Supporting InformationChiyaka et al. 10.1073/pnas.1208326109SI TextNondimensionalization. The differential equation system of themodel for development of huanglongbing in a citrus tree is givenby (see Fig. 1 for the meaning of the symbols)

dEdt

¼ α�Au þ Ai

1 þ Ai2

�HP− ðδe þ σÞE

dNu

dt¼ σE− βnN

uX þ θ2YP

− ðδn þ γÞNu

dN i

dt¼ βnN

uX þ θ2YP

− ðδn þ γÞN i

dAu

dt¼ γNu − βaA

uX þ θ2YP

− ðδa þ κÞAu

dAi1

dt¼ βaA

uX þ θ2YP

− ðδa þ κÞAi1

dAi2

dt¼ γN i − ðδa þ κÞAi

2

dHdt

¼ ξðtÞH − βpH

�Ai1 þ θ1Ai

2

�P

− λHðX þ θ2Y Þ

P− δpH

dWdt

¼ βpH

�Ai1 þ θ1Ai

2

�P

þ λHðX þ θ2Y Þ

P−�η1 þ δp

�W

dXdt

¼ η1W −�η2 þ δp

�X

dYdt

¼ η2X − δpY :

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

[S1]

We note that P = H + W + X + Y. Let

~H ¼ HP; ~W ¼ W

P; ~X ¼ X

P; ~Y ¼ Y

P;

and

~E ¼ EP; ~N

u ¼ Nu

P; ~N

i ¼ N i

P; ~A

u ¼ Au

P; ~A

i1 ¼

Ai1

P; ~A

i2 ¼

Ai2

P: [S2]

The variables ~H, ~W , ~X , and ~Y are fractions of the total flushpopulation such that ~H þ ~W þ ~X þ ~Y ¼ 1 + and ~E, ~N

u, ~N

i, ~A

u,

~Ai1, ~A

i2 are the densities of the vectors in the respective classes

per flush. We note that the rate of change of the total flushpopulation dP/dt ≠ 0 but dP/dt = ξ(t)H − δpP. Substituting thenew variables into model system (Eq. S1) and dropping the tildes(∼), the equations for the proportion of flushes and vectorpopulations per flush become

dEdt

¼ α�Au þ Ai

1 þ Ai2

�H − ðδe þ σÞEþ �

δp − ξðtÞH�E

dNu

dt¼ σE− βnN

uðX þ θ2Y Þ− ðδn þ γÞNu þ �δp − ξðtÞH�

Nu

dN i

dt¼ βnN

uðX þ θ2Y Þ− ðδn þ γÞN i þ �δp − ξðtÞH�

N i

dAu

dt¼ γNu − βaA

uðX þ θ2Y Þ− ðδa þ κÞAu þ �δp − ξðtÞH�

Au

dAi1

dt¼ βaA

uðX þ θ2Y Þ− ðδa þ κÞAi1 þ

�δp − ξðtÞH�

Ai1

dAi2

dt¼ γN i − ðδa þ κÞAi

2 þ�δp − ξðtÞH�

Ai2

dHdt

¼ ξðtÞHð1−HÞ− βpH�Ai1 þ θ1Ai

2

�− λHðX þ θ2Y Þ

dWdt

¼ βpH�Ai1 þ θ1Ai

2

�þ λHðX þ θ2Y Þ− ðη1 þ ξðtÞHÞWdXdt

¼ η1W − ðη2 þ ξðtÞHÞXdYdt

¼ η2X − ξðtÞHY :

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; [S3]

The disease-free equilibrium of the model system (Eq. S3) is

ε0 ¼�E; N

u; N

i; A

u; A

i1; A

i2; H; W ; X ; Y

�¼

�E;

σEδn þ γ þ �ξ− δp

; 0;γN

u

δa þ κ þ �ξ− δp; 0; 0; 1; 0; 0; 0

�; [S4]

where E is the egg population at the disease-free equilibrium, andðδn þ γÞ þ �ξ− δp > 0; ðδa þ κÞ þ �ξ− δp > 0; and δe þ σ þ �ξ− δp > 0:The expression (ξ− δp) gives the net growth rate constant of

the flush population in a disease-free population. In the absenceof HLB infection, the population size PðtÞ declines exponentiallyto 0 if (ξ− δp) < 0, remains constant if (ξ− δp) = 0, and growsexponentially if (ξ− δp) > 0.

Reproductive Number.The expression for the reproductive numberR0 of the plant-vector model (Eq. S3) is given by

R0 ¼ Q1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2 þQ2

1

q; [S5]

where

Q1 ¼ η1λð�ξþ η2θ2Þ2�ξðη1 þ �ξÞðη2 þ �ξÞ and

Q2 ¼ βpη1ðη2θ2 þ �ξÞ�βnγθ1Nu þ βa�γ þ δn þ �ξ− δp

�Au�

�ξ�γ þ δn þ �ξ− δp

�ðη1 þ �ξÞðη2 þ �ξÞ�κ þ δa þ �ξ− δp�:

The parameter �ξ ¼ 1=TR T0 ξðvÞdv is the average value of the

maturation rate of the flush population taken over a cycle ofperiod T. It has been shown in several studies that in somenonautonomous epidemic models, expressed in terms of differ-ential equations the basic reproductive number can be obtainedfrom the corresponding autonomous model by using time aver-ages of the coefficients (1–4).The reproductive number R0 can also be derived using heu-

ristic arguments as follows. If an infected adult vector (in the Ai1

state) is introduced into a population of susceptible flush, itinfects the flush at a rate βp over its infectious period1=ðκ þ δa þ �ξ− δpÞ. The probability that the infected flush be-comes infectious is η1=ðη1 þ �ξÞ. The average number of flush thatbecome infectious is

RA1P ¼ βp

1�κ þ δa þ �ξ− δp

� η1ðη1 þ �ξÞ: [S6]

If, on the other hand, there is an infectious asymptomatic flushand a population of adult vectors, the vectors become infected ata rate βa A

u over the infectious period of the flush 1=ðη2 þ �ξÞ.The asymptomatic infectious flush becomes symptomatic ata rate η2. In the symptomatic state the flush infects vectors ata rate βaθ2 A

u over the infectious period 1=�ξ. Therefore, thenumber of adult vectors that become infectious is

RPA1

¼ βa Au 1η2 þ �ξ

þ βaθ2 Au η2η2 þ �ξ

1�ξ: [S7]

Following similar arguments, the number of infected adults thatwere infected in the nymph stages is

Chiyaka et al. www.pnas.org/cgi/content/short/1208326109 1 of 4

RPA2

¼ βnNu γ�γ þ δn þ �ξ− δp

� 1ðη2 þ �ξÞ

�1þ η2θ2

�ξ

�; [S8]

and the number of flush infected by the adults in the Ai2 class is

RA2P ¼ βpθ1

1�κ þ δa þ �ξ− δp

� η1ðη1 þ �ξÞ: [S9]

Therefore, the average number of infected vectors that arise froma single infected vector without the infection from flush to flushis Q2 ¼ RA1

P RPA1

þRA2P RP

A2. Following the above arguments, it is

clear that the average number of infected flush from a singleinfected flush is 2Q1 in the absence of the vector population.A similar approach of finding the reproductive number fora vector-transmitted disease was considered elsewhere (5).

Parameter Values. To obtain some of the parameter values, weused information for mean developmental periods and survi-vorship of immature stages of D. citri from Table S1, part ofwhich was extracted from Liu and Tsai (6). To estimate the rateof mortality of nymphs, the sum of the survivorship proportion ofeach of the five nymphal stages was divided by five to obtain theaverage survivorship for the nymphs. The result was subtractedfrom 1 to get the probability of death and divided by the numberof days taken by the nymphs to become adults. The value ob-tained gives the mortality rate of nymphs per day, δn. The sameapproach was used to obtain the parameter for the rate at whicheggs become nonviable, δe. Several studies have shown that itmight take 15–30 min for adults to acquire the bacteria (7, 8) butreports on acquisition vary widely. We assume that nymphs wouldtake much longer to acquire the bacteria than the adults becauseof the shorter piercing mouths of young nymphs. Therefore, theacquisition time for nymphs, πn, is assumed to be 2 h.

Sensitivity Analysis. Relative importance of model parameters todisease transmission and prevalence is determined throughsensitivity analysis. Sensitivity analysis of parameters is commonlycarried out to determine robustness of model predictions tovariation in parameter values. The parameters that are crucial inthe model are those that appear in the reproductive number R0because these parameters influence the number of secondaryinfections either positively or otherwise. They should be con-sidered whenever intervention strategies are to be implemented.The ratio of the relative change in a variable to the relativechange in a parameter is the normalized forward sensitivity indexof the variable to that parameter. The variable in this case is thereproductive number, which is a differentiable function of theparameters. Therefore, sensitivity indexes may alternatively bedefined using partial derivatives (9). For example, the computa-tion of the sensitivity index of R0 with respect to the within-treetransmission rate λ using parameter values in Table 1 is given by

YR0λ ¼

�∂R0

∂λ

��λ

R0

�¼ 0:9373> 0: [S10]

The value 0.9373 shows that R0 is an increasing function of λ.The sensitivity indexes of the remaining parameters are shownin Table 2.Three parameters associated with tree characteristics, namely

maturation rate of flush (�ξ), internal movement of the pathogenwithin the tree λ, and latent period 1=η1, are all very influential onthe value of R0 (Table 2). When λ is reduced, the dynamicsof HLB development change drastically from continued expan-sion of HLB symptoms to a dynamic equilibrium between healthy,asymptomatic, and symptomatic flush (Fig. 3). When the latentperiod is varied between 30 and 180 d, there are large differencesin the maximal proportions of asymptomatic flush attained (35–60%) (Fig. S1A), but there are only slight differences in the timeneeded to reach 100% symptomatic flush (Fig. S1B).

Scenario Analysis. If an insecticide is applied starting from 360 d or450 d after initial infection, increases in the proportion of healthyflush and decreases in the proportion of asymptomatic infectedflush ensue (Fig. 4 A and B). Although early spraying reduces thetemporal increase in infected flush, it does not eliminate thepathogen from a tree (Fig. 4B). As expected, spraying the vectorsreduces the psyllid populations in all categories, but they con-tinue to oscillate (Fig. S2).If insecticides are applied when λ = 0.25, all flush and vector

categories continue to oscillate similar to the nonsprayed sce-nario (Fig. S3). However, the populations of adult psyllids arereduced from a maximum of 4.5 to 4.3 individuals per flush(Fig. S3A) and the maximum proportion of symptomatic flush isreduced from 0.35 to 0.29 (Fig. S3B). Thus, if internal resistanceto infection is high, spraying the psyllids will not have a greatimpact on the dynamics of the populations.The effects of varying internal movement of the bacteria λ, on

both healthy and symptomatic flush are shown in Fig. S4. Thegreat variability in the flush population caused by small changesin λ shows how sensitive the model is to that parameter. Theresults shown in Fig. S4 confirm the analysis obtained fromsensitivity analysis on sensitivity of R0 to λ.When initial conditions of flush populations are varied with and

without periodic forcing, different end points are obtained. Fig.S5 shows how changing the initial conditions of the healthy andlatent flush affect the end point of the dynamics of the modelwithout and with periodic forcing. In the absence of periodicforcing, that is, when strength of seasonality υ ¼ 0, varying initialconditions always results in the 100% symptomatic state. How-ever, when periodic forcing is present, changing the initial con-ditions results in either an oscillatory state (all populationspresent and oscillating) or a 100% symptomatic state. Thus,periodic forcing affects the dynamics of the model. However,periodic forcing for growth of flush was included, because it isobserved in the field that flush appearance and growth are sea-sonal (10).

1. Wesley CL, Allen LJS (2009) The basic reproductive number in epidemic models withperiodic demographics. J Biol Dyn 3:116–129.

2. Moneim IA (2007) Seasonally varying epidemics with and without latent period: Acomparative simulation study. Math Med Biol 24:1–15.

3. Ma J, Ma Z (2006) Epidemic threshold conditions for seasonally forced seir models.Math Biosci Eng 3:161–172.

4. Greenhalgh D, Moneim I (2003) SIRS epidemic model and simulations using differenttypes of seasonal contact rate. Syst. Anal. Model. Simul 43:573–600.

5. Chiyaka C, Garira W, Dube S (2007) Transmission model of endemic human malaria ina partially immune population. Math Comput Model 46:806–822.

6. Liu YH, Tsai JH (2000) Effects of temperature on biology and life table parameters ofthe Asian citrus psyllid, Diaphorina citri Kuwayama (Homoptera: Psyllidae). Ann ApplBiol 137:201–206.

7. da Graça JV (1991) Citrus greening disease. Annu Rev Phytopathol 29:109–136.8. Capoor SP, Rao DG, Viswanath SM (1967) Diaphorina citri Kuway., a vector of the

greening disease of citrus in India. Ind J Agric Sci 37:572–576.9. Hove-Musekwa SD, et al. (2011) Modelling and analysis of the effects of malnutrition

in the spread of cholera. Math Comput Model 53:1583–1595.10. Stansly P (2011) Living with citrus greening in Florida. Available at http://www.imok.

ufl.edu/docs/pdf/entomology/tja_cv_oct_2011.pdf. Accessed January 6, 2012.

Chiyaka et al. www.pnas.org/cgi/content/short/1208326109 2 of 4

0

0.2

0.4

0.6

0 500 1000 1500 2000 2500

60 days90 days120 days180 days

Time (days)

Asy

mpt

omat

ic fl

ush

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500

60 days90 days120 days180 days

Time (days)

Sym

ptom

atic

flush

A B

Fig. S1. (A and B) Proportion of asymptomatic flush (A) and symptomatic flush (B) when latent period is varied from 60 d to 180 d. The parameter for internalmovement of the pathogen in the tree is λ ¼ 0:33.

0

10

20

30

40

50

60

70

0 500 1000 1500 2000

Adult psyllidsNymphs

Time (days)

Psyl

lids/

flush

Fig. S2. Populations of total adult psyllids and nymphs per flush when psyllids are sprayed twice a year with an insecticide that is 0.75 effective, at 360 d afterinitial infection. The input rate of healthy flush follows a periodic function with two peaks in 1 y. The parameter for internal movement of the pathogen in thetree is λ ¼ 0:33.

0

1

2

3

4

5

0 500 1000 1500 2000

360 days450 days540 days

Time (days)

Adu

lts/fl

ush

0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000

360 days450 days540 days

Time (days)

Sym

ptom

atic

flus

h

2500

A B

Fig. S3. (A and B) Proportion of symptomatic flush (A) and adult psyllids (B) when psyllids are sprayed twice a year with an insecticide that is 0.75 effective,360 d, 450 d, and 540 d after initial infection. The parameter for internal movement of the pathogen in the tree is λ ¼ 0:25.

Fig. S4. (A and B) Proportion of healthy flush (A) and symptomatic flush (B) when λ has values of 0.22, 0.33, and 0.44. All of the other parameters are as shownin Table 1.

Chiyaka et al. www.pnas.org/cgi/content/short/1208326109 3 of 4

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.20.40.60.8

Healthy flush

Sym

ptom

atic

flus

h

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.20.40.60.8

Healthy flush

Late

nt fl

ush

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.20.40.60.8

Healthy flush

Sym

ptom

atic

flus

h

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.20.40.60.8

Healthy flush

Late

ntflu

sh

A B

C D

Fig. S5. (A–D) Phase plane portraits of healthy (H) and symptomatic (Y) flush (A and C) and of healthy (H) and latent (W) flush (B and D). A and B are obtainedwhen periodic forcing is zero and C and D are obtained when periodic forcing is present. Initial conditions of the healthy and latent flush were varied. Thevalues show the different initial conditions of healthy flush and those of latent flush are known by subtracting these values from one. All of the other pa-rameters are as shown in Table 1.

Table S1. Survivorship and developmental stages of immature stages of D. citri at 28 °C

Variable* Egg First instarSecondinstar Third instar

Fourthinstar Fifth instar Egg to adult

Parametersobtained

Mean ± SE developmentalperiods of immature stages(in days) of D. citri

3.46 ± 0.09 1.57 ± 0.08 1.43 ± 0.08 1.91 ± 0.09 2.29 ± 0.08 3.40 ± 0.09 14.06 ± 0.21 δnγ

δaσ

δp% survivorship ofimmature stages of D. citri

96.2 93.4 96.2 99.4 99.1 98.7 83.9 δnδe

*These values are extracted from tables 1 and 2, respectively, in ref. 6.

Chiyaka et al. www.pnas.org/cgi/content/short/1208326109 4 of 4