The disease-free equilibrium is locally asymptotically stable whenever the

basic reproduction number is less than one ([R.sub.0a] < 1).

Basic Reproduction Number for the Multiple Control Model.

In epidemiology, one of the most important parts,

basic reproduction number, [R.sub.0] is obtained by the next generation method.

In Figure 3, when the

basic reproduction number [R.sub.0] > 1, we examine the impact of d on the evolution of infection prevalence, which is defined as the proportion of infected peers with respect to the total number of the peers.

The

basic reproduction number [R.sup.0] for (3) is given by

By calculating we can know that the

basic reproduction number of SIWR model [R.sub.0SIWR] = [alpha] + [[beta].sub.1] + [[beta].sub.2]/[alpha] + [mu] = 2.2 > 1, and there exists positive equilibrium point [E.sup.+.sub.SIWR] = (0.454,0.171,0.171,0.374); the

basic reproduction number of SIR model SIR model [R.sub.0SIR] = [alpha] + [[beta].sub.2]/[alpha]+[mu] = 1.6 > 1, and there exists a positive equilibriumpoint [E.sup.+.sub.SIR] = (0.625,0.15,0.225) .Apparently, [R.sub.0SIR] < [R.sub.0SIWR] succeed (theorem2).

In the following part, we will give two examples to show changes of equilibria with the

basic reproduction number [R.sub.0] by numerical simulation when [a.sub.2] = 0 and [a.sub.2] = 0.

We know that the

basic reproduction number of the model [R.sub.0] is proportional to the total number of the host tree population available as oviposition sites for the vector beetles and the number of vector population and host infectious rates a and vector infectious rate p, respectively.

I also investigated that the

basic reproduction number for this model and On the basis of

basic reproduction number [R.sub.0], the Disease Free Equilibrium (DFE) and Endemic Equilibrium (EE) are established.

From every epidemiological model one can compute an epidemiological quantity called

basic reproduction number or BRN.

After that, we determine the

basic reproduction number and steady states of the model.

We begin by computed the

basic reproduction number [R.sub.0] of model (2).