i Λ = SΛ S Λ + I Λ C(S Λ + I Λ ) (1 + μ t)[1+ t(μ + Λ )] iλ T ; which implies that Λ (1 + μ t)[1 + t(μ + Λ )] C(S Λ + I Λ )

11 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 11 Looking at (3.1), (3.6), (3.7), (3.9), and (3.1), we see that the trivial equilibrium S Λ = Λ μ ; iλ j for j ; I Λ = Λ = R Λ =; always exists. On the other hand, (3.1), (3.9), and (3.11) can be combined to yield a single equation for Λ for non trivial equilibria. First of all we see from (3.1) that (3.12) S Λ = Λ μ + Λ : Combining (3.9) and (3.11) we obtain Λ i Λ =(1+μ t) 1+ t(μ + Λ ) S Λ : This can be rewritten, using also (3.12), in the more useful form (3.13) i Λ = so that from (3.7) we obtain (3.14) I Λ = 1 X Λ 1+μ t ß j t Λ (μ + Λ )[1+ t(μ + Λ )] ; Λ 1+μ t Therefore, combining (3.12) and (3.14) we see that» (3.15) S Λ + I Λ = Λ X 1 μ + Λ 1+ ß j t Λ (μ + Λ )[1 + t(μ + Λ )] : Λ (1 + μ t)[1 + t(μ + Λ )] Finally, substituting (3.13) and (3.15) into (3.9), we see with some simple algebra that, if Λ 6=, then (3.16) (1+μ t) 2 + Λ» t(1+μ t)+» Λ ß j t = T C μ + Λ : P Λ 1 1+ ß j t (1 + μ t)[1 + t(μ + Λ )] The left hand side of (3.16) is linearly increasing in Λ. As for its right hand side, recall that we have assumed that C( ) is a non-decreasing function. Its argument can be written as ΛF ( Λ ) where with F (x) = 1 μ + x 1+ Ax B + tx P 1 ß j t :

12 12 F. A. MILNER 1 A. PUGLIESE 2 One easily sees that F B(B μa)+ tx(x(a + t)+2b) (x) = (μ + x) 2 (B + tx) 2 is negative for x as long as B>μA. In order to check this inequality, note that by its definition (2.6), ß j» (1 + μ t) j. Therefore, (3.18) ß j t» 1 μ + t: Using (3.17) and (3.18) we then see that B>1 μa so that B >μaand F (x) <. We have therefore established that the right hand side of (3.16) is a non-increasing function of Λ. This fact easily yields the following threshold phenomenon. Theorem 3.1. The steady state equations (3.1) (3.5) always admit the trivial equilibrium This is the only non-negative equilibrium if S Λ =Λ=μ; i Λ j (j ) = I Λ = Λ = R Λ =: (3.19) C( Λ μ )T» (1 + μ t)2 : holds. On the other hand, if (3.19) does not hold, then there exists a unique positive equilibrium. In case of a constant contact rate, this can be computed as (3.2) Λ = CT (1 + μ t) 2 t(1 + μ t)+ P 1 ß j t ; together with (3.13), (3.12), (3.6), (3.11), and (3.9). IV. Conditions for Instability. Thieme and Castillo-Chavez [15] have shown that the endemic equilibrium of (1.1) may be unstable. Here, we build an explicit example of that situation. In order to do this, we start from the study of the characteristic equation. We know from [15] that the characteristic equation in z is the following: (4.1) 1= ffο 1+z where 1 1 ο ffi οffil(p)(z)+l(q)(z); (4.2) ff = 1 R 1 ß 1 ( )d ; ffi = M (U Λ ) UΛ M(U Λ ; M(U) = C(U) ) U ; U = S + I; ο = I Λ =U Λ ; ß 1 ( ) =e μ R fl(s)ds ; p( ) =ffß 1 ( ); q( ) = '( )ß 1 ( ) R 1 ;

13 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 13 L denotes the Laplace transform, and S Λ, I Λ, U Λ denote, respectively, the number of susceptibles, infected, and total active population size at the endemic equilibrium. First we look for solutions of (4.1) on the imaginary axis, that is for z = iw, w 2 R. Making this substitution in (4.1) and taking real and imaginary parts, we arrive at the following pair of real equations: (4.3) where 1= ffο 1+w ο ffi οffia 1 (w)+b 1 (w); = ffο 1+w 2 w 1 1 ο ffi + οffia 2 (w) B 2 (w); (4.4) A 1 (w) = B 1 (w) = cos( w)p( )d ; A 2 (w) = cos( w)q( )d ; B 2 (w) = sin( w)p( )d ; sin( w)q( )d : Note that, from the assumptions on the contact function C(U) and the definition of an endemic equilibrium, one necessarily has» ο;ffi» 1. Now it follows immediately from the fact that jb 1 (w)j»1 that, if A 1 (w), the first equation in (4.3) cannot have a solution. On the other hand, Thieme and Castillo-Chavez [15] showed that, if A 1 (w) <, it is possible to find a function ' concentrated sufficiently close to and values <ο<1 and ff>1 that solve (4.3). In order to build a specific example in which the endemic equilibrium is unstable, we choose a suitable form of the function fl. Let G fl 1, A>, and define ρ ;» A; fl( ) = G; >A: Then we have ρ 1;» A; (4.5) ß 1 ( ) =e μ e G( A) ; >A: Even for such a simple function, finding the roots of equation (4.1) is very hard. Therefore, we follow the opposite route: we start from a candidate solution x + iw and try to find parameter values such that x + iw is indeed a solution of (4.1). We now let μ = 1, which really only amounts to rescaling times to be measured in units of 1=μ, and we let w = 3ß 2A. After some tedious calculations we can then see that A 1 < if <A< 5: 4 Having chosen the imaginary part w, wenow fix the real part x of our putative rootasa small positivenumber, say x =:1. We then modify the equations (4.3) to be, respectively, the real and imaginary parts of (4.1) with z = x + iw. We then have (4.6) ffο(1 + x) 1 1= (1 + x) 2 + w 2 = ffο ffi 1 ο w 1 ffi οffia 1 + B 1 ; + οffia 2 B 2 ;

14 14 F. A. MILNER 1 A. PUGLIESE 2 where now (4.7) A 1 = B 1 = e x cos( w)p( )d ; A 2 = e x cos( w)q( )d ; B 2 = e x sin( w)p( )d ; e x sin( w)q( )d : As for the infectivity curve, we choose a parabola through the points (; ), (B;), and ( B ; B ) where <B<A. Specifically, 2 4 (4.8) '( ) = 1 ;»» B; B ; elsewhere: 8 < : Note that the function ' enters the equation (4.1) only through the term L(q), where q is a suitably normalized function. Therefore we can multiply ' by any constant without changing (4.1). This corresponds to the fact that only the product of ' and the contact function C(U) is present in (1.1); thus any multiplicative constant can be moved to either function without changing the model. We compute next the four coefficients A 1, A 2, B 1, and B 2 from the modifications of (4.3) just described. Next, after having fixed w = 3ß, x =:1 and A =:5 wemust choose 2A B so that there are ο and ffi between and 1 that solve (4.6). To check that this is indeed possible, we compute the Taylor series of ο and ffi in terms of B. After some algebraic manipulations, performed with Mathematica, we obtained that (4.9) ο = ο 1 B + O(B 2 ) and ffi = ffi + O(B) for B! ; where (4.1) ο 1 = (1 e A ) 9ß 2 +4A 2 (1 + x) 2Λ 9ß 2 4A 2 (1 + x)x 6Aße A(1+x) x Λ 8A 2 (9ß 2 +4A 2 (1 + x) 6Aße A(1+x) x) ß 17:47 and (4.11) ffi = 6Aße A(1+x) (1 + x) 9ß 2 4A 2 (1 + x)x 6Aße A(1+x) x ß :263: The numerical values in (4.1) and (4.11) were obtained for A =:5 and x =:1. We have therefore proved that with these choices of w, A and x, for B small enough

15 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 15 Using (4.9) we also computed numerically ο and ffi as functions of B to see the admissible range of B. The results are shown in Figures 4.1 and 4.2. Figure 4.1 Figure 4.2 Finally, given feasible values of ο and ffi we find a contact function C(U) such that, for the endemic equilibrium, I Λ =U Λ is indeed equal to ο and 1 ffi = U Λ C (U Λ ), as indicated in C(U Λ ) (4.2). We use a contact rate of Michaelis-Menten type [13]: (4.14) C(U) = ρu ; ρ; ` > : 1+`U We set Λ = 1; 7 and B = :5 and computed the values for `, ρ and U Λ to satisfy the necessary relations (4.2). Our initial distribution of infected contains just one newly infected individual; the simulation was run with a small time step t =:1 for a final time T = 4. The results are shown in Figures 4.3 and 4.4. These graphs show the convergence of the solution towards a periodic function. The values of the corresponding steady-state solution can be easily found to be U Λ ß 1; 4 and ο = :4879, which give

16 16 F. A. MILNER 1 A. PUGLIESE 2 Figure 4.3 Figure 4.4 Note that the choice of the contact rate is arbitrary. For instance, we could have also chosen C(U) =ρu 1 ffi, so that 1 ffi = U Λ C (U Λ ) C(U Λ would have been satisfied for any U Λ. ) V. Conclusions.

17 PERIODIC SOLUTIONS: A ROBUST NUMERICAL METHOD FOR AN S-I-R MODEL 17 a rate of unity with respect to the discretization parameter, unconditionally. Furthermore, the algorithm produces a discrete dynamical system which shares many important and useful properties of the continuous dynamical system defined by the model (1.1). As for the existence of periodic solutions, we have provided numerical evidence of their existence and, moreover, we should point out that the parameter values used are not unreasonable for the HIV/AIDS epidemic. Consider, for example, a population of 1,7 intra-venous drug users in which just one of them gets infected with HIV at time t =. Consider the mean permanence time in this population to be 2 years, which means μ = :5 measuring time in years. Since we assumed μ = 1, this means that our time unit is 2 years; then the simulation performed corresponds to having an infectivity function ffi supported in the interval 1 year, with a peak value B/4 at.5 years, just as the first peak in the usual curve used in most of the literature [4]. The transition function from HIV + to AIDS is taken to essentially describe a ten-year incubation period (A = :5=μ = 1) after which everyone infected develops AIDS. This is close to the average incubation period [3]. The contact rate C(U) at equilibrium is approximately 4,85, which corresponds to about 4.6/wk. This value can be scaled down by increasing the maximum value of ffi, since only the product of the two appears in the model. A not unreasonable value for the peak transmission probability in needle sharing maybe 1 per cent [7], which is 8 times B/4. If we divide C(U Λ ) by 8 we obtain a contact rate of about.575/wk, or about 3 partners per year. This is a high value but seems close to values estimated for certain communities. A place where our simulation differs quite essentially from real life" parameters and functions, is in having an infectivity function with only one initial peak. We shall investigate numerically under which conditions periodic solutions can also be found in the presence of an infectivity function which, after the initial peak and a moderately long plateau, becomes monotone increasing. Note that [15] has established that periodic solutions may exist also for infectivity functions with multiple peaks. It should also be observed that the period of the oscillations in this example is :68=μ, that is 13.6 years, which is about the same time we have been aware of and collecting prevalence data for AIDS. The main epidemiological implication of this work, is that one should be very careful in becoming enthusiastic if the number of new HIV infections and/or of new AIDS cases starts dropping, because this could just represent a downswing in one of the oscillations rather than a steady decrease towards a low prevalence steady-state or extinction. REFERENCES 1. L. M. Abia and J. C. López-Marcos, Runge-Kutta methods for age-structured population models, preprint. 2. R. M. Anderson, The epidemiology of HIV infection: variable incubation plus infectious periods and heterogeneity in sexual activity, J. Royal Stat. Society A 151 (1988), P. Bacchetti and A. Moss, Incubation period of AIDS in San Francisco, Nature 338 (1989), S. P. Blythe and R. M. Anderson, Variable infectiousness in HIV transmission models, IMA J. Math. Appl. Med. Biol. 5 (1988), C. Castillo-Chavez, K. Cooke, W. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). 1. Single population models, J. Math. Biol. 27 (1989), C. Castillo-Chavez, K. Cooke, W. Huang and S. A. Levin, Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (Ed.), Lecture Notes in Biomathematics, vol. 83, Springer, M. Iannelli, R. Loro, F. A. Milner, A. Pugliese, and G. Rabbiolo, An AIDS Model with Distributed Incubation and Variable Infectivity: Applications to IV-Drug Users in Latium, Europ. J. Epidem. 8

18 18 F. A. MILNER 1 A. PUGLIESE 2 8. M. Iannelli, F. A. Milner, and A. Pugliese, Analytical and Numerical Results for The Age- Structured S I S Model with Mixed Inter-Intracohort Transmission, SIAM J. Math. Anal. 23 (1992), W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc.A 115 (1927), M. Kubo and M. Langlais, Periodic solutions for a population dynamics problem with age dependence and spatial structure., J. Math. Biol. 29 (1991), T. Lafaye and M. Langlais, Threshold methods for threshold models in age dependent population dynamics and epidemiology, Calcolo 29, I. M. Longini jr., W. S. Clark, M. Haber and R. Horsburgh jr., Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (Ed.), Lecture Notes in Biomathematics, vol. 83, Springer, L. Michaelis and M. I. Menten, Die Kinetik der Invertinwirkung, Biochem. Z. 49 (1913), F. A. Milner and G. Rabbiolo, Rapidly Converging Numerical Algorithms for Models of Population Dynamics, J. Math. Biol. 3 (1992), C. Castillo-Chávez and H. Thieme, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math. 53 (1993), H. Thieme and C. Castillo-Chavez, Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (Ed.), Lecture Notes in Biomathematics, vol. 83, Springer, 1989, pp