Some remarks on Bartoszyński's rabies model

Some Remarks on Bartoszyirski’s Rabies Model* WOLFGANG

J. BUHLER

AND GUNTER KELLER Fachbereich Mathemaiik der Johannes-Gutenberg Saarstrasse 21, Maim, Germany

Received 2 7 December

Uniwrsitat

1977

ABSTRACT In this paper human organism a straightforward

we show that Bartoszynski’s model for the development of rabies in the is in essence a branching process model, and we contrast this model with branching process approach. In this way we are able to simplify some of

the theoretical discussion and give slight improvements the risk of developing rabies after a bite.

1.

for some inequalities

relating

to

INTRODUCTION

Bartoszynski [l] has presented a model to describe the development of rabies in a human organism after infection by a rabid animal. This model describes the population of rabies virus in the body as a stochastic process {X(t), t b 0}, whose realizations are continuous from the right and decrease exponentially with decay parameter c between jumps. These jumps occur at random times T,, T2,. . . such that the conditional probability for a jump occuring between times t and t + h, given the development of X up to time t, is given by uX(t)h + o(h). Jump sizes are independent and identically distributed with distribution function H. Some of Bartoszynski’s results were derived under the additional assumption (A) that the jump size distribution is degenerate at r. The occurence of recognizable symptoms of the disease (usually leading to death) is modeled by the passage of X(t) to an upper boundary A. The discussion in [l] centers around the probability *This paper was prepared for publication with partial support of the Stiftung Volkswagenwerk and, during the first author’s stay at the Statistical Laboratory, Univ. of California, Berkeley, with the partial support of USPHS NIH ROl ES01299-13. MATHEMATICAL

BIOSCIENCES

0 Elsevier North-Holland,

Inc., 1978

273

39,273-279(1978) 0025-5564/78/0039-0273SO1.75

214

WOLFGANG

J. BtiHLER

AND

GihTER

KELLER

of this event and the time of its occurence, if indeed it does occur. The process X(t) in many ways behaves quite analogously to a Markov branching process. This analogy is described in more detail in [4]. One of the purposes of the present note is to display this analogy: to show that the Bartoszynski process can be obtained by a limiting procedure from a Markov branching process and that it is indeed a Markov branching process with continuous state space (C.B. process) in the sense of Jiiina [3] and Lamperti [5, 61. In our view it is really the branching property that makes the model mathematically tractable. Our second purpose is to simplify some of the theoretical discussion and to improve the Bartoszynski bounds for the probabilities of nonoccurence of symptoms. For a detailed discussion of the model and its implications we refer to Bartoszynski’s original publication [ 11. 2.

THE MODEL

AS A C.B. PROCESS

Markovian continuous state space branching processes were first introduced by Jii-ina [3]. For a detailed discussion see Lamperti [5, 61. It is clear that the only condition that needs verification for the Bartoszynski model to be a C.B. process is the branching condition. Thus let {X,(t)}, {X2(t)} and {X3(t)} be three independent Bartoszynski processes involving the same parameters; let Xi(O)= zi, i = 1,2,3, with z, + z2= zs; and denote X,(t)+X,(t)= U(t). If T,;, i= 1,2,3 is the time of the first jump in the ith process, then the process { U(t)) jumps first at f, =min( T,,, Ti2). We then have: (a) Since z, e -E’+z2e-C1=z 3 e-c’ > the value of U(t) given fp, > t coincides with the value of X,(t) given T,, > t. (b) Denoting by P, (or E_,) probabilities (or expectations) under the condition X (0) = z, we easily see that P,(T,>r)=exp(

-oi’ze-“di)=exp(

-$(1-e-“))

Therefore

P(f,>tlX,(O)= z,,X*(0)=z*)=PI,(TLI>~)Pz*(TL2>~) = exp

-~(~-e~“))exp( ( - Q(zi;z2)

=exp 1 =

pz,(Tn > t).

(1 _e-ct))

-?(I-e-“))

(2.1)

SOME REMARKS ON BARTOSZYIiSKI’S

(c) Clearly the distribution { U(t)} is the same.

RABIES MODEL

of jump

sizes in the processes

275 .(X3(t)} and

Using the strong Markov property in addition to (a), (b) and (c) yields the equivalence of the processes {X3(t)} and {u(t)}, i.e., the branching property. 3.

THE APPROXIMATION

For each n > a/c we define a Markov branching process { U,(t), t > 0}, whose family size distribution is given by p&“)= 1 -a/q p$“)=O, pi’;‘, = (l-p&“)) [H(j/n)-H((j--1)/n)], j=l,2,..., and whose branching rate is b, = c/pp). Let U,,(O)= [nz], and denote the time of the first positive jump in this process by T,@‘j. Then we have PROPOSITION

3.1.

The sequence of processes {( 1/n) U, (t)) converges to the Bartoszytiski process {X (t)} in the sense of convergence of the finite dimensional distributions. Proof: Again by the Markovian argument it suffices to show that (a) things converge properly before the first jump, (b) the distribution of the time of the first jump converges and (c) the distribution of jump size converges. To show (a), observe that under the condition T,c”j> t any individual in the &-process can either be alive at time t [with probability e-“‘1 or dead without descendants [with probability pp)( 1 - e-“‘)I. This makes plausible a fact which can be easily demonstrated, namely that under T$“) > t the random variable U,(t) is binomially distributed with parameters e-b”‘{P6”)(l-e-b,‘)+e-b.t}-i and [nz]. From this (a) follows by seeing that all moments have the appropriate limit behavior. For (b), take the limits in P(f, > t/U,(O) = [nz]) = ((1 - a/cn)(l - e-c’(l--(I/cn)) + e-C’(‘--a/c”)}tnrl, whose proof will be deferred to the next section (Proposition 4.1). Finally, (c) is clear from the definition of the pi’“).

4.

THE BRANCHING

PROCESS

MODEL

We compare now the Bartoszynski process with a continuous time Markov branching process, which (without danger of confusion) we shall also denote by {X(t)}. Let its branching rate be b and its family size be distributed according to a probability vector p0,p2,p3,. . . . By condition (A) we shall now mean that po+pr+ 1= 1. The results derived will be mostly quite analogous to those of Bartoszynski. PROPOSITION

4.1

PZ(T,>t)=[pO+(l-pO)e-b’]Z.

(4.1)

276

WOLFGANG J. BiJHLER AND GiJNTER KELLER

Proof. The expression in the bracket is just the probability that a single virus will die or survive time t (thus not causing a positive jump up to time t). Using (2.1) or (4.1) one can give an explicit expression for the conditional density f, of T,, given T, is finite. PROPOSITION

4.2

d zP,-,(T,x)$‘,(T,Gt) f,(t)=

P,(T,
(4.2)

’

both for the Bartoszyriski process and for our model. Proof

Observe P, (T, > t) = { P,( T, > t)}‘, and differentiate.

The conditional distribution truncated at z in the Bartoszyriski case can be derived more easily in our case. Noting that z - X ( T, - 0) = y means that there are exactly y virus deaths before the first positive jump we obtain the corresponding result by replacing the exponential with a geometric. Using this fact and assumption (A), observe that X (T, + 0) > t if and only if either z < r, with T, < co, or z > r, with T, < co, and there are at most r - 1 deaths before the first positive jump. We have thus shown PROPOSITION

4.3

(I-Po)PB P,(z-X(T,-0

)-ylT,
if

y
1 -PO’ 0

P,(T,z)=

if

l--PO’

if

z>r,

I-P,’

if

z
3

(4.3)

y>z, (4.4)

Denoting by m(z) and u(z) the conditional expectation and variance of z -X( T, - 0) given T, < co, both can be evaluated by differentiating the probability generating function corresponding to (4.3). While the result is quite analogous to that in [l], this procedure of obtaining it seems somewhat simpler. PROPOSITION

4.4

m(z)=

PO

--21-p,” 1 -Po

PO u(z) = 2 (1 -Po)

PO’

-z2

(4.5) PO’

(1 -Po,Y

*

(4.6)

SOME REMARKS ON BARTOSZtiSKI’S

277

RABIES MODEL

As z becomes large these expressions tend to the mean and the variance of the geometric distribution, as should be expected. While estimates for the expected time L(z) of the first jump analogous to those of BartoszyIiski have been obtained [4], we shall not reproduce them here, as we do not need them to show that X(t) can almost surely have only finitely many jumps in any finite interval: this fact is well known in our present setup (see e.g. Harris [2]). We shall however give a related result using a different method. PROPOSITION

4.5

lim L(z) =O. r+m

(4.7)

Proof: Under X(0) = z, let U,, U,, . . . be the waiting times for the first event, between the first and second events.. . in the branching process. Then, if T, < co, we see that T, = U, + U, + . . . + Uy+ ,, where Y has the distribution given by (4.3) and the q are independent of each other and of Y, having exponential distributions with parameters (z -j)b. Thus

=E

b(z_l)

bz+ 1

+..-

1

b(z-

+

1

Y)

1 1 T1
<;E(slT,
=-l y+l =- 1 c -. b

y=o

z---Y

which goes to zero by dominated 5.

BOUNDS

PJ(l-P0) I-p,’

’

convergence.

FOR THE PROBABILITY

OF SYMPTOMS

Both [I] and [4] investigate in some detail the probability W(z, t) that the critical level A is not reached up to time t and the probability N(z) that this bound will never be reached. Keller [4] notes that both his and Bartoszyriski’s expressions [valid under assumption (A) and for t sufficiently large for the lower bound of W(z,t)] can be written as P,(T,>t) Pz(T,>t)+P,(X(T,+O)>z,

T,
278

WOLFGANG J. BiJHLER AND GiJNTER KELLER

This can be strengthened PROPOSITION In

to

5.1

both models we have, for all t > 0, even without assumption (A),

Pz(T,>t)

W(z, t) > P,(T,>tor

N(z)’P,(T,=w Proof. Observing lower than z within obtain

T1z)’

Pz(T,=w) or T,z)’

(5.1)

(5.2)

that the probability of not reaching A from a value a time shorter than t must be at least W(z,t), we

W(z,t)>P(T1>t)+P(T, t implies X(T, +0) < z, which is the case for large z and t if the jump size is bounded. In the Markov branching process, however, it is always an improvement. The two events, the probability of whose union appears in the denominator of (5.2), are always mutually exclusive. Thus this inequality has not been sharpened; however, we have removed the need for assuming (A). The bounds which follow from (5.2) if we assume (A) can alternatively be obtained by a simple argument. We shall give the version for the Bartoszynski process: Let r < z and to be such that z exp( - ct,-J+ r = z; define i=, - T, and for i > 1, z = q - q_ ,, i.e., the c are the waiting times between the successive jumps, putting z = cc if T, = co. Then

P=(~~+,>tolt,<~, P,(ZXo), and

Pz(fi+,= colt,,<

T,
t~<~p.<<)>P,(~~=w),

SOME REMARKS

ON BARTOSZtiSKI’S

since the condition

RABIES

MODEL

279

implies that X ( Fk + 0) < z. Thus

K=O which is (5.2). The upper bounds for FV(z,t) and N(z) given in [I] and [4] can be seen as expressing the fact that under assumption (A), if the level A shall not be reached, then the population size must get below A -I before the first (positive) jump. We shall now restate these bounds. Obviously, in the branching process situation the total number of deaths before the kth jump has to be at least an additional (k - I)r. The improvement reached in this way for k = 2 is also given below. PROPOSITION

I2

Under assumption (A) we have N(z)$exp(

(5.3)

- :(z+r-A)]

for the Bartoszyriski model and N(z)

& pg+r+‘--A

(5.4)

for our model. The latter bound can be improved to

N(z)

< p;+r+l-A

1

PO’+’+(r+

l)p,‘(l

+Po)].-

(5.5)

REFERENCES 1. R. Bartoszyliski, On the risk of rabies, Math. Biosci. 24, 355-377 (1975). 2. T. E. Harris, The l%eoty of Branching Processes, Springer, Berlin, 1963. 3. M. Jiiina, Stochastic branching processes with continuous statespace, Czechoslowk. Mufh. J. 8, 292-313 (1958). 4. G. Keller, Der Bartoszy&i-Prozess im Vergleich mit anderen stochastischen Prozessen, Diplomarbeit, Mainz, 1977. 5. J. Lamperti, The limit of a sequence of branching processes, Z. Wuhrscheinfichkeitstheorie und wnv. Gebiete 7, 27 l-288 (1967). 6. J. Lamperti, Continuous state branching processes, BUN. Amer. M&h. Sot. 73, 382-386 (1967).