On the Spread of a Fatal Disease

Ray Watson

Statistics Department University of MeBoume Park&e Vie 3052, Australia

Transmitted by Melvin R. Scott

ABSTRACT

Gleissner investigaten a detenuinistic epidemic model for which the disease was inevitably fatal, and derived a threshold theorem. In this paper, an alternative derivation of the threshold theorem is given, and the correspondiug stochastic model is investigated. An approximation for the distribution of the number of survivors is derived.

1. THE MODEL

Let X( t ) denote the number of susceptible individuals at time t, Y( t ) the number of infective individuals at time t, and Z(t) the number of deaths which have occurred up to time t. Let P(t) =(X(t),Y(t), Z(t)). It is assumed that ( P( t ), t >, 0) is a Markov process with initial conditions

P(O) = (n,a,O)

and transition probabilities given by

h[P(t+dt)=(r-l,y+l,z))P(t)=(r,y,a)] =Zdt,

Pr[P(t +dt)=(le,y-1,2+1)1P(t)=(r,y,z)] =ygdt.

It is seen that X(t)+Y(t)+Z(t)=n+a for t&O.

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I24 RAY WATSON

This corresponds to the model proposed by Gleissner [l] in which homoge- neous mixing among the survivors with a constant contact rate is assumed. This will be referred to OS th+? fatal epidemic model.

2. DETERMINISTIC APPROXIMATION

If (x( Q, y(t j, z(t)) denotes the deterministic approximation to (X( t ), Y( ), Z(t)), then (x( t ), y(t), z( t )) satisfy the following set of differential equations

dx lew dt= -x+y’

dY Bxy -=-- dt x+y y’s (2)

with initial conditions (x(O), y(O), z(O)) = (n, a, 0). From (1) and (3) we obtain

dz y(n+a-2) -=- dx r8 9

x (4)

from which we see that

k(t)=ylnL+ln n+a

x(t) n+a-z(t)

is a constant of the motion, i.e. k(t) = k(0) = 0. This information can be represented in a variety of ways, but it is informative to consider the determinktic path in the (x, y) plane so as: to facilitate comparison with the general epidemic model.

The deterministic path for the general epidemic model-for which the rate of infection is (/3/n)xy rather than [/3/(x + y)]xy -begins at (n, a) and proceeds to (0, - oo), crossing the x+&s at some point in the intervd (0, n), where the epidemic ceases as the number of infectives has reached zero. The position of this crossing point determines the size of the outbreak,

On the Spread of a Fatal Llisease 125

FIG. 1. Deterministic path for the general epidemic model.

i.e. the number of individuals infected during the outbreak. The proximity of this crossing point to x = n is determined by the initial behavior of the path: if it starts down, the crossing point is close to r = n and the outbreak is small; if it starts up, the crossing point is away from x = n and the outbreak is large-see Figure 1. When x1 = n the slope of the path is (y//3) - 1, so if y/j9 2 1 the outbreak is small, while if y/j3 < 1 the outbreak is large.

In the case of the fatal epidemic model, the determini& path begins at (n, a) and proceeds to (0,O). It may or may not cut the x-axis at some point in the interval (0, n). Again, whether it does or not is determined by the slope ofthepathwhenr=n.Inthiscase

dY Yb+Y) _l

-=

dx /3x ’ Thus the initial slope of the path is y(n + @/fin - 1, and the path will cut the x-axis in the interval (0,n) according as this is greater or less than a/n-see Figure 2.

This leads immediately to the threshold theorem obtained by Gleissner [l]:

Some part of the initial population will survive if and only if J3 < y.

RAY WATSON

FIG. 2. Deterministic path for the fatal epidemic model.

If j3 < y, then the number of supvivors is given by observing that at the end of the outbreak n + a - z(r) = x(r) and solving k(lr) = k(0). This gives

n

( 1 MY -8)

x(7)=n - n+a

Thus, for large n and small a, either almost none of the population survives.

. (5)

all the population survives, or

3. STOCHASTIC APPROXIMATION

The stochastic andogue of a constant of the motion is a martingle. If we d&&x y-1,2

b 1 H,(aJ)= E k’ for a, b=1,2 ,... and a&

k-a

On the Spread of a Fatal Disease

then the process

K(t)=y[H,(X(t)+l,n)] -&(n+a-Z(t)+l,n+a)

is a zero mean martingale with respect to the u-field genemted by (P(S), O,<s<t). Note that as H,(x+P,n)-ln(n/x) as x,n+oo, it is seen that the martingale K(t) bears a very close resemblance to the constant of the motion k( t ).

An unbiased estimator of the variance of K(t) is given by the sum of the squared martingale increments. Thus,

is an unbiased estimator of var(K( t )). The behavior of the martingale K( t ) indicates that the process follows the determinktic path to within a band of width O(G).

Further, if a = ne and T denotes the stopping time given by

T=inf{t:X(t)=OorY(t)=Q), l

then, as in Watson [2], it can be shown that

K(T) d - -+ N(O,l) U(T)

as n+oo.

We observe that if the stopping time T is used, then the final state cannot be (0,O): either X(T) > 0, i.e. a proportion of the population sunive the outbreak, or Y(T) > 0, i.e. d he sunddng individuals have the &ease. The first will tend to ha ppen if 48 < y, while the second will happen if /3 > y. The distribution of (X(T), Y(T)) can thus be approximated using (6) as follows: If j9 < y, then for large n, with high probabi!ity Y(T) = 0, and X(T) is such

128 RAY WATSON

that

#(X(T)) z N(W),

where

n n+a y”x-/3h~

J/(x)= 11Y’(~_~)+82(~_$.--) l

It follows that, for large n,

where 4 = X(T), as given by (5). This specifies, approximately, the distribu- tion of the number of suntivors of the epidemic.

If $ j y, then for large n, with high probability X(2’) = 0 and Y(T) is such that

g@(T)) z N(O,l),

where

n+a yhn-/3ln-

It follows that, for large n,

where 7 = n-T/@ (n + a), is obtained as the solution of +(q) = 0. This speci- fies, approximately, the distribution of the number of infectives remaining alive when the last susceptible is infected.

On the spread of a Futal Disease 129

FIG. 3. Stochastic path aud distribution of the fiual state for the fatal epidemic model.

The situation is ihstrated in Figure 3.

REFERENCES

1 W. Gleissner, The spread of epidemics, Appt. Ma&a. Cumput. 27:167-171(1988). 2 R. Watson, On tie spread of 8 raairozz, Six!~~tic Recess. A&. 27:141-149

(1988).