KERMACK McKENDRICK EPIDEMIC MODEL REVISITED

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1 K Y B E R N E T I K A V O L U M E 4 3 ( 2 7 ), N U M B E R 4, P A G E S KERMACK McKENDRICK EPIDEMIC MODEL REVISITED Josef Štěpán and Daniel Hlubinka This paper proposes a stochastic diffusion model for the spread of a susceptible-infectiveremoved Kermack McKendric epidemic (M) in a population which size is a martingale N t that solves the Engelbert Schmidt stochastic differential equation (2). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size N t. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed. Keywords: SIR epidemic models, stochastic differential equations, weak solution, simulation Mathematics Subject Classification: 37N25, 6H, 6H35, 92D25. INTRODUCTION An epidemy of a highly infectious disease with a fast recovery (or fatality) in a homogeneous population is considered, the influenza being an example of such epidemics. This classical Kermack McKendrick model [2] assumes a fixed sized population of n individuals, the population being divided into three subpopulations which change their respective sizes in the running time of the epidemic: Susceptibles (the individuals exposed to the infection), infectives (the infected individuals that are able to spread the disease) and removals (the individuals restored to health not able further to spread the infection or get themselves to be infected again) numbering by x(t), y(t) and z(t) the individuals that are susceptible, infected and removed at some time t, respectively. Hence, x(t) + y(t) + z(t) = n, x(t) and z(t) being generally a nonincreasing and nondecreasing function, respectively such that z() =. The model assumes the dynamics given by the following three dimensional differential equation ẋ(t) = βx(t)y(t), x() = x >, ẏ(t) = βx(t)y(t) γy(t), y() = y = n x >, (M) ż(t) = γy(t), z() =,

2 396 J. ŠTĚPÁN AND D. HLUBINKA where the intensity β > is higher for more infectious diseases and the parameter γ > is proportional to the average duration of the being infected status, i. e. to the average time for which an individual is infected. In general there is no explicit solution to (M), the approximation e u u + 2 u2 is known, (see [7]) to deliver a unique solution z as z(t) ρ2 x ( x ρ ) + αρ2 x ( ) tanh 2 γαt ϕ (S) with [ ( ) ] 2 /2 2x α = ρ 2 (n x x ) + ρ () [ ( )] ϕ = tanh x α ρ where ρ = γ/β denotes the relative removal rate of the disease. In some cases we may get a precise solution assuming a non constant intensity β = β(x, y, z) or adopting a more simple and still realistic choice β = β(z). Assuming, for example, that β(z) is a decreasing function we in fact propose a model in which the population grows to be more cautious and hence the epidemy will slow down its spread. See Theorem and the examples that follow. Our aim is to propose and justify a diffusion version of (M) model that allows both more general intensities β and to model the spread of epidemic in a population that changes its size N t due to a diffusion type emigration and immigration processes defined for example by the Engelbert Schmidt stochastic differential equation If this is the case we assume dn t = N t σ(n t ) dw t, N = n := x + y. (2) σ bounded such that supp(σ) [a, b], where a n b <, (3) which assumption keeps the size of population N t in the interval [a, b]. Further we assume the ratios x(t), y(t) and z(t) of susceptibles, infectives and removals, respectively, to derive their dynamics from a generalized (M) model that is given as the ordinary differential equation with random time dependent coefficients ẋ(t) = α(x(t), y(t), z(t), N t ) x(t)y(t), x() = x n ẏ(t) = α(x(t), y(t), z(t), N t ) x(t)y(t) γ y(t), y() = y, n ż(t) = γy(t), z() =, (M3) (note that x(t) + y(t) + z(t) = ), where α(x, y, z, n) is a function that is Lipschitz continuous on { (x, y, z) [, ] 3, x + y + z = uniformly for n [a, b].

3 Kermack McKendrick Epidemic Model Revisited 397 If this is the case we are able to represent uniquely (Theorem 5 and Corollary ) the size X t = x(t) N t, Y t = y(t) N t and Z t = z(t) N t of suspectibles, infectives and removals, respectively, as the solution to a three dimensional SDE dx t = β(x t, Y t, Z t )X t Y t dt + X t σ(n t ) dw t, X = x > dy t = β(x t, Y t, Z t )X t Y t dt γy t dt + Y t σ(n t ) dw t, Y = y > (M2) dz t = γy t dt + Z(t)σ(N t ) dw t, Z =, where the intensities α(x, y, z, n) and β(x, y, z) rescale each other as α(x, y, z, n) = n β(nx, ny, nz). Note that the size of population N t = X t + Y t + Z t where (X, Y, Z) solves (M2) is a solution to (2). Theorems 2, 5 and Corollary 2 offer sufficient conditions for (M2) to have a unique strong and weak solution, respectively. Section, namely Theorem, summarizes and extends well-known properties of the solution to the deterministic model (M) and also possible choices of the intensity β(z) are listed. The article is closed by Example 5 that delivers a visible computer illustration of the above results. We refer the reader to [4, 7], and to more recent [8] and [3] for the history and present state of art of stochastic modelling of epidemics. The martingale and diffusion models probably first appeared in [6] where (in our setting and notation) the stochastic process M t := Y t t (βx uy u γy u ) du is assumed to be a martingale which assumption makes it possible to estimate the constant intensity β. References [, 2, 3, 4] propose a multidimensional diffusion model built up on the top of the deterministic infection in a population that consists only of suspectibles and infectives: Having interpreted γ as the disease death rate and β as its transmission rate, denoting by N t = X t + Y t the size of the population we may simplify this model to the dimension one as Ẋ t = β N t X t Y t, Ẏ t = β N t X t Y t γy t, hence Ṅ t = γy t. The authors propose its diffusion version given by the non-linear stochastic differential equation dx t = β X ty t N t dt + b (t) dw t + b 2 (t) dw 2 t, dy t = β X ty t N t dt γy t dt + b 2 (t) dw t + b 22 (t) dw 2 t, where (W, W 2 ) is a two-dimensional standard Wiener process and ( ) ( b b β 2 N B = B(X, Y ) = = XY β N XY ) 2 b 2 b 22 β N XY. β N XY + γy Phillips Saranson theorem [6, 2.2 Theorem, p. 34] proves that the equation has a unique strong solution and authors perform its sophisticated and instructive simulations. Note that the size of population N t is a supermartingale in this case.

4 398 J. ŠTĚPÁN AND D. HLUBINKA Obviously, while the (M2) model expects an epidemic or even a pandemic with only negligible fatal consequences, the model proposed by [2] may be relevant when the infections as HIV AIDS in humans are studied. [9] proved a unique existence theorem for a differential equation that, if generalized to R 3, would cover the equation (M3). 2. KERMACK McKENDRIC DETERMINISTIC EPIDEMIC Here we shall treat (M) deterministic model with an intenzity β = β(z) which values are subject to changes dependent on the size of the removals subpopulation. Information on a solution (x(t), y(t), z(t)) to (M) we are able recover in this case is summarized by Theorem. Consider γ > and assume β = β(z) to be a nonnegative, bounded and locally Lipschitz continuous function on R +. Then (M) has a unique solution (x, y, z) C (R +, R 3 ) that is positive on (, ) and such that x = X(z), y = Y (z) and ż = γ[n z X(z)], z() =, (4) hold, where X(z) = x exp { γ z β(u) du and Y (z) = n z X(z), z n. (5) The number of susceptibles individuals x(t) and the number of removals z(t) is a positive nonincreasing and increasing function on R +, respectively, such that < x( ) < n and < z( ) n while y( ) exists and equals to zero. The limit z( ) is a solution to the equation n z = X(z) (6) and if β(z) is nonincreasing on R + then it is a unique solution to the equation on the interval [, n]. γ If β() >, β() < x and β(z) is again a nonincreasing function then the number of infectives has a unique maximum y + = y(t + ), where y + = n z + γ β(z + ), t + = γ z+ du (7) Y (u) and z + is a unique < z < z( ) such that β(z)x(z) = γ. (8) The number of infectives y(t) is increasing on [, t + ] and decreasing on the interval [t +, ]. All the above statements are proved for a constant β(z) in [7], for example.

5 Kermack McKendrick Epidemic Model Revisited 399 P r o o f. The unique existence part follows from a more general Theorem 3 choosing there σ =. Also, apply (2), (3) and (4) in Lemma to verify (4) and that (x, y, z) > on (, ), hence (x, y, z) [, n] 3 on R +. It follows that ż = γy >, hence z is an increasing function on R + with z( ) (, n]. Similarly ẋ and x is seen as nonincreasing on R + with n > x x( ) = X(z( )) >. Thus, y(t) = n x(t) z(t) has a finite limit y( ). Assuming y( ) > we conclude that lim t ż(t) > which implies z( ) =, hence a contradiction. Finally, z( ) = n x( ) < n. The limit z( ) solves (6) because n z( ) = x( ) = X(z( )), according to (4). Assuming that β(z) is a nonincreasing function we get X(z) to be convex on R +. Hence if < z < z 2 n are two distinct solutions to (6), then the graph of X(z) on [, z 2 ] is bellow the segment that connects the points (, x ) and (z 2, n z 2 ), which segment is further strictly bellow the segment that joins points (, n) and (z 2, n z 2 ). It follows that n z > X(z ) which is a contradiction. Compute Y (z) = + β(z) γ X(z), and Y ( + ) = + β() γ x >. (9) It follows that there is a z (, z( )) such that Y (z) = as Y () = y > and Y (z( )) =. Assuming that Y (z ) = Y (z 2 ) = for a pair < z < z 2 < z( ), or equivalently that (7) has not a unique solution in (, z( )), it follows that /X(z ) = β(z )/γ β(z 2 )/γ = /X(z 2 ). This and the inequality X(z ) X(z 2 ) imply that X(z ) = X(z 2 ) and further that β(z) = on [z, z 2 ]. Hence, X(z ) = that contradicts the definition of X(z). Thus, there is unique z + (, z( )) such that + β(z +) γ X(z + ) = Y (z + ) = and Y (z) increases and decreases on (, z + ) and (z +, z( )), respectively, Y (z + ) being its unique maximum. Finally, let t = t(z) be the inverse function to z = z(t). It follows by (M) that dt dz = γy (z), hence t(z) = γ of max t< y(t) = Y (z + ). z Y (u) du and t + = γ z+ Y (u) du is the only argument Remark. The number of removals z(t) is a unique solution to the differential equation (4). Having computed the integral t = t(z) = γ z [ { n u x exp γ and putting z(t) = t (z) we get the solution to (4). u β(v) dv] du, () Remark 2. The size of relative removal rate ρ = γ/β is considered by epidemiologists as a good measure of the virulence of an epidemic: If x exceeds ρ only by a small quantity then we model something as the common cold or a weak influenza. On the other hand, its values ρ x /2 indicate the danger of a pandemic spread of infection.

6 4 J. ŠTĚPÁN AND D. HLUBINKA Example. We may illustrate this by choosing a constant β, n =, x = 9995, γ = 3 and ρ x, and ρ 2 2 x, respectively. See Figures and 2 for the numerical illustration of the epidemics dynamics during first 6 days. size time Fig.. The time dynamics of susceptibles (dotted line), infected (solid line) and removed (dashed line), ρ = 4995 (pandemic). size time Fig. 2. The time dynamics of susceptibles (dotted line), infected (solid line) and removed (dashed line), ρ = 8995 (mild epidemic). We get ρ z( ) y + t where y + = max t y(t) = y(t + ). The limits z( ) = z are given by (6), i. e. as the solutions to n z = x exp{ z/ρ, the maxima y + are equal to n z + ρ where z + = ρ ln(x /ρ) by (7) and (8). To establish the arguments of maxima t + we apply the procedure (7) and compute t + = γ z+ n u X(u) du, where X(u) = x exp { u. ρ

7 Kermack McKendrick Epidemic Model Revisited 4 We have used program [9] in Mathematica R for numerical calculation (and interpolation) of x( ), y( ), z( ) and t +. Remark 3. Having postulated that β(z) is a nonincreasing function supported by a compact [, z ], γ where z > and such that β() < x, we assume that the intensity of infection monotonously decreases with the increasing number of removals and becomes negligible at the time t when z(t ) = z. Note that ρ() = γ/β() < x if and only if ẏ(+) >, and the latter inequality is a condition necessary and sufficient for the outbreak of the epidemic. Also note that the number of susceptibles x(t) = X(z(t)) is decreasing on [, t ] and then remains at the level x(t ) = X(z ) for ever. The number of removals will be of course still increasing to z( ) = n X(z ) z. Example 2. Choose n, x, β, γ and z positive such that ρ = γ β < x and put β(z) = β ( z z ) for z [, z ] and β(z) = if z z. We get X(z) = x exp{ ρ z( z 2z ) and z( ) = n X(z ). The time t + when the number of infectives achieves its maximum generally precedes the time t of the first entry of z(t) to z since ẏ(t ) = γy(t ) <. The maximum y + and its argument t + are computed by the procedure suggested by (7) and (8) that are specified as y + = n z + z γ, t + = z+ (z z + )β γ Y (u) du, z z + β X(z + ) = γ. z The differential equation (M) reads in this case as ẋ(t) = ẏ(t) = { z β z(t) z x(t)y(t), z(t) < z x() = x >,, z(t) z { z β z(t) z x(t)y(t) γy(t), z(t) < z y() = y = n x >, γy(t), z(t) z ż(t) = γy(t), z() =, Note that β(z(t)) = βγ z ẏ(t) for < t < t in this case. For a numerical illustration we have chosen similar values as in Example, in particular n =, γ = 3 and ρ = We have chosen z = 4. The time course of the epidemics is shown in Figure 3. Compare Figure 3 with decreasing intensity of infection and Figure to see that the total number of infected is much smaller now. We have again used Mathematica R to calculate the limit z( ). = 333 and y +. = 33, z + = 344 and t + = 9.42.

8 42 J. ŠTĚPÁN AND D. HLUBINKA size time Fig. 3. The time dynamics of susceptibles (dotted line), infected (solid line) and removed (dashed line), β(z) decreasing function. Example 3. Kendall [] proposes (see also [7]) a more sophisticated choice for β(z) in the form β(z) = { 2β ( z/ρ )+( z/ρ ), z ρ, ρ = γ β, γ, β >. =, z ρ, Note that β(z) is chosen such that the assumptions of Remark 3 are satisfied, in particular z = ρ, β() = β, and z( ) = n X(ρ ) = (x /ρ + α)ρ 2 /x, with α given by (). 3. KERMACK McKENDRICK WITH A STOCHASTIC EMIGRATION AND IMMIGRATION A random three dimensional dynamics L t = (X t, Y t, Z t ), t > of the number X t, Y t and Z t of susceptibles, infectives and removals, respectively, with the size of population N t = X t + Y t + Z t whose stochastic nature is caused by small random emigration-immigration perturbations is proposed by the stochastic differential equations (M2) where β(x, y, z) and σ(n) are suitable functions on R 3 and R, respectively. Note that the size N t is a solution to the Engelbert Schmidt stochastic differential equation (2). The nature of β(x, y, z), or perhaps more simply of β(z), and the constant γ > that enter (M2) is explained in Section. The diffusion coefficient σ(n) is designed to control the global size of the population N t inside reasonable bounds. It can be seen, for example, by choosing σ(n) as in (3) since a N t b holds almost surely in this case. Note that choosing a = b = n and σ = (M2) becomes (M). Let us agree that L t = (X t, Y t, Z t ) is a solution to (M2) if (M2) makes sense and if it holds for a standard Brownian motion W t defined on a complete probability space (Ω, F, P ), i. e. if X, Y and Z are continuous F t -semimartingales that satisfy (M2), having denoted by F t the augmented canonical filtration of the process W t.

9 Kermack McKendrick Epidemic Model Revisited 43 Recall that (M2) is said to have a unique strong solution if there is an almost surely determined solution on arbitrary (Ω, F, P, W ). Also recall that (M2) is said to have a unique weak solution if there is a setting (Ω, F, P, W ) on which a solution X, Y, Z may be constructed and if L(X, Y, Z) = L(X, Y, Z ) whatever solutions to(m2) (X, Y, Z) and (X, Y, Z ) might be chosen. By Itô formula, or more precisely by [, Proposition 2.2] on Doleáns equation, we prove easily the following lemma. Lemma. Assume β(z, y, z) and σ(n) bounded. Then (X, Y, Z) is a solution to (M2) if and only if { t N t = n exp σ(n u ) dw u t σ 2 (N u ) du, () 2 X t = x { t exp β(x u, Y u, Z u )Y u du N t, (2) n Y t = y { t exp β(x u, Y u, Z u )X u du γt N t, (3) n Z t = γ t Y u N u du N t (4) hold for all t almost surely. Especially, processes X,Y and N are positive on R + and Z is a process positive on (, ). Theorem 2. Assume β nonnegative and bounded, let σ satisfy (3) and consider arbitrary solution (X, Y, Z) to (M2). Then the following statements hold; (i) The size of population N t is a bounded martingale such that a N b holds and such that the limit N := lim N t exists. Especially, EN t = n holds for t all t. (ii) If supp(σ) = (a, b) and a + equals to a and b with probability b n b a a u 2 σ 2 (u) du = b b u 2 σ 2 (u) du =, the limit N and n a b a, respectively. (iii) The number of susceptibles < X b and the number of removals Z b is a supermartingale and submartingale, respectively. The process Z is positive on (, ). Especially, EX s EX t and EZ s EZ t holds for s t. (iv) All limits X, Y and Z exist, Y = and X > if and only if N >, hence assuming a > we get X as a positive random variable. Observe that all the above statements, equalities and inequalities are meant to hold almost surely, for example X > is to be read as X t > holds for all t almost surely. The stated martingale and semimartingale properties are w.r.t. the augmented canonical filtration of W t denoted by F t. For example, the statement X is a supermartingale is precisely as X is an F t -supermartingale.

10 44 J. ŠTĚPÁN AND D. HLUBINKA P r o o f. The size of population N t is a positive local martingale by definition and by (). It is a constant on a bounded interval (s, t) if and only if its quadratic variation [N] t = t N(u)2 σ 2 (N u ) du has the same property [6, 3.4 Theorem, p. 54]. Since [σ 2 > ] = (a, b) we reason that a N b and therefore N t is a bounded martingale. Thus, according to [5, Theorem 69., p. 76] a N b exists. We have proved (i). The proof of (ii) will be postponed until Theorem 5. Since N t is a martingale and t β(x u, Y u, Z u )Y u du a nonincreasing process we get X b as a bounded supermartingale by (2) in Lemma. Similarly, we verify that Z t b is a bounded submartingal by (4). This concludes the proof of (iii). It follows by the supermartingale and submartingale property of bounded processes X t and Z t, respectively, applying [5, Th. 69.] again, that X and Z exist. To prove that Y = N X Z = we shall verify that Y u du < : Observe first that t Z2 uσ 2 (N u ) du < holds for all t >. This implies that t Z uσ(n u ) dw u defines an L 2 -martingale. Hence, E t Y u du = γ EZ t by (M2) and E Y u du = γ EZ b γ. In particular, Y u du <. Finally, observing that β(x u, Y u, Z u )Y u dy c Y u du < for a c (, ), it follows by (2) that X > if and only if N > and (iv) is proved completely. A unique strong solution to (M2) exists under fairly mild requirements on the coefficients β(x, y, z) and σ(n). Putting L = (X, Y, Z) we write (M2) as dl t = b(l t ) dt + a(l t ) db t, L = (x, y, ), (5) where b(x, y, z) = b (x, y, z) xyβ(x, y, z) b 2 (x, y, z) = xyβ(x, y, z) γy : R 3 R 3, b 3 (x, y, z) γy a(x, y, z) = a (x, y, z) xσ(x + y + z) a 2 (x, y, z) = yσ(x + y + z) : R 3 M 3 a 3 (x, y, y) zσ(x + y + z) and B t = (Wt, Wt 2, Wt 3 ) is a three dimensional Brownian motion with Wt = W t. By M 3 we have denoted the space of the 3 3-matrices endowed with the Eucleidian norm. A standard of stochastic analysis [6, V.2. Theorem] says that (5) has a unique strong solution provided that the maps b and a are locally Lipschitz and of a linear growth. Even though our particular case (M2) involves a coefficient b = (b, b 2, b 3 ) that is not of a linear growth whatever nontrivial bounded β(x, y, z) may be chosen, we are able to prove Theorem 3. Let σ(n) be a Lipschitz continuous function such that (3) holds for some a b. Further, assume that β(x, y, z) is a nonnegative function that is Lipschitz continuous on ab := { (x, y, z) [, b] 3, a x + y + z b.

11 Kermack McKendrick Epidemic Model Revisited 45 Then (M2) has a unique strong solution L = (X, Y, Z) that is a positive process on (, ) such that a N = X + Y + Z b. P r o o f. Choose bounded locally Lipschitz ϕ i (x, y, z) : R 3 R such that ϕ (x, y, z) = β(x, y, z)y, ϕ 2 (x, y, z) := β(x, y, z)x holds for (x, y, z) ab and consider the stochastic differential equations dx t = ϕ (X t, Y t, Z t )X t dt + X t σ(n t ) dw t, X = x dy t = ϕ 2 (X t, Y t, Z t )Y t dt + Y t σ(n t ) dw t γy t dt, Y = y (6) dz t = γy t dt + Z t σ(n t ) dw t, Z =, where N t = X t + Y t + Z t. This is the case of equation (5) that has coefficients b(x, y, z) and a(x, y, z) that are locally Lipschitz and of a linear growth. Hence, by [6, V.2. Theorem], (6) has a unique strong solution L = (X, Y, Z) that is positive on (, ): { t X t =x exp ϕ (X u, Y u, Z u ) t 2 σ2 (N u ) du + σ(n u ) dw u, and { t Y t =y exp ϕ 2 (X u, Y u, Z u ) 2 σ2 (N u ) du γt + { t Z t = exp t exp σ(n u ) dw u 2 { u t σ(n w ) dw w + 2 σ 2 (N u ) du u t σ(n u ) dw u, σ 2 (N w ) dw γy u du by [, Theorem 2.2] again. Denote by τ := inf{t > : L t / ab the first entry of L = (X, Y, Z) to the complement of ab. Obviously, (6) yields equations X t τ = x + Y t τ = y + Z t τ = γ t τ t τ t τ β(x u, Y u, Z u )X u Y u du + t τ β(x u, Y u, Z u )X u Y u γy u du + t τ Y u du + Z u σ(n u ) dw u. X u σ(n u ) dw u, t τ Y u σ(n u ) dw u (7) Hence, N t = n + t N uσ(n u ) dw u for t τ and therefore a N t b for any such t as supp(σ) [a, b]. Obviously there is no t > for which either N t / [a, b] or min(x t, Y t, Z t ) < would hold, consequently there is no t > such that max(x t, Y t, Z t ) > b. Hence τ =. Reading (7) again we conclude that (X, Y, Z) solves (M2). Finally, if (X, Y, Z) is a solution to (M2) it follows from Theorem 2 that (X t, Y t, Z t ) ab at any time t. It yields that (X, Y, Z) is a solution to (6). The stochastic differential equation (M2) has a unique strong solution since (6) has the property. Later on we shall appreciate even a more general result.

12 46 J. ŠTĚPÁN AND D. HLUBINKA Theorem 4. Consider (M2) model with a time dependent intensity β(x, y, z, t) that is Lipschitz continuous on ab uniformly for t (especially bounded on ab ). Also let a Lipschitz continuous σ(n) to satisfy (3) for some a b. Then (M2) has a unique strong solution and any solution L = (X, Y, Z) to (M2) is a positive process on (, ) with a N = X + Y + Z b. The proof goes along the lines of the reasoning employed in the proof of of Theorem 3 only a finer construction of Lipschitz extensions ϕ i has to be performed: Define ϕ i : C(R +, R 3 ) R R for t, (x., y., z. ) C(R +, R 3 ) by ϕ (x., y., z., t) = β(x t τ, y t τ, z t τ, t τ) y t τ ϕ 2 (x., y., z., t) = β(x t τ, y t τ, z t τ, t τ) x t τ, where τ = τ(x., y., z. ) = inf {t > : (x t, y t, z t ) / ab : C(R +, R 3 ) [, ] is time of the first entry of the C(R +, R 3 )-canonical process l t = (x t, y t, z t ) to the complement of ab. It easy to check that ϕ and ϕ 2 are bounded progressive path functionals on C(R +, R 3 ) R + that are Lipschitz continuous on C(R +, R 3 ), i. e. such that ϕ i (l, t) ϕ i (l, t) C l l t, l, l C(R +, R 3 ), t (8) holds for a constant C and l t denotes the sup-norm in C([, t], R 3 ). Putting b(l, t) = ϕ (l, t)x t ϕ 2 (l, t)y t γy t, a(l, t) = x tσ(x t + y t + z t ) y t σ(x t + y t + z t ) γy t z t σ(x t + y t + z t ) for l t = (x t, y t, z t ) C(R +, R 3 ) and t we propose a three dimensional SDE dl t = b(l t, t) dt + a(l t, t) db t, L = (x, y, ) (9) where L t = (X t, Y t, Z t ) and B t = (W t, W 2 t, W 3 t ) is a three dimensional standard Brownian motion as before. Because b and a are progressive path functionals such that b : C(R +, R 3 ) R + R 3, a : C(R +, R 3 ) R + M 3 b(l, t) b(l, t) + a(l, t) a(l, t) C N l l t, holds for l N, l N, N N and t and b(l., t) + a(l., t) C ( + l ) holds for all l C(R +, R 3 ) and t according to (8), where C N and C are constants. We have denoted by b, a, the Eucleidian norms in R 3 and M 3, respectively and l the sup-norm in C(R +, R 3 ). It follows by [6, V.2. Theorem], p. 3 that (9) has a unique strong solution. Choosing a solution L = (X, Y, Z) to (9) we proceed the proof exactly as in the proof of Theorem 3 putting there τ = τ(l) where τ : C(R+, R 3 ) [, ] is defined as the first entry of l. to R 3 \ ab.

13 Kermack McKendrick Epidemic Model Revisited 47 Remark 4. Saying that b(l., t) is a progressively measurable map we mean that b is an C t -progressive process on C(R +, R 3 ) where C t is the minimal right continuous filtration built up on the top of the canonical filtration H t in C(R +, R 3 ). It is easy to check that V.2. Theorem of [6], that is formulated for H t -progressive (previsible) coefficients, is applicable more generally for the C t -progressive path functionals a and b. Our reason for this enrichment is that τ, being a C t -Markov time, lacks this property with respect to the canonical filtration H t. 4. WEAK SOLUTIONS TO KERMACK McKENDRICK STOCHASTIC EPIDEMIC MODEL The following theorem suggests a two-step procedure for obtaining a solution (X, Y, Z) to (M2). Theorem 5. A semimartingale L t = (X t, Y t, Z t ) is a solution to (M2) if and only if X t = x(t) N t, Y t = y(t) N t, Z t = z(t) N t, t, almost surely, (2) where N t is is a solution to Engelbert Schmidt stochastic differential equation (2) and l(t) = ( x(t), y(t), z(t) ) is a semimartingale that solves almost surely the ordinary differential equation with random coefficients (M3) where α(x, y, z, n) = n β(nx, ny, nz) for (x, y, z, n) R 4. P r o o f. Let L = (X, Y, Z) to solve (M2) and put l(t) = (x(t), y(t), z(t)) := N t (X t, Y t, Z t ) where N = X + Y + Z. Then N t is a solution to (2) and it follows by (2), (3) and (4) in Lemma that (x(t), y(t), z(t)) is a solution to (M3) almost surely. On the other hand, consider a solution N t to (2) and ( x(t), y(t), z(t) ) a solution to (M3). Note that x(t)+y(t)+z(t) = and put (X t, Y t, Z t ) := ( x(t), y(t), z(t) ) N t. It follows that { t x(t) = x() exp α(x(u), y(u), z(u), N u ) y(u) du and t = x { exp n { t y(t) = y() exp z(t) = γ = y { t exp n t β(x u, Y u, Z u ) Y u du, α(x(u), y(u), z(u), N u ) x(u) du γ t β(x u, Y u, Z u ) X u du γt t Y u y(u) du = γ du, N u that is exactly as to say that (X t, Y t, Z t ) is a solution to (M2) according to Lemma.

14 48 J. ŠTĚPÁN AND D. HLUBINKA We need sufficient conditions for (M3) to have a unique solution almost surely. Observe the following requirements on β(x, y, z) and α(x, y, z, n), respectively. (i) β(x, y, z) is a Lipschitz continuous function on ab. (ii) α(x, y, z, n) is Lipschitz continuous on = { (x, y, z) [, ] 3, x + y + z = uniformly for n [a, b]. (iii) For arbitrary N. C(R + ) such that a N. b the function α(x, y, z, N t ) is Lipschitz continuous on uniformly for t. (iv) For arbitrary N C(R + ) such that a N b and N = n, the equation (M3) has a unique solution l(n, t) = (x(n, t), y(n, t), z(n, t)) C(R +, R 3 ). It is obvious that (i) = (ii) = (iii) = (iv), (2) where the last implication is proved by Theorem 4 with a = b =, especially with σ =. Consequently, the proposed two-step construction of a solution to (M2) is good one from the mathematical point of view. Corollary. Let either β(x, y, z) to satisfy (i) or α(x, y, z, n) to satisfy (ii) and assume that σ(n) follows (3) for some a b. Then (M2) has a unique strong solution if and only if the equation (2) has the property. Moreover, assuming that (2) has a unique weak solution (M2) inherits the property. P r o o f. The first assertion is mostly a simple consequence of Theorem 5 applying the chain of implications (2). It remains to assume that (M2) has a unique strong solution and prove that N = N almost surely for any pair of solutions to (2). Let l(t) = (x(t), y(t), z(t)) be the unique solution to (M3) generated by N t and l (t) = (x (t), y (t), z (t)) the unique solution to (M3) generated by N t. Then l(t) N t = l (t)n t holds for t almost surely. Hence, N t = c(t) N t, where c(t) := x(t) x (t), c() =, is a continuous process of finite variation. Since N t is a martingale, it follows by Itô per partes formula that E t N u dc(u) = for all t almost surely, hence c is a process with constant trajectories. Thus, c(t) = for all t and therefore N = N. Note that (iv) defines a map F : { N C(R + ), a N b, N = n C(R +, R 3 ) that sends any such N to F (N)(t) = l(n, t), i. e. to a unique solution to (M3) generated by N. The map F is easily seen to be Borel measurable and therefore the probability distribution of N t uniquely determines the probability distribution of N F (N) that is a solution to (M2) according to Theorem 5. We suspect that there might be situations when (M2) has a unique weak solution while there need not to exist a unique strong solution to the Engelbert Schmidt equation (2).

15 Kermack McKendrick Epidemic Model Revisited 49 B Example 4. Choose β(x, y, z) such that β(x, y, z) = A(x+y+z)+z holds on ab. Then α(x, y, z, n) = B A+z is Lipschitz on and the equation (M3) adopts the deterministic form of Kermack McKendric model scrutinized in Theorem. Write p = x /n and q = y /n = p. We get (M) as We easily compute and X(z) = p exp ẋ = B A + z x y, x() = p ẏ = B A + z x y γy, y() = q ż = γ y, z() =. { γ z ( ) B/γ A Y (z) = z p. A + z ( ) B/γ A α(u) du = p A + z Choosing B = γ and A = p 2 we get γ α() = A < p and observe that Theorem is applicable completely with α(z) = 2γ p + 2z, X(z) = p2 p + 2z and Y (z) = z p2 p + 2z. The above differential equation has a unique solution (x(t), y(t), z(t)) on R +, z(t) being computed by solving ( ) ż = γ z p2, z() =. p + 2z There is no explicit solution to this differential equation nevertheless some features of the solution can be stated. The limit z = lim t z(t) [, ] may be computed by solving z = X(z ) = p 2 /(p + 2z ). We get z = 2 p p 7p 2 4, x = 2 + p 4 + 4p 7p 2. 4 Also solve the equation 2p 2 = (p + 2z + ) 2 to establish a z + (, ) and compute to get y + = z + p 2 p + 2z +, t + = γ ) y + = max y(t) = p ( 2 2 z+ 2 u p 2 (p du + 2u) 2 2, z + = p, x + = p 2 2 and t + > such that y + = y(t + ).

16 4 J. ŠTĚPÁN AND D. HLUBINKA 2γ p (x+y+z)+2z We summarize that having (M2) model with β(x, y, z) = and a Lipschitz continuous function σ(n) that satisfies () for a pair a b, we get a model that has a unique strong solution X t = x(t) N t, Y t = y(t) N t, z t = z(t) N t, where N t solves the equation (2). We easily compute that EX t = x(t) n, EY t = y(t) n, EZ t = z(t) n. The above two-step procedure obviously generates a couple of problems: (A) To solve Engelbert Schmidt equation () by which we mean to receive an information as complex as possible about the probability distribution of its solution N t with the aim to deliver simulations of its trajectories and formulas for the moments EN k t. To complicate things even more epidemiologists prefer models with possibly non-lipschitz weight functions σ(n) σ(n) = K(n a) µ (b n) ν, a n b, < a < n < b <, µ, ν >. (B) Having done this we are of course expected to solve the ordinary equation (M3) with the similar aims as proposed by (A). To this end precise requirements (2) as σ bounded with supp(σ) = (a, b), < a < n < b <, (22) a + a n 2 σ 2 (n) dn = b b n 2 σ 2 (n) dn = (23) and note that (23) is satisfied for Lipschitz continuous σ(n) that satisfy (22). Consider a Brownian motion B(t) such that B() = n, denote A(s) = s B(u) 2 σ 2 (B(u)) du, s, τ t = inf {s > : A(s) > t, t and recall Engelbert Schmidt Theorem [, Theorem 23., Lemma 23.2, p. 45 2] to get (looking also through the proofs) a simple Corollary 2. Assume (22). Then the condition (23) is necessary and sufficient for the equation (2) to have a unique weak solution. Arbitrary solution N t to (2) is equally distributed in C(R + ) as the continuous stochastic process B(τ t ). Moreover, τ := inf {s : A(s) = = lim t τ t = inf {s : B(s) {a, b. (24) In a combination with Corollary we also prove

17 Kermack McKendrick Epidemic Model Revisited 4 Corollary 3. Assume (22) and (23) for σ(n) and consider arbitrary α(x, y, z, n) = n β(nx, ny, nz) that is Lipschitz continuous on = { (x, y, z) [, ] 3 : x + y + z = uniformly for n [a, b]. Then the equation (M2) has a unique weak solution. Engelbert Schmidt Theorem in a more detailed form of Corollary 2 also proves the (ii) statement in Theorem 2: Consider arbitrary solution N t to the equation (2). It follows by (24) that the limit N and transformation B(τ ) are equally distributed random variables. Because B(τ ) is a {a, b-valued random variable almost surely and τ a Markov time such that B(t) [a, b] if τ t we may apply the Wald equalities ap [B(τ ) = a] + b P [B(τ ) = b] = n, a 2 P [B(τ ) = a] + b 2 P [B(τ ) = b] = Eτ to verify (ii) in Theorem 2. The reader may find in [7, 8] a source of information relevant to the problems proposed by (A). Example 5. Consider the same β(x, y, z) as in Example 4, where B = γ = /4, p =.995 and A = p /2. Recall that p is the initial ratio of the susceptibles in the population. The numerical solution of the population ratio of the susceptibles, infected and removed dynamics is shown in Figure 4. We can also calculate z x y + z + t size time Fig. 4. The time dynamics of susceptibles (dotted line), infected (solid line) and removed (dashed line) proportions of the population size. For the population size itself we use the Engelbert Schmidt equation (2), where (n a)(b n) σ(n) = 6, a =, b = 5, n = 25. (25) a + b

18 42 J. ŠTĚPÁN AND D. HLUBINKA Fig. 5. The simulated population size. size time Fig. 6. The susceptibles and removed at the end of the epidemic size time Fig. 7. The infected during the period of highest infection. A simulated sample path of such a population up to the time 6 is given in Figure 3. Parts of the simulated solutions of the stochastic model (M2), (X t, Y t, Z t ),

19 Kermack McKendrick Epidemic Model Revisited 43 are shown in Figures 6 and 7. The population dynamics is, however, quite small with respect to the dynamics of the size of susceptibles, infected and removed. The population size is between, and,5 to keep the emmigration/immigration dynamics realistic. On the other hand the number of susceptibles changes from 995 to 5 quite fast. ACKNOWLEDGEMENT This work was partially supported by the Ministry of Education, Youth and Sports of the Czech Republic under research project MSM The work of J.Š. was partially financed by the Czech Science Foundation under grant 2/3/27, the work of D.H. was partially financed by the Czech Science Foundation under grant 2/3/p38. The authors thank for the interesting problem proposed to them by the Roche Company. (Received April 8, 26.) R E F E R E N C E S [] E. J. Allen: Stochastic differential equations and persistence time for two interacting populations. Dynamics of Continuous, Discrete and Impulsive Systems 5 (999), [2] L. J. S. Allen and N. Kirupaharan: Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens. Internat. J. Num. Anal. Model. 2 (25), [3] H. Andersson and T. Britton: Stochastic Epidemic Models and Their Statistical Analysis. (Lecture Notes in Statistics 5.) Springer Verlag, New York 2. [4] N. T. J. Bailey: The Mathematical Theory of Epidemics. Hafner Publishing Comp., New York 957. [5] F. Ball and P. O Neill: A modification of the general stochastic epidemic motivated by AIDS modelling. Adv. in Appl. Prob. 25 (993), [6] N. G. Becker: Analysis of infectious disease data. Chapman and Hall, London 989. [7] D. J. Daley and J. Gani: Epidemic Modelling; An Introduction. Cambridge University Press, Cambridge 999. [8] D. Greenhalgh: Stochastic Processes in Epidemic Modelling and Simulation. In: Handbook of Statistics 2 (D. N. Shanbhag and C. R. Rao, eds.), North Holland, Amsterdam 23, pp [9] J. Hurt: Mathematica R program for Kermack McKendrick model. Department of Probability and Statistics, Charles University in Prague 25. [] O. Kallenberg: Foundations of Modern Probability. Second edition. Springer Verlag, New York 22. [] D. G. Kendall: Deterministic and stochastic epidemics in closed population. In: Proc. Third Berkeley Symp. Math. Statist. Probab. 4, Univ. of California Press, Berkeley, Calif. 956, pp [2] W. O. Kermack and A. G. McKendrick: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London Ser. A 5 (927), [3] N. Kirupaharan: Deterministic and Stochastic Epidemic Models with Multiple Pathogens. PhD Thesis, Texas Tech. Univ., Lubbock 23. [4] N. Kirupaharan and L. J. S. Allen: Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality. Bull. Math. Biol. 66 (24),

20 44 J. ŠTĚPÁN AND D. HLUBINKA [5] L. C. G. Rogers and D. Williams: Diffusions, Markov Processes and Martingales. Vol. : Foundations. Cambridge University Press, Cambridge 2. [6] L. C. G. Rogers and D. Williams: Diffusions, Markov Processes and Martingales. Vol. 2: Itô Calculus. Cambridge University Press, Cambridge 2. [7] J. Štěpán and P. Dostál: The dx(t) = Xb(X)dt + Xσ(X) dw equation and financial mathematics I. Kybernetika 39 (23), [8] J. Štěpán and P. Dostál: The dx(t) = Xb(X)dt + Xσ(X)dW equation and financial mathematics II. Kybernetika 39 (23), [9] R. Subramaniam, K. Balachandran, and J. K. Kim: Existence of solution of a stochastic integral equation with an application from the theory of epidemics. Nonlinear Funct. Anal. Appl. 5 (2), Josef Štěpán and Daniel Hlubinka, Department of Probability and Statistics, Faculty of Mathematics and Physics Charles University, Sokolovská 83, Praha 8. Czech Republic. s: stepan@karlin.mff.cuni.cz, hlubinka@karlin.mff.cuni.cz