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The exact solution of the general stochastic rumour
MATHEMATICAL COMPUTER MODELLING
PERGAMON
Mathematical
and Computer
Modelling 31 (2000)
289-298 www.elsevier.nl/locate/mcm
The Exact Solution of the General Stochastic Rumour C. E. M. PEARCE Applied
Mathematics
Department,
Adelaide
SA 5005,
University
of Adelaide
Australia
Abstract-A characterization is given of the complete time-dependent evolution of a general stochastic rumour which includes the two Daley-Kendall models and the Maki-Thompson model as special cases. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Stochastic
rumour, Epidemics,
Daley-Kendall
models, Maki-Thompson
model.
1. INTRODUCTION The mathematical
theory
of epidemics
has proved a fertile area.
The need for several substantial
literature reviews had already been manifested two decades ago. We note Dietz [l] (who mentions a number of earlier reviews), Bartholomew [2], Bailey [3], and Mollison [4]. A number of books also have been written in the area. We note [5-8] as well as the volume of papers edited by Gabriel
et al. [9].
A number
of these
surveys
also address
rumours.
Rumours
bear
an immediate
superficial
resemblance to epidemics, and the two are often thought of together. Both are processes operating through individuals transferring amongst subpopulations of susceptibles, infectives and immunes, or removed individuals, who are sometimes referred to as susceptibles, spreaders, and stiffers in the context individuals number.
of rumours. become
However,
immunes
the differences
through
In the rumour-mongering
common encounter of spreaders rate of the latter is proportional
death,
process,
isolation,
are significant. or recovery
the production
In an epidemic, at a rate proportional
of immunes
infectious to their
occurs either through
or through the encounter of a spreader with an immune. to the product of the numbers of spreaders and immunes,
the The while
that of the former, at least in the seminal model of Daley and Kendall [lo], is proportional to (r) where Y is the number of spreaders. Furthermore, in the basic Daley-Kendall model, the former
,
results in two members of the spreader subpopulation rather than one becoming immunes. These differences led Daley and Kendall to comment on the relation between epidemics and rumours: “Investigation soon showed the parallel to be a misleading one, and in fact the two phenomena could hardly be more different.” The literature for rumours is much
less developed
than
that
for epidemics,
partly
because
we now know of a large variety of distinct detailed mechanisms applying to different types of epidemics and also perhaps because of a perceived disparity in importance between epidemics The author thanks the referees for useful comments. 0895-7177/00/L - see front matter PII: SO895-7177(00)00098-4
@ 2000 Elsevier Science Ltd. All rights reserved.
Typeset
by J&G-WJ
C. E. M. PEARCE
290
and rumours to the human condition. Further, production of immunes makes the mathematical of epidemics.
Thus, and
although
Siskind
[ll]
hitherto
been resolved.
The rumour
Gani
of rumour model
politics,
earlier literature
it is generally
has concentrated to their
warfare, recognized
processes,
model
on rumours that
versions.
of Daley
in the tide of human
structure
and
and Kendall
spreader
and a susceptible
also proposed resulting
of Daley and Kendall
Daley and Kendall
were unable
approximations
a more elaborate in the spread
of the number
of contact
the rumour. See also [24]. A simplification of Maki and Thompson immune
to be produced
has not
a significant
model
from the encounter
events
to deterministic [lo], the literature
to find an exact solution
via a new technique
that
[20] gives an elaboration
allowing
of the rumour
unity and for a contact between two spreaders resulting which is distributed over the integers 0, 1, 2. Dunstan which keeps track
Kendall
ago by
models (see [14-191).
relevant ideas. Watson [21] has given a rigorous treatment of the Daley-Kendall under the condition that the initial proportion of spreaders is not negligible. complemented by Pittel [22] w h o examined the case when the initial proportion small. Daley
years
alfairs, playing
is lost by going
of arbitrary constants. Barbour
of the di&sion
thirty
makes use of deterministic
important
and since the publication
on stochastic
was solved
and in the career and social life of the individual.
but were able to derive excellent
called the ptinciple
epidemic
stochastic
is, however, far from unimportant
As with epidemics, versions
stochastic
[12,13], the paradigm
role in the stock market, However,
the general
the relative complexity of the process of the analysis of rumours more difficult than that
approximation This was later of spreaders was
for a contact
with
they of the
involving
a probability
a
less than
in a number of removals, the value of 1231 has studied a deterministic model
a spreader
is removed
from the initiators
of
[25] (see also [6, Chapter 21) provides for only one of two spreaders. This leads to a significantly easier
analysis, comparable with that of epidemic processes. With this restriction, Sudbury [26] was able to give a rigorous derivation of a result of Daley and Kendall relating to the proportion of the population
never hearing
a rumour.
See also [21]. Using a martingale
method,
Lefevre and
Picard [27] have been able to characterize for this class of rumour process the distribution of the number who ultimately hear the rumour. This they do in terms of Gontcharoff polynomials. However,
they note that the double
removal transition
in the Daley and Kendall
the application of their martingale method to that situation. In this paper, we propose a generalization of the more elaborate
stochastic
model precludes rumour
model
of
Daley and Kendall. Our generalization includes the Maki-Thompson model, which is actually a degenerate limiting case of that Daley-Kendall model. In Section 2, we set up the model, formulate the basic equations, and proceed with some preliminary discussion. In Section 3, we make a reduction of the equations and identify two cases, the general case (which is treated in Section 4) and a degenerate case (which is treated in Section 5). We address the determination of the complete time-dependent behaviour of the process in both cases. The arguments
of Sections
4 and 5 are related
to the treatment
by Gani
[13] of the general
stochastic epidemic, though that treatment is somewhat simpler, reflecting the additional complexity of rumour models. In particular, we need to make use of a critical parameter < to effect our solution. In an epidemic context, this parameter can be taken as zero.
2. THE
MODEL
We assume a closed population of N + a -t b individuals with homogeneous mixing. At time t 2 0 there are S(t) individuals who are ignorant of the rumour, I(t) who are spreading the rumour, and R(t) individuals who have heard the rumour but have ceased to spread it, so that S(t) + 1(t) + R(t) = N + a + b. Here, we suppose
N = S(O), a = I(0) 2 1, and b = R(0).
General Stochastic
The interaction
of a susceptible
with probability disseminate
p.
the rumour
(with probability
with probability
one of each is assumed the further
parameter
q = (ql + q2)/2.
the product
involving
can be taken
effect.
of size Y, it is proportional
unity. The parameter
By suitable
P -
(1 -P)
their
more sophisticated
q2 = 0. For nonnegative
integers
i, j
or between
Kolmogorov
and
we require p > 0 and q + T > 0.
subpopulations
is proportional
involving
[lo], the constants
of proportionality
of encounters
of an arbitrarily can be chosen as
for 0 < i 5 N.
equations
for the process
Pti, j, t)yj
P(i,j
+2,t)
1 j + 1, t)
(
P(i, j + 1, t) >
P(i,j,t),
can be expressed
in terms
of the generating
functions
as
+ [r(N+a+b-i
I
+ (q - T)Y
- l)(l
-y)
t)
&fi(Y,
= p(i +
i-(q l)Yz~.f~+l(Y~ S)+ (1 - y) iq2
+ a + b - i - l)(l
- y)
-
O
-piy]-$f,(y,t),
The solution procedure will be effected in terms of Laplace transforms. Laplace transform of fi(y, t) for Res 2 0. Then, for 0 5 i < N we have
+[r(N
j+l 2
+qi
1
$fiht>=di + l)w2~k+l(Y,t) +(1- Y) [ $2
6N,iya
at time t 1 0.
will then be
+rj(N+a+b-i-j) relations
and j spreaders
or if either i or j is negative.
+ a + b - i - j - l)P(i,
above
while
(2.1)
j+a 2
[pij + 29(32) The
model,
- o) with 0 < p 5 1 and 0 < IY 5 1 yields model arises with p = q1 = T = 1 and
that there are i susceptibles
( + r(j + l)(N
to
two individuals
i+jjN+u,
gP(i,j,t)=p(i+l)(j--l)P(i+l,j-l,t)+q2
sf,*tYTs) -
in places to use
with
P(i, j, t) as zero if (2.1) is not satisfied
fi(y, t) := Cj2c
two susceptibles,
choice of the unit of time, this constant
q1 = [l - (1 - ~)~]2a(l model. The Maki-Thompson
by P(i, j, t) the probability
The forward
the rumour the spreader
convenient
of the frequency
T = pa,
i
We interpret
(i) . A s in
to
converts
while for transitions
to
ceasing
to disseminate
choices p = q2 = T = 1 with q1 = 0 give the basic Daley-Kendall 2 I a2,
q2 =
we denote
processes,
of two different
a spreader
individuals
with a removal
of two immunes,
To avoid trivial members
becoming
in both
It will be notationally
as the same, each being a measure
chosen pair of individuals.
in the susceptible
can result
of a spreader
of the sizes of the two subpopulations,
of a subpopulation
291
42) or in one only ceasing
T. The interaction
to be without
The rate for interactions
results
of two spreaders
q1 5 1 - q2). The interaction
(with probability into a removal
and a spreader
The interaction
Rumour
piy]$f:(y,
Denote
T)y -$fl(YT 1
by f,*(y, s) the
s)
(2.2)
s).
At this stage we note that since fT(y, s) is a polynomial in y of degree N + a - i, the functions f; To begin the procedure, take i = N, so that (2.2) can be obtained by a downward recursion. gives Sfifv(Y,
3) - Ya = (1 - Y>
C. E. M. PEARCE
292
Substituting My,
s) = 2 olv,e(s)yye e=o
provides the recursive relation [S+(q+T)e(~-l)+{r(a+b-l)+pN}e]
eIV,(+[4(~+l)(r-_qr)
-(C+1)7(a+b-l)]an,e+l OIe
- $L?(~ + 2)(e + l)cN,e+z = &,e,
which can be used for the successive determination of the functions aN,e(s) for C = a, a - 1, . . . , 0. On substituting a+1 fi;-r(y, s) = C alv-r,e(s)ye e=o in (2.2) with i = iV - 1, we can now find the coefficients aN_r,e(s), and so on. While straightforward, this procedure is not very insightful. In the following sections, we proceed to a solution in a more structured form. 3. REDUCTION
OF THE
EQUATIONS
First, we encapsulate the functions ft(y, s) as an (IV + 1)-column vector
WY,s>:=(fi;(Y> s),fi;-l(Y,
s>, .** fo*(Y, s)>‘. 7
We denote by A = A(y) the (N + 1) x (IV + 1) matrix
where ,f$ = r (N + a + b - 1 - i) (1 - y ) - piy and ^(i = piy2, and define the (N + 1)-element column Vector EN+1 as ,?++I = (l,O,. . . ,o)T. Equation (2.2) can now be written as (1 - Y)
Differentiation
$2 + (4 - T-)Y] &F
+ A’F
ay
- SF = -$EN+~.
(3.1)
C times with respect to y at any point y = < provides
v-r> [’2Q2+(q-r)S Fe+2+
I[{
A(t)
+e
1 0:
(r - !I)1 + eA’(c) - ~1 Fe +
541
--T
A”(E)Fe_l
-2(q-~-)[ =
for C 2 0, where Fe 3 Fe([, S) := $F(Y+)
v=E
.
>I I
~~~~
General Stochastic
The argument so that
Here,
now branches
the coefficient
the three
addition vector.
Fe+2
= Ce(We+l
coefficient
matrices
The matrix
entries,
Then,
s)Fe+ Le(Pe-1
+ De(E,
E can be chosen
(3.2) can be expressed +
term
as
He(J)-
are all (N + 1) x (N + 1) and lower triangular, He is an (N + 1)-element
Le has in column
for C = 0,1, so there is no need to define Fe-1 in (3.3) for C = 0..
in the sequel we put F-1 = 0.
(3.3) may be written
more conveniently
as
4 L 0,
+e+i = Be$e + he, where &+i
293
first that a (real) number
and the inhomogeneous
Le vanishes
for convenience,
Equation
into two cases. Suppose
g(c) of F !+2 in (3.2) is nonvanishing.
zero diagonal
However,
Rumour
and he are the (3N + 3)-column
(3.4)
vectors He 0
he=
#Je+i =
,
0 0 and the (3N + 3) x (3N + 3) matrix
Be is given by
Be z B~([,s) :=
Equation
whereas
(3.4) may be iterated
subsequently
the empty
to give
product
is interpreted
as the identity
matrix
and we adopt
the
convention fIB+fi j=o for continued
matrix
products.
hold also for e = -1. In the next section,
1 Bj := BeBe_1..
. Bo,
j=O
By defining
empty
matrix
sums as zero, we can extend
(3.6) to
we find $0 and use the value found to solve for F(y, s).
This leaves the degenerate case in which g(c) vanishes identically, that is, where q2 = 0 and r = q both occur. This case includes notably the model of Maki and Thompson. These two conditions imply that the coefficient q + T > 0 and q = r yield T > 0. Since of the section that for (real) 5 suitably Then, for each e satisfying 0 5 fJ < N
of Fe+1 in (3.2) reduces to A(c). Further, the conditions p, r > 0, we have from the form of A as given at the head chosen, A(<) is invertible. Suppose such a choice is made. + a, (3.2) can be cast as
Fe+1 = ue(C, Parallel are (N+
s)Fe
+ Ve(SF’e-1
+ we(E).
to the main case, we have VI = Vi = 0 and take F-1 = 0. Also, the coefficient matrices 1) x (N+l) and lower-triangular and Ve has, in addition, its entries on the main diagonal
all equal to zero. More compactly,
we set
[F;l] =[? !] [iE]+[?I)
C. E. M. PEARCE
294
or OIZ
tie+1 = KelCle+ we, Relation
(3.6)
(3.7) may be iterated to give
As for the main case, this holds also for e = -1. In Section 5, we consider the determination
of $0 = (Fo,O)~ and the consequent solution for
F(K s).
4. THE
GENERAL
SOLUTION
We now pursue the solution in the general situation. We shall need the following preliminary result, which is a generalization of a result of Roughan and Pope [28]. PROPOSITION 1. Suppose’s
square matrix M is partitioned
where
,. I: s(U)
191
s(i,j)
=
...
0
s(Q)
s(Q)
%l
22
as
0
*.
-
’
’
.
*.
..
s(h) ’ s(i’j; O .$$I L,l **. L&-l ) -
Then,
IMI = fi
(A(+)/ ,
i=l
where
s+l) A&j)
=
‘h3
5992)
...
bj
~324 273
PROOF. By paired row interchanges
and column interchanges,
A&l) A&‘)
A03 A(?,?)
A(Lv1) A@3
lIi,j<_L.
1
*.
M can be rearranged as
... ..,
A&L) A&L)
..,
A(L.L)
1
Because the formation of N from M involves an even numbe! qf interchanges, IMI = INI. F’rom the structure of the matrices S(“$j), we have A(“*j) = 0 for j 5 i, so N is block lower-triangular, giving the desired result. I We are now ready to give our main result. It will be convenient to write (elm to denote the truncation of a matrix to its leading m x m submatrix and [.lm to denote the truncation of a column vector to its first m rows. With a slight abuse of notation, we may denote by Y* the matrix formed from a matrix Y by deleting all the off-diagonal entries and also denote by X* the matrix formed from a block matrix X with entries Yi by replacing each Yi by Y;.
General Stochastic
THEOREM
295
Rumour
1. Suppose thatt isred and such that g(e) # 0. Then, for all s with R,e s > 0 and
Im s # 0, and for all real s suflkiently large,
(4.1) 2N+2
is invertible, with inverse B(<, s), say. For such < and s,
(4.2) and the time-dependent evolution of the rumour process is given by
F(Y,
C4e3)
s) = Fo+
where 40 is given by (4.4) PROOF. Any product of lower-triangular matrices is also lower-triangular, so since all the block entries in Be+l are lower-triangular, the same must, therefore, hold for the block entries in
X:=
By Proposition
N+a n B&s). j=O
1, det{X)zlv+z
= det({X}2N+2)*
= det{X*}zlv+s.
Further, Ni-a
x*= j=O n q(t,s), and since Ls = 0, we have {x*}ZN+Z = Hence,
det{X*}m+s = and again invoking Proposition 1,
Thus, det{X}2N+z or
is nonvanishing if and only if (De)i,i # 0 for 1 5 i 5 Nf
2e0
2
1 and 0 < e < iV +a,
(T--)-e[T(N+a+b-i-i)+pi]-sfo,
for 0 5 i < N and 0 < e 5 N + a. The invertibility part of the enunciation follows.
C. E. M. PEARCE
296
Since fT(y, s) is a polynomial
in y of degree N + a - i, the vector 0
4N+a+l
=
0
[ FN+a Further,
~~50assumes
Restricting
(2N + 2) subvector
Solution (4.2) for 40 follows. Define 4-1 = (Fo,O,O) T. Then,
on both sides gives
we have component-wise
= 1
the three Taylor
expansions
from (3.6) to obtain
du Truncating
1
form (4.4). Hence, we have from (3.6) that
to the leading
We may substitute
~PJ+~+I is of the form
to the leading
REMARK 1. In the study
N + 1 subvectors
on both sides yields (4.3).
of Daley and Kendall,
q2 > 0. In this event, < = 0 is an obvious
choice which suffices to make g(c) # 0. If q2 = 0 and q # T, any choice of < # 0,l
5. THE We now proceed
I
DEGENERATE
simple
suffices.
CASE
to our second result.
that qz = 0 and q = r and 5 (real) is chosen so that A(<) is invertible. Then for all s with Res > 0 and Ims # 0, and for all real s sufficiently large,
THEOREM 2. Suppose
N+l
is invertible,
with inverse
K([, s), say. For such c and s
(5.1)
General Stochastic
and the time-dependent
F(Y,s> = PROOF.
An argument KIy$ Kj)N+l from (3.2) that
evolution of the rumour process is given by
N+a+l (y
1 I! e=o
-
,c)[
({tjK~}N+lfi+~{j~lK
to the first N + 1 rows of this equation
parallel
to that
is nonsingular
involving
The statement
about
invertibility
ity, (5.2) gives (5.1). Further,
lower-triangular
matrices
in Theorem
is nonzero.
Since
1 shows that
q =
T,
we have
[!A’(<) - sI] .
follows readily
we have the paired
from the form of A. In the event of invertibilTaylor expansions
[I 1 F(Y,
s>
YF(u,s)do F
Substitution
to obtain
if and only if flz’(Ue)i,i Ue = -A([)-1
=
e=o
from (3.8) leads to F(Y, ii c
and truncation
E=
297
Since FN+~+I = 0, we have
We now restrict
REMARK
Rumour
2.
’ F(u,
s) s) du
1 =
of this result In the degenerate
0 is also inadmissible
to the first N + 1 rows completes case, < = 1 is incompatible
the proof.
with A(<) being invertible.
I The choice
if a + b = 1.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
K. Dietz, Epidemics and rumours: A survey, J. Roy. Statist. Sot. Ser. A 130, 505-528, (1967). D.J. Bartholomew, Stochastic Models for Social Processes, Second Edition, John Wiley, London, (1973). N.T.J. Bailey, The Mathematical Theory of Infectious Diseases, Charles Griffin, London, (1975). D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Sot. Ser. B 39, 283-326, (1977). N.T.J. Bailey, The Mathematical Theory of Epidemics, Charles Griffin, London, (1957). J.C. fiauenthal, Mathematical Modelling in Epidemiology, Springer-Verlag, (1980). H.A. Lauwerier, Mathematical models of epidemics, In Mathematical Centre Tracts, Vol. 138, Mathematical Centre, Amsterdam, (1981). N.G. Becker, Analysis of Infectious Disease Data, Chapman and Hall, London, (1989). J.-P. Gabriel, C. Lefkre and P. Picard, Stochastic processes in epidemic theory, In Lecture Notes in Biomathematics, Vol. 86, Springer-Verlag, (1990). D.J. Daley and D.G. Kendall, Stochastic rumours, J. Inst. Math. Applic. 1, 42-55, (1965). V. Siskind, A solution of the general stochastic epidemic, Biometrika 52, 613-616, (1965). J. Gani, On a partial differential equation of epidemic theory I, Biometrika 52, 617-622, (1965). J. Gani, On the general stochastic epidemic, In Proc. gh Berkeley Symp. on Math. Stat. and Prob., Vol. 4, (Edited by L.M. LeCam and J. Neyman), pp. 271-279, Univ. of Calif. Press, Berkeley, (1967).
298
C. E. M. PEARCE
14. A. Bapoport and L.J. Hebhun, On the mathematical theory of rumour spread, Bull. Math. Biophys. 14, 375-383, (1952). 15. A. Ftapoport, Spread of information through a population with socio-structural bias: I. Assumption of transitivity, Bull. Math. Biophys. 15, 523-533, (1953). 16. A. Hapoport, Spread of information through a population with socio-structural bias: I. Various models with partial transitivity, Bull. Math. Biophys. 15, 535-546, (1953). 17. H.G. Landau and A. Ftapoport, Contribution to the .mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population, Bull. Math. Biophys. 18, 173-183, (1953). 18. H.D. Landahl, On the spread of information with time and distance, Bull. Math. Biophys. 15, 367-381, (1953). 19. D.J. DaIey, Concerning the spread of news in a population of individuaIs who never forget, Bull. Math. Biophys. 29, 373-376, (1967). 20. A.D. Barbour, The principle of the diffusion of arbitrary constants, J. Appl. Prob. 9, 519-541, (1972). 21. Ft. Watson, On the size of a rumour, Stoch. Proc. and Applic. 27, 141-149, (1988). 22. B. Pittel, On a Daley-Kendall model of random rumours, J. Appl. Prob. 27, 14-27, (1990). 23. Ft. Dunstan, The rumour process, .I. Appl. Prob. 19, 759766, (1982). 24. V.R. Cane, A note on the size of epidemics and the number of people hearing a rumour, J. Roy. Statist. Sot. Ser. B 28, 487-490, (1966). 25. D.P. Maki and M. Thompson, Mathematical Models and Applications, Prentice-Hall, %nglewood Cliffs, NJ, (1973). 26. A. Sudbury, The proportion of the population never hearing a rumour, J. Appl. Prob. 22, 443-446, (1985). 27. C. Lef&vre and P. Picard, Distribution of the final extent of a rumour process, J. Appl. Prob. 31, 244-249, (1994). 28. M. Floughan and K. Pope, The determinant of a triangular-block matrix, SIAM Review 38, 513-514, (1996).
PERGAMON
Mathematical
and Computer
Modelling 31 (2000)
289-298 www.elsevier.nl/locate/mcm
The Exact Solution of the General Stochastic Rumour C. E. M. PEARCE Applied
Mathematics
Department,
Adelaide
SA 5005,
University
of Adelaide
Australia
Abstract-A characterization is given of the complete time-dependent evolution of a general stochastic rumour which includes the two Daley-Kendall models and the Maki-Thompson model as special cases. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Stochastic
rumour, Epidemics,
Daley-Kendall
models, Maki-Thompson
model.
1. INTRODUCTION The mathematical
theory
of epidemics
has proved a fertile area.
The need for several substantial
literature reviews had already been manifested two decades ago. We note Dietz [l] (who mentions a number of earlier reviews), Bartholomew [2], Bailey [3], and Mollison [4]. A number of books also have been written in the area. We note [5-8] as well as the volume of papers edited by Gabriel
et al. [9].
A number
of these
surveys
also address
rumours.
Rumours
bear
an immediate
superficial
resemblance to epidemics, and the two are often thought of together. Both are processes operating through individuals transferring amongst subpopulations of susceptibles, infectives and immunes, or removed individuals, who are sometimes referred to as susceptibles, spreaders, and stiffers in the context individuals number.
of rumours. become
However,
immunes
the differences
through
In the rumour-mongering
common encounter of spreaders rate of the latter is proportional
death,
process,
isolation,
are significant. or recovery
the production
In an epidemic, at a rate proportional
of immunes
infectious to their
occurs either through
or through the encounter of a spreader with an immune. to the product of the numbers of spreaders and immunes,
the The while
that of the former, at least in the seminal model of Daley and Kendall [lo], is proportional to (r) where Y is the number of spreaders. Furthermore, in the basic Daley-Kendall model, the former
,
results in two members of the spreader subpopulation rather than one becoming immunes. These differences led Daley and Kendall to comment on the relation between epidemics and rumours: “Investigation soon showed the parallel to be a misleading one, and in fact the two phenomena could hardly be more different.” The literature for rumours is much
less developed
than
that
for epidemics,
partly
because
we now know of a large variety of distinct detailed mechanisms applying to different types of epidemics and also perhaps because of a perceived disparity in importance between epidemics The author thanks the referees for useful comments. 0895-7177/00/L - see front matter PII: SO895-7177(00)00098-4
@ 2000 Elsevier Science Ltd. All rights reserved.
Typeset
by J&G-WJ
C. E. M. PEARCE
290
and rumours to the human condition. Further, production of immunes makes the mathematical of epidemics.
Thus, and
although
Siskind
[ll]
hitherto
been resolved.
The rumour
Gani
of rumour model
politics,
earlier literature
it is generally
has concentrated to their
warfare, recognized
processes,
model
on rumours that
versions.
of Daley
in the tide of human
structure
and
and Kendall
spreader
and a susceptible
also proposed resulting
of Daley and Kendall
Daley and Kendall
were unable
approximations
a more elaborate in the spread
of the number
of contact
the rumour. See also [24]. A simplification of Maki and Thompson immune
to be produced
has not
a significant
model
from the encounter
events
to deterministic [lo], the literature
to find an exact solution
via a new technique
that
[20] gives an elaboration
allowing
of the rumour
unity and for a contact between two spreaders resulting which is distributed over the integers 0, 1, 2. Dunstan which keeps track
Kendall
ago by
models (see [14-191).
relevant ideas. Watson [21] has given a rigorous treatment of the Daley-Kendall under the condition that the initial proportion of spreaders is not negligible. complemented by Pittel [22] w h o examined the case when the initial proportion small. Daley
years
alfairs, playing
is lost by going
of arbitrary constants. Barbour
of the di&sion
thirty
makes use of deterministic
important
and since the publication
on stochastic
was solved
and in the career and social life of the individual.
but were able to derive excellent
called the ptinciple
epidemic
stochastic
is, however, far from unimportant
As with epidemics, versions
stochastic
[12,13], the paradigm
role in the stock market, However,
the general
the relative complexity of the process of the analysis of rumours more difficult than that
approximation This was later of spreaders was
for a contact
with
they of the
involving
a probability
a
less than
in a number of removals, the value of 1231 has studied a deterministic model
a spreader
is removed
from the initiators
of
[25] (see also [6, Chapter 21) provides for only one of two spreaders. This leads to a significantly easier
analysis, comparable with that of epidemic processes. With this restriction, Sudbury [26] was able to give a rigorous derivation of a result of Daley and Kendall relating to the proportion of the population
never hearing
a rumour.
See also [21]. Using a martingale
method,
Lefevre and
Picard [27] have been able to characterize for this class of rumour process the distribution of the number who ultimately hear the rumour. This they do in terms of Gontcharoff polynomials. However,
they note that the double
removal transition
in the Daley and Kendall
the application of their martingale method to that situation. In this paper, we propose a generalization of the more elaborate
stochastic
model precludes rumour
model
of
Daley and Kendall. Our generalization includes the Maki-Thompson model, which is actually a degenerate limiting case of that Daley-Kendall model. In Section 2, we set up the model, formulate the basic equations, and proceed with some preliminary discussion. In Section 3, we make a reduction of the equations and identify two cases, the general case (which is treated in Section 4) and a degenerate case (which is treated in Section 5). We address the determination of the complete time-dependent behaviour of the process in both cases. The arguments
of Sections
4 and 5 are related
to the treatment
by Gani
[13] of the general
stochastic epidemic, though that treatment is somewhat simpler, reflecting the additional complexity of rumour models. In particular, we need to make use of a critical parameter < to effect our solution. In an epidemic context, this parameter can be taken as zero.
2. THE
MODEL
We assume a closed population of N + a -t b individuals with homogeneous mixing. At time t 2 0 there are S(t) individuals who are ignorant of the rumour, I(t) who are spreading the rumour, and R(t) individuals who have heard the rumour but have ceased to spread it, so that S(t) + 1(t) + R(t) = N + a + b. Here, we suppose
N = S(O), a = I(0) 2 1, and b = R(0).
General Stochastic
The interaction
of a susceptible
with probability disseminate
p.
the rumour
(with probability
with probability
one of each is assumed the further
parameter
q = (ql + q2)/2.
the product
involving
can be taken
effect.
of size Y, it is proportional
unity. The parameter
By suitable
P -
(1 -P)
their
more sophisticated
q2 = 0. For nonnegative
integers
i, j
or between
Kolmogorov
and
we require p > 0 and q + T > 0.
subpopulations
is proportional
involving
[lo], the constants
of proportionality
of encounters
of an arbitrarily can be chosen as
for 0 < i 5 N.
equations
for the process
Pti, j, t)yj
P(i,j
+2,t)
1 j + 1, t)
(
P(i, j + 1, t) >
P(i,j,t),
can be expressed
in terms
of the generating
functions
as
+ [r(N+a+b-i
I
+ (q - T)Y
- l)(l
-y)
t)
&fi(Y,
= p(i +
i-(q l)Yz~.f~+l(Y~ S)+ (1 - y) iq2
+ a + b - i - l)(l
- y)
-
O
-piy]-$f,(y,t),
The solution procedure will be effected in terms of Laplace transforms. Laplace transform of fi(y, t) for Res 2 0. Then, for 0 5 i < N we have
+[r(N
j+l 2
+qi
1
$fiht>=di + l)w2~k+l(Y,t) +(1- Y) [ $2
6N,iya
at time t 1 0.
will then be
+rj(N+a+b-i-j) relations
and j spreaders
or if either i or j is negative.
+ a + b - i - j - l)P(i,
above
while
(2.1)
j+a 2
[pij + 29(32) The
model,
- o) with 0 < p 5 1 and 0 < IY 5 1 yields model arises with p = q1 = T = 1 and
that there are i susceptibles
( + r(j + l)(N
to
two individuals
i+jjN+u,
gP(i,j,t)=p(i+l)(j--l)P(i+l,j-l,t)+q2
sf,*tYTs) -
in places to use
with
P(i, j, t) as zero if (2.1) is not satisfied
fi(y, t) := Cj2c
two susceptibles,
choice of the unit of time, this constant
q1 = [l - (1 - ~)~]2a(l model. The Maki-Thompson
by P(i, j, t) the probability
The forward
the rumour the spreader
convenient
of the frequency
T = pa,
i
We interpret
(i) . A s in
to
converts
while for transitions
to
ceasing
to disseminate
choices p = q2 = T = 1 with q1 = 0 give the basic Daley-Kendall 2 I a2,
q2 =
we denote
processes,
of two different
a spreader
individuals
with a removal
of two immunes,
To avoid trivial members
becoming
in both
It will be notationally
as the same, each being a measure
chosen pair of individuals.
in the susceptible
can result
of a spreader
of the sizes of the two subpopulations,
of a subpopulation
291
42) or in one only ceasing
T. The interaction
to be without
The rate for interactions
results
of two spreaders
q1 5 1 - q2). The interaction
(with probability into a removal
and a spreader
The interaction
Rumour
piy]$f:(y,
Denote
T)y -$fl(YT 1
by f,*(y, s) the
s)
(2.2)
s).
At this stage we note that since fT(y, s) is a polynomial in y of degree N + a - i, the functions f; To begin the procedure, take i = N, so that (2.2) can be obtained by a downward recursion. gives Sfifv(Y,
3) - Ya = (1 - Y>
C. E. M. PEARCE
292
Substituting My,
s) = 2 olv,e(s)yye e=o
provides the recursive relation [S+(q+T)e(~-l)+{r(a+b-l)+pN}e]
eIV,(+[4(~+l)(r-_qr)
-(C+1)7(a+b-l)]an,e+l OIe
- $L?(~ + 2)(e + l)cN,e+z = &,e,
which can be used for the successive determination of the functions aN,e(s) for C = a, a - 1, . . . , 0. On substituting a+1 fi;-r(y, s) = C alv-r,e(s)ye e=o in (2.2) with i = iV - 1, we can now find the coefficients aN_r,e(s), and so on. While straightforward, this procedure is not very insightful. In the following sections, we proceed to a solution in a more structured form. 3. REDUCTION
OF THE
EQUATIONS
First, we encapsulate the functions ft(y, s) as an (IV + 1)-column vector
WY,s>:=(fi;(Y> s),fi;-l(Y,
s>, .** fo*(Y, s)>‘. 7
We denote by A = A(y) the (N + 1) x (IV + 1) matrix
where ,f$ = r (N + a + b - 1 - i) (1 - y ) - piy and ^(i = piy2, and define the (N + 1)-element column Vector EN+1 as ,?++I = (l,O,. . . ,o)T. Equation (2.2) can now be written as (1 - Y)
Differentiation
$2 + (4 - T-)Y] &F
+ A’F
ay
- SF = -$EN+~.
(3.1)
C times with respect to y at any point y = < provides
v-r> [’2Q2+(q-r)S Fe+2+
I[{
A(t)
+e
1 0:
(r - !I)1 + eA’(c) - ~1 Fe +
541
--T
A”(E)Fe_l
-2(q-~-)[ =
for C 2 0, where Fe 3 Fe([, S) := $F(Y+)
v=E
.
>I I
~~~~
General Stochastic
The argument so that
Here,
now branches
the coefficient
the three
addition vector.
Fe+2
= Ce(We+l
coefficient
matrices
The matrix
entries,
Then,
s)Fe+ Le(Pe-1
+ De(E,
E can be chosen
(3.2) can be expressed +
term
as
He(J)-
are all (N + 1) x (N + 1) and lower triangular, He is an (N + 1)-element
Le has in column
for C = 0,1, so there is no need to define Fe-1 in (3.3) for C = 0..
in the sequel we put F-1 = 0.
(3.3) may be written
more conveniently
as
4 L 0,
+e+i = Be$e + he, where &+i
293
first that a (real) number
and the inhomogeneous
Le vanishes
for convenience,
Equation
into two cases. Suppose
g(c) of F !+2 in (3.2) is nonvanishing.
zero diagonal
However,
Rumour
and he are the (3N + 3)-column
(3.4)
vectors He 0
he=
#Je+i =
,
0 0 and the (3N + 3) x (3N + 3) matrix
Be is given by
Be z B~([,s) :=
Equation
whereas
(3.4) may be iterated
subsequently
the empty
to give
product
is interpreted
as the identity
matrix
and we adopt
the
convention fIB+fi j=o for continued
matrix
products.
hold also for e = -1. In the next section,
1 Bj := BeBe_1..
. Bo,
j=O
By defining
empty
matrix
sums as zero, we can extend
(3.6) to
we find $0 and use the value found to solve for F(y, s).
This leaves the degenerate case in which g(c) vanishes identically, that is, where q2 = 0 and r = q both occur. This case includes notably the model of Maki and Thompson. These two conditions imply that the coefficient q + T > 0 and q = r yield T > 0. Since of the section that for (real) 5 suitably Then, for each e satisfying 0 5 fJ < N
of Fe+1 in (3.2) reduces to A(c). Further, the conditions p, r > 0, we have from the form of A as given at the head chosen, A(<) is invertible. Suppose such a choice is made. + a, (3.2) can be cast as
Fe+1 = ue(C, Parallel are (N+
s)Fe
+ Ve(SF’e-1
+ we(E).
to the main case, we have VI = Vi = 0 and take F-1 = 0. Also, the coefficient matrices 1) x (N+l) and lower-triangular and Ve has, in addition, its entries on the main diagonal
all equal to zero. More compactly,
we set
[F;l] =[? !] [iE]+[?I)
C. E. M. PEARCE
294
or OIZ
tie+1 = KelCle+ we, Relation
(3.6)
(3.7) may be iterated to give
As for the main case, this holds also for e = -1. In Section 5, we consider the determination
of $0 = (Fo,O)~ and the consequent solution for
F(K s).
4. THE
GENERAL
SOLUTION
We now pursue the solution in the general situation. We shall need the following preliminary result, which is a generalization of a result of Roughan and Pope [28]. PROPOSITION 1. Suppose’s
square matrix M is partitioned
where
,. I: s(U)
191
s(i,j)
=
...
0
s(Q)
s(Q)
%l
22
as
0
*.
-
’
’
.
*.
..
s(h) ’ s(i’j; O .$$I L,l **. L&-l ) -
Then,
IMI = fi
(A(+)/ ,
i=l
where
s+l) A&j)
=
‘h3
5992)
...
bj
~324 273
PROOF. By paired row interchanges
and column interchanges,
A&l) A&‘)
A03 A(?,?)
A(Lv1) A@3
lIi,j<_L.
1
*.
M can be rearranged as
... ..,
A&L) A&L)
..,
A(L.L)
1
Because the formation of N from M involves an even numbe! qf interchanges, IMI = INI. F’rom the structure of the matrices S(“$j), we have A(“*j) = 0 for j 5 i, so N is block lower-triangular, giving the desired result. I We are now ready to give our main result. It will be convenient to write (elm to denote the truncation of a matrix to its leading m x m submatrix and [.lm to denote the truncation of a column vector to its first m rows. With a slight abuse of notation, we may denote by Y* the matrix formed from a matrix Y by deleting all the off-diagonal entries and also denote by X* the matrix formed from a block matrix X with entries Yi by replacing each Yi by Y;.
General Stochastic
THEOREM
295
Rumour
1. Suppose thatt isred and such that g(e) # 0. Then, for all s with R,e s > 0 and
Im s # 0, and for all real s suflkiently large,
(4.1) 2N+2
is invertible, with inverse B(<, s), say. For such < and s,
(4.2) and the time-dependent evolution of the rumour process is given by
F(Y,
C4e3)
s) = Fo+
where 40 is given by (4.4) PROOF. Any product of lower-triangular matrices is also lower-triangular, so since all the block entries in Be+l are lower-triangular, the same must, therefore, hold for the block entries in
X:=
By Proposition
N+a n B&s). j=O
1, det{X)zlv+z
= det({X}2N+2)*
= det{X*}zlv+s.
Further, Ni-a
x*= j=O n q(t,s), and since Ls = 0, we have {x*}ZN+Z = Hence,
det{X*}m+s = and again invoking Proposition 1,
Thus, det{X}2N+z or
is nonvanishing if and only if (De)i,i # 0 for 1 5 i 5 Nf
2e0
2
1 and 0 < e < iV +a,
(T--)-e[T(N+a+b-i-i)+pi]-sfo,
for 0 5 i < N and 0 < e 5 N + a. The invertibility part of the enunciation follows.
C. E. M. PEARCE
296
Since fT(y, s) is a polynomial
in y of degree N + a - i, the vector 0
4N+a+l
=
0
[ FN+a Further,
~~50assumes
Restricting
(2N + 2) subvector
Solution (4.2) for 40 follows. Define 4-1 = (Fo,O,O) T. Then,
on both sides gives
we have component-wise
= 1
the three Taylor
expansions
from (3.6) to obtain
du Truncating
1
form (4.4). Hence, we have from (3.6) that
to the leading
We may substitute
~PJ+~+I is of the form
to the leading
REMARK 1. In the study
N + 1 subvectors
on both sides yields (4.3).
of Daley and Kendall,
q2 > 0. In this event, < = 0 is an obvious
choice which suffices to make g(c) # 0. If q2 = 0 and q # T, any choice of < # 0,l
5. THE We now proceed
I
DEGENERATE
simple
suffices.
CASE
to our second result.
that qz = 0 and q = r and 5 (real) is chosen so that A(<) is invertible. Then for all s with Res > 0 and Ims # 0, and for all real s sufficiently large,
THEOREM 2. Suppose
N+l
is invertible,
with inverse
K([, s), say. For such c and s
(5.1)
General Stochastic
and the time-dependent
F(Y,s> = PROOF.
An argument KIy$ Kj)N+l from (3.2) that
evolution of the rumour process is given by
N+a+l (y
1 I! e=o
-
,c)[
({tjK~}N+lfi+~{j~lK
to the first N + 1 rows of this equation
parallel
to that
is nonsingular
involving
The statement
about
invertibility
ity, (5.2) gives (5.1). Further,
lower-triangular
matrices
in Theorem
is nonzero.
Since
1 shows that
q =
T,
we have
[!A’(<) - sI] .
follows readily
we have the paired
from the form of A. In the event of invertibilTaylor expansions
[I 1 F(Y,
s>
YF(u,s)do F
Substitution
to obtain
if and only if flz’(Ue)i,i Ue = -A([)-1
=
e=o
from (3.8) leads to F(Y, ii c
and truncation
E=
297
Since FN+~+I = 0, we have
We now restrict
REMARK
Rumour
2.
’ F(u,
s) s) du
1 =
of this result In the degenerate
0 is also inadmissible
to the first N + 1 rows completes case, < = 1 is incompatible
the proof.
with A(<) being invertible.
I The choice
if a + b = 1.
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K. Dietz, Epidemics and rumours: A survey, J. Roy. Statist. Sot. Ser. A 130, 505-528, (1967). D.J. Bartholomew, Stochastic Models for Social Processes, Second Edition, John Wiley, London, (1973). N.T.J. Bailey, The Mathematical Theory of Infectious Diseases, Charles Griffin, London, (1975). D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Sot. Ser. B 39, 283-326, (1977). N.T.J. Bailey, The Mathematical Theory of Epidemics, Charles Griffin, London, (1957). J.C. fiauenthal, Mathematical Modelling in Epidemiology, Springer-Verlag, (1980). H.A. Lauwerier, Mathematical models of epidemics, In Mathematical Centre Tracts, Vol. 138, Mathematical Centre, Amsterdam, (1981). N.G. Becker, Analysis of Infectious Disease Data, Chapman and Hall, London, (1989). J.-P. Gabriel, C. Lefkre and P. Picard, Stochastic processes in epidemic theory, In Lecture Notes in Biomathematics, Vol. 86, Springer-Verlag, (1990). D.J. Daley and D.G. Kendall, Stochastic rumours, J. Inst. Math. Applic. 1, 42-55, (1965). V. Siskind, A solution of the general stochastic epidemic, Biometrika 52, 613-616, (1965). J. Gani, On a partial differential equation of epidemic theory I, Biometrika 52, 617-622, (1965). J. Gani, On the general stochastic epidemic, In Proc. gh Berkeley Symp. on Math. Stat. and Prob., Vol. 4, (Edited by L.M. LeCam and J. Neyman), pp. 271-279, Univ. of Calif. Press, Berkeley, (1967).
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14. A. Bapoport and L.J. Hebhun, On the mathematical theory of rumour spread, Bull. Math. Biophys. 14, 375-383, (1952). 15. A. Ftapoport, Spread of information through a population with socio-structural bias: I. Assumption of transitivity, Bull. Math. Biophys. 15, 523-533, (1953). 16. A. Hapoport, Spread of information through a population with socio-structural bias: I. Various models with partial transitivity, Bull. Math. Biophys. 15, 535-546, (1953). 17. H.G. Landau and A. Ftapoport, Contribution to the .mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population, Bull. Math. Biophys. 18, 173-183, (1953). 18. H.D. Landahl, On the spread of information with time and distance, Bull. Math. Biophys. 15, 367-381, (1953). 19. D.J. DaIey, Concerning the spread of news in a population of individuaIs who never forget, Bull. Math. Biophys. 29, 373-376, (1967). 20. A.D. Barbour, The principle of the diffusion of arbitrary constants, J. Appl. Prob. 9, 519-541, (1972). 21. Ft. Watson, On the size of a rumour, Stoch. Proc. and Applic. 27, 141-149, (1988). 22. B. Pittel, On a Daley-Kendall model of random rumours, J. Appl. Prob. 27, 14-27, (1990). 23. Ft. Dunstan, The rumour process, .I. Appl. Prob. 19, 759766, (1982). 24. V.R. Cane, A note on the size of epidemics and the number of people hearing a rumour, J. Roy. Statist. Sot. Ser. B 28, 487-490, (1966). 25. D.P. Maki and M. Thompson, Mathematical Models and Applications, Prentice-Hall, %nglewood Cliffs, NJ, (1973). 26. A. Sudbury, The proportion of the population never hearing a rumour, J. Appl. Prob. 22, 443-446, (1985). 27. C. Lef&vre and P. Picard, Distribution of the final extent of a rumour process, J. Appl. Prob. 31, 244-249, (1994). 28. M. Floughan and K. Pope, The determinant of a triangular-block matrix, SIAM Review 38, 513-514, (1996).