Citrus plant is one of the biggest production fruit plants in the world as well as in Indonesia. Citrus plant can be infected with a disease by virus, such as Citrus Tristeza Virus (CTV) propagated by Toxoptera citricida. CTV spreads some affects for citrus plant, for example dwarf, leaf clorosis, fall prematurely of flower, decreasing production, and decreasing of fruit size. Based on the well-known SIR-SI model, it was derived the CTV propagation model of the citrus plant. The model obtained involved citrus plant which consisted of subpopulation susceptible S(t) , infected I(t), and recovered R(t). While the vector’s population were susceptible insect S_X(t) and infected insect I_Y(t).The model without any delay is given by

dS/dt= Gamma- miu*S(t)-beta*I_Y/(1+alfa*I_Y(t)*S(t)

dI/d= beta*I_Y/(1+alfa*I_Y(t)*S(t)-v*I-miu*I-gamma*I

dR/dt=gamma*I-miu*R(t)

dS_X/dt=lambda-beta1*I/1+alfa1*I*S_X(t)-mS_X

dI_Y/dt=beta1*alfa1/1+alfa1*I*S_X-m*I_Y

Assuming that the susceptible citrus plant S(t) can be infected by the infected insect I_Y(t) the dynamic model with delays were obtained by involving the terms, namely   and .

Based on the previous researches and the chosen value for all parameters   then two equilibrium points were determined for the model without delays, i.e. the disease free equilibrium E^0=(1000;0;0;40;0) and the endemic equilibrium point E^*=(944,68;2,66;2,66;39,23;0,77). Hence, the stability analysis of the model, around the disease-free equilibrium point and endemic without delays, were asymptotic stable. Moreover, the basic reproduction number (rho) was determined for the parameter value m. The numerical simulation without and using delays were demonstrated that the response of the model were asymptotic stable.

Applying time delays, it is concluded that the spread of CTV on citrus plant can be inhibited by accelerating the harvest before the citrus plant is infected to the virus, so that the harvest citrus fruits can be obtained more better than without giving delays.