The assumptions underlying models based on discrete 

 time usually consider all susceptible and infested in- 

 dividuals to be mixed together homogeneously. This 

 situation is most nearly represented by small groups of 

 trees but does not hold for large stands. However, in- 

 cubation and latent periods are not variable, and the in- 

 festing of a tree can be considered as a relatively short 

 period of time. As tbas model was refined, habitat type 

 and volume yield factors were included. These factors 

 govern tree and stand susceptibihty and affect the life 

 processes of beetle populations. 



One important problem with the chain binomial model 

 is that substantial departures from the assumptions of 

 constant incubation and latent periods and a very short 

 infectious period would invalidate the model. Another 

 problem is failure to properly identify the Unks of the 

 chain. However, if a highly variable incubation period oc- 

 curs, or the symptoms cannot be identified correctly, 

 there is still an alternative— to base our analysis on the 

 total number of cases occurring during the course of the 

 epidemic. Some precision is lost when the parameters are 

 estimated. However, if the number infested is large, fre- 

 quencies based on this number can be calculated directly 

 and will probably be more accurate than those derived 

 from the probabihties of the individual chains. 



THE RATE OF LOSS MODEL 



If p is the probability of a tree becoming infested in a 

 given time interval, then q = 1 — p is the probabihty of 

 a tree not becoming infested. The probabUity of a tree 

 becoming infested depends on the susceptibility or 

 resistance of the tree, the infestivity of the beetle, the 

 length of attack period, and the size of the attacking 

 beetle population, as well as the environmental condi- 

 tions of the stand. 



If Dt is the number of trees infested at time t, then 

 q^t is the probability that a specified tree wiU not be in- 

 fested, and 1 -qDt is the probability that the tree will be 

 infested. If there are green trees capable of being in- 

 fested in the population at time t, the expected number 



of infested trees produced at the time t + 1 is times 

 the probability of at least one tree being infested. Or: 



Dt+i = Gt (1 - qDt)and Gj+i = G^qOt. 

 This equation provides a method of stepwise .calculation 

 of trees infested at successive time periods as shown in 

 table 1. 



If Gt = 0, all the trees are dead— no more susceptible 

 trees are left— and the epidemic subsides due to food 

 depletion. If = 0, there are no more trees successfully 

 producing beetles— and the epidemic subsides. 



The Greenwood model postulates that the probabihty 

 of a susceptible tree being infested is a constant and is 

 not related to the number of infested trees. In other 

 words, a susceptible tree in a stand with one infested 

 tree is as likely to be attacked as the same tree sur- 

 rounded by many infested trees. This is obviously not 

 the case. Thus, we adopted the Reed-Frost model for 

 susceptibihty because it accounts for the increase in in- 

 festation pressure due to the number of infested trees. 

 In the Reed-Frost model, the probability of a tree not 

 being infested from only one source is taken to be a con- 

 stant, q. The probabihty of not being infested from two 

 sources is thus (q) (q), and consequently from n sources 

 it is qn. 



The value of q, the probabihty of a tree not being in- 

 fested from one source, can be calculated by solving the 

 equation of G^+i for q. This yields: 

 q = (Gt+i/G,)(l/Dt) 



Theoretically, q will be a constant, but the real world 

 is never constant. Thus the q for time t (q^) varies shght- 

 ly with t, and may be determined for each time interval. 

 However, we found a closer prediction of Dj+i was ob- 

 tained if several values for q^ were calculated, and q was 

 estimated by q^ for several stands. We also found that 

 precision of prediction increased with decreasing size of 

 diameter classes. Estimates of tree mortahty over time 

 approximated true losses more closely when predicted by 

 2-inch (5.1-cm) diameter classes than by larger diameter 

 classes. 



Table ^.— Calculation of a theoretical epidemic from the Reed-Frost 



model (p = 0.5) 





Number 



Number of 





Time 



of dead 



susceptible 



Calculation of D,+i and 0,+., 



period 



trees 



trees 







1 



100 



□i = 100 (1 - 0.95) = 5.00 = 5 

 Gi = 100 - 5 = 95 



1 



5 



95 



□2 = 95 (1 - 0.955) = 21.49 = 21 

 G2 = 95 - 21 = 74 



2 



21 



74 



D3 = 74 (1 - O.9521) = 48.80 = 29 

 G3 = 74 - 49 = 25 



3 



49 



25 



D4 = 25 (1 - 0.9549) = 22.97 = 23 

 G4 = 25 - 23 = 2 



4 



23 



2 



D5 = 2 (1 - 0.9523) = 1.39 = 1 

 G5 = 2 - 1 = 1 



5 



1 



1 



Dg - 1 (1 - O.951) = 0.05 = 

 Gg = 1 - = 1 



6 







1 





2 



