Diekmann, O. (Odo)

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. stichting mathematisch centrum AFDELING TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) 0. DIEKMANN TW 170/77 ON A NONLINEAR INTEGRAL EQUATION ARISING IN MATHEMATICAL EPIDEMIOLOGY Preprint ~ MC SEPTEMBER 2e boerhaavestraat 49 amsterdam i,IBL\.OTHEEK MATHEMATISCH O:.NW.U(IA AMS Tc.RD AM PJunted a;t :the Ma:thema.:tlcal. Cmtll.e, 49, 2e BoeJt.haa.vu.tltaA.:t, Am.6:teJtdam. The Ma:thema.:tlcal. Centlr.e, 6ou.nded :the 11-:th 06 FeblluaJLy 1946, ,ii, a non-pllo6U w.tltution a.,i.m,in.g a;t :the pJLomoUon 06 pWte ma:thema.:tlC-6 and .lt:6 appUc.ailoYL¢. I:t ,ii, .6poYt¢olled by :the Ne:theJri.a.nd.6 Gove/l.nment :thllough :the Nethe/Lf..a.nd.6 OJt.ga.nlza.:tlon 6oll :the Advanc.ement 06 PUite RueaJLc.h (Z.W.O). AMS(MOS) subject classification scheme (1970): 92AIS,45G99,45M99 . . 1 . h . 1 "d . 1 *) On a nonlinear integra equation arising in mat ematica epi emio ogy by O. Diekmann ABSTRACT In this note we study a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. The emphasis is on qualitative aspects. We prove that the equation exhibits the hair-trigger effect if a certain parameter exceeds a threshold value. Subsequently we pay attention to travelling wave solutions. These are found as solutions of a certain nonlinear convolution equation on the line. We show that there exists c0 , 0 < c0 < 00 , such that a travelling wave solution with speed c exists if lei > c0 • Moreover, using Tauberian methods we prove that this solution is unique (modulo translation) and that no travelling wave solutions with speed c do exist if lei < c0 . In this note most proofs are merely outlined. More general results and complete proofs will be published elsewhere. KEY WORDS & PHRASES: spread of infection in space and time; nonlinear integral equation cf mixed Volterra-Fredholm type; threshold phenomenon; hair-trigger effect; travelling waves; nonlinear convolution equation; existence and uniqueness (modulo translation) or nonexistence of nontrivial solutions; Tauherian methods. *) To appear in: Proceedings of the Third Scheveningen Conference on Differential Equations, W. Eckhaus and E.M. de Jager Editors, North-Holland. I. INTRODUCTION In this note we shall study some qualitative aspects of the development of an epidemic in space and time. The mathematical problems that we shall come across are mainly those of proving existence or nonexistence of solu-tions of nonlinear convolution equations. We shall give a survey of some of the results that we obtained in [2] and, jointly with H.G. Kaper, in [3]; all details that we omit here can be found there. In the space-independent Kermack and McKendrick model the evolution of the epidemic is governed by the equation (I.I) u(t) = s0 f~ g(u(t-T))A(T)dT+ f~ h(T)dT, O:c:: t< 00 , where u(t) g(y) = -£n S(t) Sis s ' 0 I - e -y, the density of susceptibles, and A,h are given nonnegative functions describing, respectively, the infec-tivity of an individual which has been infected at t = 0 and the influence of the history up to t = 0. Suppose I; A(T)dT = y < 00 , f; h(T)dT H(oo) < oo, then, under appropriate hypotheses, u(t) + u( 00 ) as t + 00 and (1.2) u(00 ) = yS0g(u(00 )) + H(oo). Equation (1.2) has a unique positive solution u( 00 ) for each positive H( 00 ). Let~ be defined by u inf u(oo) H(oo)>O then clearly ( I. 3) u The most important qualitative feature of the model is the so-called tPes-hold phenomenon u > 0 if and only if yS0 > I. (Plotting a picture will make this evident.) The fact that this result is biologically significant appears from -u < e -or, in words: the fraction of the susceptible population that escapes from getting the disease is less than exp(-~) for any initial infectivity (no matter how small). 2 Following Kendall we introduce space-dependence in the model by assumine that the infectivity is in fact a weighted spatial average (1.4) u(t,x)= s0 J~ A(T) J ]Rn g(u(t-T,E;))V(x-Ud~dT + Jg h(T,x)dT, where V: ]Rn ➔ lR is a nonnegative radial function, and J V(x)dx = I. ]Rn Art obvious question is now: is there an analogous threshold phenomenon for equation ( 1.4 )? We shall show that the answer is yes if n = I or n = 2. In particular we find that, if yS0 > I, the equation exhibits the hair-trigger effect: no matter how little infectivity is introduced inan arbitrarily small subset of lRn, eventually there will be a large effect at every point. Subsequently we shall investigate the possibility of travelling wave solutions. Instead of equation (1.4), describing an initial value problem, we then consider the time-translation invariant equation (1.5) u(t,x)= s0 J~ A(T) J]Rn g(u(t-T,E;))V(x-Od~dT, -oo< t<oo. Our main result is that, under appropriate hypotheses, there exists c0 , 0< c0 < 00 , such that (1.5) has a travelling wave solution u(t,x) = w(x+ct) if lei> c0 and no such solution if lei< c0 • In the following we shall normalize A and incorporate the constants y and s0 in the function g. 2. THE HAIR-TRIGGER EFFECT ( 2. I ) where HA: H : g Let us consider the equation u(t,x)= fot A(T) J g(u(t-T,E;))V(x-Od~dT ]Rn +_ f(t,x), u: ]R+ X ]Rn ➔ lR 1.S unknown and A, g, V and f satisfy A: ]R+ ➔ ]R 1. s nonnegative; A E LI (lR+) and 1; A(T )dT = I. g: ]R ➔ ]R 1.S Lipschitz continuous (unHormly on JR+), monotone non-decreasing and bounded from above; g(0) = o. V: ]Rn ➔ JR is nonnegative; VE L1 (1Rn) and J n V(x)dx radial function. JR I; V 1.s a f: lR x 1Rn ➔ JR is nonnegative and continuous; f(•,x) is monotone non-decre+asing for each x E lRn; {f(t, ·) I t ~ 0} is uniformly bounded and equicontinuous. Let BC(Q) denote the Banach space of the bounded continuous functions on Q, equipped with the supremum norm. THEOREM 2.1. There exists a unique continuous solution u: lR x ]Rn ➔ lR of equation (2.1). Moreover u is nonnegative~ u(•,x) is monoton~ nondecreasing for each x E ]Rn and there exists u( 00 ,•) E BC(lRn) satisfying (2.2) u( 00 ,x) = f g(u(00 ,~))V(x-()ds + f(oo,x), ]Rn 3 and such that u(t,x) ➔ u( 00 ,x) as t ➔ 00 uniformly on compact subsets of lR.n. SKETCH OF THE PROOF. The local (i.e., t E [0,T], T sufficiently small) exis-tence and uniqueness of a solution follows from a straightforward application of the Banach contraction mapping principle. The nonnegativity and the mono-tonicity follow from the construction of the solution as the limit of a se-quence that is obtained by iteration. The global existence can be established by a continuation procedure and the boundedness by a simple estimate. The boundedness and the monotonicity yield pointwise convergence to a limit as t ➔ 00 and by application of the Arzela-Ascoli theorem this can be strengthen-ed as stated. Then it is easy to show that the limit has to satisfy equation (2.2). □ The mapping N defined by Nf = u relates the introduced infectivity to the thereby caused effect. Note that Nf = 0 if f = 0. For each finite T, N is continuous as a mapping from BC([O,T] x lR.n) into itself. However, as a mapping from BC(lR.+ x lR.n) into itself, N need not be continuous at f = 0. In order to show this we shall investigate the final state equation (2.2). Firstly we state some lemmas which are crucial. Let w* k denote the convolution of,.w and k and km* the (m-1)-times itera-ted convolution of k with itself. Let k denote the Fourier transform of k. Suppose n ~: k E L 1 (JR. ), k is nonnegative and J k(x)dx = I. lR.n Consider the inequality (2.3) LEMMA 2.2. The following statements are equivalent (i) there exists w E BC(lR.n) such that (2.3) is satisfied with strict 1,n-equality in some point; (ii) there exists h > 0 such that loo f m* k (x)dx. m=I I xl:s;h converges; (iii) there exist a> 0, C < 00 such that f lxl:s;a Rec+ E ~ k(x)) dx :,; C for alt E > 0. It is not easy to check directly ~hether a given k satisfying Hk has proper-ty (iii). Note that k(O) = I and k(x) f I for x f 0. So for E = 0, the inte-grand in (iii) has a singularity at the origin and nowhere else. The next lemma establishes conditions on k such that the singularity is integrable. LEMMA 2.3. (a) If n = (b) if n = and ( 00 lxlk(x)dx < 00 then (iii) is equivalent to J:00 xk(x)dxf O; 2 and f lR.2 lxl 2k(x)dx< 00 then (iii) is equivalent to a J ,.., x 1k(x)dx + b J 2 x2k(x)dx f 0 ]R_L JR. for some a and b; (c) if n ~ 3 then every k satisfying Hk has property (iii). A proof of lemmas 2.2 and 2.3 can be found in Essen [4] and in Feller [5, sections VI.JO and XVIII.7]. The following lemma deals with the case that the inequality (2.3) is in fact an equality. 4 LEMMA 2.4. Suppose w E BC(1Rn) satisfies (2.4) w = W* k., then w _ C for some constant C. For a proof see Rudin [6, Theorem 9.13] or Feller [5, section XI.2]. ,. THEOREM 2. 5. Suppose that (a) there exists p > 0 such that g(y) > y for O < y < p and g(p) = p, (B) n = 1 or n = 2 and f]Rn lxlnV(x)dx < 00., (y) f( 00 ,•) is not identically zero., then u(00 ,x) 2:: p for every x E ]Rn. PROOF. Let w be defined by w(x) = min{u( 00 ,x),p}, then (2.2) implies that w satisfies (2.3). Since Vis a radial function, lemmas 2.2 and 2.3 imply that w has to satisfy (2.4) and hence that w = C, 0 ~ C ~ p. If O ~ C < p then (2.2) would not be satisfied. Hence C = p. D In the special case that g(y) = ySo(l - exp(-y)), the condition (a) of Theorem 2.5 is equivalent to yS0 > 1. The lower bound pis independent of the function f. Thus we have demonstrated the hair-trigger effect if yS0 > 1. 3. TRAVELLING WAVES Throughout the remaining part of the paper we shall assume that n = I. g is two times continuously differentiable and there exists p > 0 such that g(y) > y for O < y < p and g(p) = p. Vis continuous and has ~ompact support. Our results are actually valid under conditions on g and V which are weaker and which may be different for different theorems. Substitution of u(t,x) = w(x+ct) into equation ( 1.5) yields, upon some rearranging, (3. 1) where (3. 2) Since w ( O = J:00 g ( w ( n ) )V c ( ~ -n) dn, ~ x + ct, Ve(~)= J~ A(T)V(~-cT)dT, -oo < ~ < oo r V (l;)d~ = 1, -oo C (3.1) has, for every c, the constant solutions w = 0 and w = p. By a non-trivial solution of (3.1) we shall mean a continuous function w satisfying (3.1), 0 ~ w(x) ~ p and neither being identically O nor identically p. Theorem 2.5 shows that for c = 0 no nontrivial solutions do exist. As c creases, the mass of V shifts to the right and in Theorem 3. 1 we shall in-show has that a nontrivial solu~ion exists if the distribution of the mass of V become lopsided enough. Because of the symmetry of V we can restrict oSr at-tention to positive c. THEOREM 3.1. Suppose that g(y) ~ g'(O)y for O ~ y ~ p., then there exists c0 , 0 < c0 < 00 ., such that for every c > c0 (3.1) has a monotone nondecreasing solution w satisfying w(-00 ) = 0, w( 00 ) = p. SKETCH OF THE PROOF. With the linearized equation (3.3) v(~) = g'(O) J:00 v(n)Vc(~-n)dn, there is associated the characteristic equation (3 .4) L (>..) = I , where C (3 .5) L (>..) = g I (O) f' e-u,; V (Ods C -oo C g' (O) J°" -)..cT J°" -)..f,; 0 e A(T)dT _00 e V(f,;)df,;. Note that L (0) = g'(O) > I (since y < g(y) ~ g'(O)y for O < y ~ p). Real roots of (3~4) yield sign-definite solutions of (3.3). The constant c0 is defined by c0 = inf{c I there exists>..> 0 such that Lc(>..) = !}. 5 The nonnegativity of A and V guarantees that this definition makes sense (for fixed c, L (>..) is a convex function of ).. and for fixed ).. it is a mono-tone decreasingcfunction of c). The basic idea of the proof is to use infor-mation obtained from L (>..) and the properties of gin the construction of two functions¢ and W ~uch that¢~ W, T¢ ~ ¢, and Tw ~ W, where T denotes the formal integral operator that is associated with the right-hand side of (3. I). The existence of a solution having the asserted properties is then established by means of an iterative process (Tis monotone). D Similar results have been obtained by Atkinson and Reuter [I] and by Wein-berger [7]. Weinberger has also constructed functions¢ and w for the case C = c0 • 4. NONEXISTENCE AND UNIQUENESS The first step towards a proof of the nonexistence of travelling waves with speed less than c0 is provided by the following lemma concerning the convolution inequality (2.3). LEMMA 4. I. Suppose k satisfies Hk for n = I and J:00 lxlk(x)dx < 00 , J:00 xk(x)dx 'f 0. Let w be a bounded and uniformly continuous solution of Then ( i) w - w * k E: LI (JR), (ii) lim w(x) and lim w(x) both exist, x-+-oo x-++oo J:00 (w - W*k) (x) dx (iii) w(oo) - w(-oo) = J:00 xk(x)dx SKETCH OF THE PROOF (see Essen [4] for a detailed proof). Define f I: k(Ods, X > 0 n(x) l-I~°" k(Ods, X ~ 0, then n E: LI (JR) and ( 4 . I ) w * n ( x ) - w * n ( y ) = J x ( w - W* k) ( 0 d C y From the monotonicity of the right-hand side and the boundedness of the left-hand side of (4. I) there follows (i) and (4.2) W* n( 00 ) - W* n(-00 ) = J:00 (w-w*k)(Ods-6 Since -I -it(),) for II # 0 and n(O) J:00 xk(x)dx we know that n(11) # 0 for all 11. Then (ii) and (iii) follow from (4.2) and Pitt's form of Wiener's general Tauberian Theorem (see for example Rudin [6, Theorem 9.7] or Widder [8, Theorem V. !Oa]. D Assuming that Hi: J~ TA(T)dT < 00 we have COROLLARY 4.2. Let w be a nontrivial solution of (3.1) for some c > 0~ then lim w(x) = 0 and lim w(x) = p. x--+--co x++oo PROOF. A bounded solution of (3. I) is necessarily uniformly continuous. Since J:00 xVc(x)dx = c J; TA(T)dT > 0, we deduce from Lemma 4. I that w( 00 ) - w(-00 ) > 0. From (3. I) and the properties of git follows that only O and pare candidates for being limits. D THEOREM 4.3. Let the assumptions of Corollary 4.2 be satisfied and suppose g'(O) >I.Then ther•e exists a> 0 such that J:<X, w(x) e -11X dx converges for O < 11 < a. SKETCH OF THE PROOF. There exists£> I such that g(w(x)) > £w(x) for x+-00 • Using this inequality and the same kind of arguments as those leading to (i) of Lemma 4. I, one can prove that w € L1 ((-00 ,0)) and subsequently by an in-duction process that Jo m -m _ 00 lxl w(x)dx ~ m! a for some a> 0. D Motivated by Theorem 4.3 we define I !00_00 -,\x -;ll: = sup{ 11 € JR w(x) e dx converges} and The function W(11) is analytic in the strip O < Re 11 < 7:. As a consequence of the nonnegativity of w(x) we have (see Widder [8, Theorem II.Sb]) LEMMA 4.4. If*< 00 ~ then W(11) is singular in 11 = 7:. Writing (3.1) as w(x) where g' (O) W* V (x) + r(x), C we obtain by Laplace transformation ( 4. 3) WC\) = W(11)L (:\) + R(11). C If 7: < 00 then Lemma 4.4 implies that L (11) = I (note that R(11) is regular in a neighbourhood of 11 = 7:). The possibility that 7: = 00 can be excluded by a straightforward but technical proof. Thus we have established the following nonexistence result. THEOREM 4.5. Suppose c > 0 and L (A) > I for A 2 0~ then (3. I) has no non-e trivial solution. 7 Suppose, on the contrary, that the equation L (A)= I has a positive real root, then (4.3) can be used to obtain inform~tion concerning the asymptotic behaviour, as x ➔ - 00 , of solutions of (3.1). THEOREM 4.6. Suppose c > c0 and g(y) s g' (O)y for O s y s p. Let o denote the smallest positive root of L (A) = 1. Let w be a nontrivial monotone non-decreasing solution of (3.1)~ t~en there exists C > 0 such that lim -ox w(x)e = C. x➔-oo SKETCH OF THE PROOF. Note firstly th~t L~(o) j O and R(o) j 0. From (4.3) we obtain R(o) W(A) ~ L'(o)(o-A) ' At o. C By a complex variable TaubP.rian theorem of the Ikehara type (see for instance Widder [8. Theorem V.17]) we can deduce from this formula the asymptotic behaviour of w(x) as x ➔ - 00 • D If w(x) is a solution of (3.1), then so is every translate w-(x) = w(x+~) of w. Our final theorem establishes a condition on g such th~t every mono-tone nondecreasing nontrivial solution of (3. 1) is obtained by translating one specific solution. THEOREM 4.7. Suppose c > c0 and lg(y 1)-g(y 2) I s g'(O)ly 1-y2 1 for 0 s y I ,Y 2 s p. Then there is modulo translation one and only one monotone nondecreasing nontrivial solution of (3.1) .. SKETCH OF THE PROOF. Let w1 and w2 be two monotone nondecreasing solutions. By Theorem 4.6 we can find x such that v defined by -ox -v ( x) = e { w I ( x) - w 2 ( x+x) } satisfies lim v(x) = lim v(x) = 0. x-+-oo From one can deduce that lvl cannot assume a maximum. Hence v - 0. 0 nontrivial It is an open problem whether every nontrivial solution is monotone non-decreasing. REFERENCES [1] ATKINSON, C. & G.E.H. REUTER, Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc. 80(1976) 315-330. [2] DIEKMANN, o., Thresholds and travelling waves for the geographical spread of infection~ Mathematical Centre Report TW 166/77, Amsterdam, 1977. [ 3] DIEKMANN, 0. & H. G. KAPER, On the bounded solutions of a nonlinear eon-volution equation, in preparation. [4] ESSEN, M., Studies on a convolution inequality~ Ark. Mat. 5(1963) I 13-152. [5] FELLER, W., An Introduction to Probahility Theory and Its Applications, Vol. II (Wiley, New York, 1966). [6] RUDIN, W., Functional Analysis (McGraw-Hill, New York, 1973). 8 [7] WEINBERGER, H.F., Asymptotic behavior of a model in population genetics, to appear in: J. Chadam, ed., Indiana University Seminar in Applied Mathematics, Springer Lecture Notes. [8] WIDDER, D.V., The Laplace Transform (University Press, Princeton, 1946). REMARK. We also draw attention to the recent paper D.G. ARONSON, The asymptotic speed of propagation of a simple epidemic, to appear in: W.E. Fitzgibbon & H.F. Walker, eds., Nonlinear Diffusion (Research Notes in Mathematics, Pitman Publishing Co., 1977). There it is shown that for a special case of the model c0 is the asymptotic speed of propagation of infection. 

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