A PHLID Model for Tomato Bacterial Canker Predicting on Epidemics of the Pathogen

A pathogen, healthy, latently infected, infectious, and diseased plant (PHLID) model for botanical epidemics was defined for tomato bacterial canker (TBC) caused by the pathogenic plant bacteria, Clavibacter michiganensis subsp. michiganensis (Cmm). First, the incubation period had to be defined to develop this type of model. To estimate the parameter of incubation period, inoculation experiments were conducted in which it was assumed that infection is transferred to healthy plants by cutting with contaminated scissors after cutting infected plants with early symptoms or symptomless. The concentration of Cmm was increased over 1 × 106 cells/g plant tissue at 20 cm away from the inoculated point on the stem 10 days after inoculation, and then the approximate incubation period of TBC in symptomless infected plants was defined as 10 days. The developed PHLID model showed the dynamics of diseased plants incidence and fitted the curve of the proportion of diseased plants observed in fields well. This model also contains the factors of pathogen and disease control, and it was able to simulate the control effects and combined two different control methods, which were the soil and scissors disinfections to prevent primary and secondary transmissions, respectively. Thus, this PHLID model for TBC can be used to simulate not only the increasing number of diseased plants but also suppressing disease increase.

In commercial greenhouses, the number of diseased plants increases rapidly after pruning [2,11]. Therefore, the authors have previously verified that infected plant debris acts as the primary inoculum, and unwittingly pruning latently infected tomato plants without disinfection serves as a source of secondary spread, which occurs sooner than the spread from infected debris in commercial greenhouses in Japan [2,9]. To prevent TBC outbreaks, it is crucial to disinfect agricultural equipment, such as pruning shears and gloves, before pruning [2,9]. However, despite such efforts, TBC continues to occur globally [1,3,13]. Therefore, a comprehensive understanding of TBC epidemics is necessary to effectively prevent future outbreaks.
Previously, the authors reported that the healthy-latently-diseased (HLD) model, which was developed for TBC epidemics and can simulate the increase in the number of diseased plants and the duration of disease incidence, was based on the theory of the susceptible-infected-recovered (SIR) model [3,14]. In the field of botanical epidemiology, SIR and the susceptible-exposed-infected-recovered (SEIR) models have been commonly used to understand the mechanisms of plant disease epidemics [3,[15][16][17].
In general, the SIR model is used to model diseases in which individuals have permanent immunity or at least a very long period of temporary immunity, while the SEIR model is used to model diseases with a long incubation period and a long period of immunity [18][19][20]. The authors developed the HLD model for TBC epidemics, but this model did not take the incubation period into account [3]. Furthermore, the factor of primary pathogen should be considered in the epidemic model because the infection mechanism of plant diseases such as TBC involves both primary and secondary inoculum and spread [16]. To gain a more precise understanding of the epidemics of TBC in commercial greenhouses, a new epidemic model should be developed.
Thus, the objectives of this study were to develop a new epidemic model called the pathogen-healthy-latently-infectious-diseased (PHLID) model, based on the SEIR theory of TBC. We aimed to analyze the relationship between disease incidence and the probability of infection from primary sources such as seeds or soil, as well as secondary sources such as pruning with infected scissors. Additionally, we added a new parameter to estimate the control effects of preventing primary and secondary infections. To develop the PHLID model, we assessed the incubation periods of Cmm in latently infected tomato plants.

Incubation Period of Cmm in Latently Infected Tomato Plants
The Cmm strain CMM16-3 [3], which is pathogenic to tomato (Solanum lycopersicum cv. Momotaro), was used in this study. Cell suspensions of the Cmm strain were prepared from cultures grown on potato dextrose agar medium for 48 h and adjusted to an OD 600 of 0.01 (1 × 10 7 cells/mL). Scissors were dipped in the cell suspension for 1 s, and 50 tomato plants (2 months old) growing in plastic pots with soil (20 cm depth × 9 cm diameter; 1 plant per pot) were inoculated by pruning two compound leaves. Those plants were grown in a greenhouse at 25-30 • C. To confirm infection, stem samples were collected from five asymptomatic plants at 0, 1, 3, 7, and 10 days after inoculation (dai). The stem samples were collected from the inoculated point of the plant (0 cm), as well as three parts of the stem located 10, 20, and 40 cm away from the inoculated point (0.2 g fresh weight per plant, one sample per plant).
The population of Cmm in inoculated plants was assessed using the serial dilution plate method [3,14]. First, the samples were initially washed with sterile distilled water and subsequently crushed in 1 mL of sterile distilled water using an autoclaved mortar and pestle. Then, 10-fold serial dilutions (100 µL) of the samples were plated and spread onto Cmm-selective medium SMCMM [2,3,21], following which the plates were incubated at 25 • C for 5 days. The colony growth was monitored on five plates for each dilution, and the numbers of colony-forming units (CFUs) were determined. To verify the identity of the suspected colonies grown on SMCMM plates, ImmunoStrip for Cmm (Agdia, Elkhart, IN, USA), a rapid immune-chromatographic strip test, was used [22]. The bacterial populations in plants (CFU/g of plant tissue) were log 10 -transformed before statistical analysis. This assay was independently performed twice. Statistical analysis was performed using the RStudio user interface (RStudio, Inc., version 1.2.5001) for R software (R Foundation for Statistical Computing, version 3.6.1). ANOVA and Tukey's HSD tests (n = 10, 5 plants per experiment) were conducted using RStudio.

PHLID Model, Basic Reproduction Number (R 0 ), and Effective Reproduction Number (R t )
In this study, the PHLID model was defined based on the SEIR model. The TBC system was represented by a set of linked differential equations for a compartmental system, Plants 2023, 12, 2099 3 of 10 describing the changes in the status of plants from healthy (H) to latently infected (L), infectious (I), and ultimately to diseased (D) with symptoms of TBC, such as canker, wilt, and dead plants ( Figure 1). Growers in commercial greenhouses often cut infectious latently infected tomato plants without visible symptoms [8]. In this study, latently infected plants (L) were defined as non-infectious and symptomless plants and assumed to be infected by soil or seed-borne pathogens. Infectious plants (I) were defined as plants that were infectious and still symptomless, and ultimately diseased plants (D) were completely wilted and dead plants and often removed by farmers to avoid touching [2,3,9,11]. Generally, although plants showing symptoms of disease can be infectious, farmers tend to refrain from touching visibly diseased plants and instead remove them from the greenhouse [2,3,9,11]. In the common commercial greenhouses in Japan, wind was blocked, and the soil was covered with plastic films over tubes to avoid water splashing during irrigation. Although insects might be possible vectors of Cmm infection, there is currently no evidence to support this. The fertilization method used was irrigation using water mixed with nutrients. Therefore, these factors were not considered in this study [2,11]. Although plant debris from Cmm-infected and/or dead plants inside the soil can become the primary inoculum for soil infection in the next cultivation year [2,9], this study only considers one term of cultivation, from planting seedlings to the end of cultivation. Therefore, the interactions between the causal infections described in this study were absent. Thus, in this study, D was defined as post-infectious individuals. In this study, the PHLID model was defined based on the SEIR model. The TBC system was represented by a set of linked differential equations for a compartmental system, describing the changes in the status of plants from healthy (H) to latently infected (L), infectious (I), and ultimately to diseased (D) with symptoms of TBC, such as canker, wilt, and dead plants ( Figure 1). Growers in commercial greenhouses often cut infectious latently infected tomato plants without visible symptoms [8]. In this study, latently infected plants (L) were defined as non-infectious and symptomless plants and assumed to be infected by soil or seed-borne pathogens. Infectious plants (I) were defined as plants that were infectious and still symptomless, and ultimately diseased plants (D) were completely wilted and dead plants and often removed by farmers to avoid touching [2,3,9,11]. Generally, although plants showing symptoms of disease can be infectious, farmers tend to refrain from touching visibly diseased plants and instead remove them from the greenhouse [2,3,9,11]. In the common commercial greenhouses in Japan, wind was blocked, and the soil was covered with plastic films over tubes to avoid water splashing during irrigation. Although insects might be possible vectors of Cmm infection, there is currently no evidence to support this. The fertilization method used was irrigation using water mixed with nutrients. Therefore, these factors were not considered in this study [2,11]. Although plant debris from Cmm-infected and/or dead plants inside the soil can become the primary inoculum for soil infection in the next cultivation year [2,9], this study only considers one term of cultivation, from planting seedlings to the end of cultivation. Therefore, the interactions between the causal infections described in this study were absent. Thus, in this study, D was defined as post-infectious individuals. In this study, post-infectious individuals were defined as R. Comparing the definitions of PHLID and SEIR, H, L, I, and D correspond to S, E, I, and R, respectively. However, the PHLID model differs from the original SEIR model. Madden and van den Bosch [16] showed that the botanical SEIR model was created by adding new parameters of primary inoculum (P) and primary infection rate (v) ( In this study, post-infectious individuals were defined as R. Comparing the definitions of PHLID and SEIR, H, L, I, and D correspond to S, E, I, and R, respectively. However, the PHLID model differs from the original SEIR model. Madden and van den Bosch [16] showed that the botanical SEIR model was created by adding new parameters of primary inoculum (P) and primary infection rate (v) ( Figure 1). P represents the amount of inoculum such as spores, bacterial cells or other infectious units, or amount of inoculum, although the precise amount of inoculum is often unknown in commercial fields [16]. For the purposes of this study, P was set at 1.0. Furthermore, in this study, new original parameters were defined to simulate any control effects, specifically the interruption of primary and in which N is the proportion of tomato plants in total (=1, or 100%), D is diseased plants, β is the infection rate of plants, B is the infection rate per contact, k is the number of contacts per day, d is the duration of infection (days), 1/d is the removal rate of susceptible plants, e is the rate of latent individuals becoming infectious (i.e., the incubation period), and 1/e is the average incubation period. Some of the parameters used in this study were taken from previous reports (Table 1) [3,9,23,24], while others, such as 1/e, were determined by the experiment conducted in this study ( Table 1). The parameter v represents the primary infection rate and includes v seed and v soil , which are the infection rates from infected seeds and soil infected by infected plant debris, respectively. Additionally, the model includes two parameters, c 1 and c 2 , which represent the control effects against primary and secondary inoculum, respectively. These parameters were defined using the integrated relative risk (IRR) obtained through meta-analysis procedures (Table 1) [3,24]. If the IRR value is 0.1, the control treatment can decrease to 10% of that of the non-treatment, and the control effect is considered extremely high in the greenhouse. If no diseased management is conducted, c 1 and c 2 are given a neutral value of 1.0. However, if diseased management is conducted, c 1 and c 2 are given values of 0.48 and 0.12, respectively ( Table 1). The c 1 value was obtained from the IRR value of the control effect by soil disinfection treatment using commercial dazomet (C 5 H 10 N 2 S 2 ) micro granule [24], while the c 2 value was obtained from the IRR value of the disinfection for contaminated scissors by dipping them into a 70% ethanol solution [3].
To compare the PHLID model with the actual dynamics of TBC incidences previously reported, disease incidence data from six different commercial greenhouses (with over 1000 tomato plants per greenhouse), which experienced an outbreak of TBC in Okayama, Japan in 2005-2006 [2,3,11], were utilized. The actual dynamics of TBC incidences in seven different commercial greenhouses in Okayama from 2005 to 2008 were also used, where soil disinfection treatment was conducted using commercial dazomet (C 5 H 10 N 2 S 2 ) micro granules, and scissors and groves were disinfected using 70% ethanol isolation [2,11,24].
Moreover, the basic reproduction number (R 0 ) and the effective reproduction number (R t ) of TBC were calculated. The equations for R 0 and R were presented below: In epidemiology, R 0 refers to the expected number of cases directly generated by one infected individual in a population where all individuals are susceptible to infection. On the other hand, R t refers to the number of cases generated in the current state of a population [25]. In commonly used infection models, the infection can spread in a population when R 0 or R t > 1, but not when R 0 or R t < 1. R 0 is constant, while R t is variable and can gradually or rapidly decrease as the disease progresses. Generally, a higher R 0 value makes it harder to control the epidemic. Based on the parameters of the PHLID model, both R 0 and R t were estimated. Integrated relative risk of scissors disinfection from four independent field experiments 0.12 [3] when R0 or Rt > 1, but not when R0 or Rt < 1. R0 is constant, while Rt is variable and can gradually or rapidly decrease as the disease progresses. Generally, a higher R0 value makes it harder to control the epidemic. Based on the parameters of the PHLID model, both R0 and Rt were estimated. Infection rate by infected seeds as primary inoculum 0.067 [9] vseed Infection rate by infected seeds as primary inoculum 0.04 [23] c1 Integrated relative risk of soil disinfection from five independent field experiments 0.48 [24] c2 Integrated relative risk of scissors disinfection from four independent field experiments 0.12 [3]

Population Dynamics of Cmm in Latently Infected Tomato Plants and Incubation Period
At the inoculation point (0 cm) of latently infected tomato plants, a Cmm population of 5.07 (95% CI: 4.63-5.51) log 10 CFU/g plant tissue, which is approximately 1.0 × 10 5 CFU/g plant tissue, was detected immediately after inoculation (0 dpi) (Figure 2). The Cmm population continued to increase, and at 10 dpi, 9.40 (95% CI:9.30-9.50) log 10 CFU/g plant tissue was detected at the inoculation point (0 cm) (Figure 2). In contrast, no Cmm cells were detected at three points on the stem located at 10, 20, and 40 cm away from the inoculation point at 0 dpi. However, Cmm populations of approximately 3.0 log 10 CFU/g plant tissue were detected at these points at 7 dpi (Figure 2). At a point on the stem located 10 cm away from the inoculation point at 10 dpi, a Cmm population of 7.41 (95% CI: 7.33-7.49) log 10 CFU/g plant tissue, which is approximately 2.6 × 10 7 CFU/g plant tissue, was detected (Figure 2). At a point on the stem located 20 cm away from the inoculation point at 10 dpi, a Cmm population of 6.05 (95% CI: 5.90-6.20) log 10 CFU/g plant tissue, which is approximately 1.0 × 10 6 CFU/g plant tissue, was detected ( Figure 2). According to Kawaguchi et al. [3], a Cmm population of over 1.0 × 10 6 CFU/g plant tissue can cause TBC symptoms. Therefore, the incubation period of TBC was assumed to be 10 days.

Development of PHLID Model
The parameters used to develop the PHLID model of TBC were estimated and defined using data obtained from this study and several reports ( Figure 2 and Table 1). Based on the variables listed in Table 1   In model A, R 0 was estimated as 3.2, and it could take up to 103 days from the beginning of transmission to decrease by R t < 1.0 (as shown in Figure 3A). In model B, R 0 was estimated as 2.9, and it could take up to 110 days from the beginning of transmission to decrease by R t < 1.0 ( Figure 3B). Although the disease incidence level of model A is slightly higher than that of model B, both models are similar. When comparing the two types of PHLID models (A and B) with the observed disease incidence with no disinfection in commercial greenhouses (Figure 4), the curved shape of the dynamics of D in both models is closely similar to that of the observed disease incidence (Figures 3A,B and 4).  . TBC incidence dynamics observed in commercial greenhouses are presented. Two types of greenhouses are compared: non-disinfection and disinfection. The latter refers to the average population disease incidence (%) in commercial greenhouses where soil disinfection treatment was conducted using commercial dazomet micro granules, and scissors and gloves were disinfected using 70% ethanol isolation.

Estimating Control Effects Using PHLID Model
To simulate the impact of soil disinfection on primary inoculum control, PHLID model C was developed based on model A, with variables c1 and c2 adjusted (c1 = 0.48, c2 = 1), as shown in Figure 3C. When compared to PHLID models A and C, model A showed a disease incidence rate of 70.0% (as seen in the curve of D, which shows the dynamics of disease plants), while model C showed a disease incidence rate of 56.9% at 127 days after infection and an R0 value of 2.8. These results suggest that treatment with soil disinfection alone may not be sufficient to effectively control the occurrence of TBC during cultivation ( Figure 3A,C).
On the other hand, PHLID model D was developed to simulate the impact of disinfecting contaminated scissors on secondary inoculum control, based on variables c1 and c2 adjusted (c1 = 1, c2 = 0.12) ( Figure 3D). When compared to PHLID models C and D, model D showed a disease incidence rate of 8.4% at 127 days after infection and an R0 value of 1.1. These results suggest that disinfecting contaminated scissors could significantly inhibit the transmission of Cmm and reduce TBC incidence much more effectively than soil disinfection alone ( Figure 3C,D). Furthermore, PHLID model E was developed to simulate the combined effects of soil disinfection and disinfection of contaminated scissors, based on variables c1 and c2 adjusted (c1 = 0.48, c2 = 0.12) ( Figure 3E). Model E showed a disease incidence rate of 4.2% at 127 days after infection and an R0 value of 0.7, indicating that combining both treatments of soil and scissors disinfections strongly inhibited the transmission of Cmm by scissors in greenhouses and demonstrated the best control effects, almost perfectly inhibiting the expansion of TBC ( Figure 3E). When compared to the two types of PHLID model E and the . TBC incidence dynamics observed in commercial greenhouses are presented. Two types of greenhouses are compared: non-disinfection and disinfection. The latter refers to the average population disease incidence (%) in commercial greenhouses where soil disinfection treatment was conducted using commercial dazomet micro granules, and scissors and gloves were disinfected using 70% ethanol isolation.

Estimating Control Effects Using PHLID Model
To simulate the impact of soil disinfection on primary inoculum control, PHLID model C was developed based on model A, with variables c 1 and c 2 adjusted (c 1 = 0.48, c 2 = 1), as shown in Figure 3C. When compared to PHLID models A and C, model A showed a disease incidence rate of 70.0% (as seen in the curve of D, which shows the dynamics of disease plants), while model C showed a disease incidence rate of 56.9% at 127 days after infection and an R 0 value of 2.8. These results suggest that treatment with soil disinfection alone may not be sufficient to effectively control the occurrence of TBC during cultivation ( Figure 3A,C).
On the other hand, PHLID model D was developed to simulate the impact of disinfecting contaminated scissors on secondary inoculum control, based on variables c 1 and c 2 adjusted (c 1 = 1, c 2 = 0.12) ( Figure 3D). When compared to PHLID models C and D, model D showed a disease incidence rate of 8.4% at 127 days after infection and an R 0 value of 1.1. These results suggest that disinfecting contaminated scissors could significantly inhibit the transmission of Cmm and reduce TBC incidence much more effectively than soil disinfection alone ( Figure 3C,D). Furthermore, PHLID model E was developed to simulate the combined effects of soil disinfection and disinfection of contaminated scissors, based on variables c 1 and c 2 adjusted (c 1 = 0.48, c 2 = 0.12) ( Figure 3E). Model E showed a disease incidence rate of 4.2% at 127 days after infection and an R 0 value of 0.7, indicating that combining both treatments of soil and scissors disinfections strongly inhibited the transmission of Cmm by scissors in greenhouses and demonstrated the best control effects, almost perfectly inhibiting the expansion of TBC ( Figure 3E). When compared to the two types of PHLID model E and the observed disease incidence with disinfection in commercial greenhouses (Figure 4), the average observed disease incidence in commercial greenhouses was 21.1% at 127 days, which was higher than the simulation result of model E (Figures 3E and 4).

Discussion
Previously, the authors developed the HLD model, which is a simulation model of TBC that increases disease based on an SIR model. This model was able to estimate the dynamics of disease incidence caused by infected scissors [3]. In this study, a new PHLID model was developed that estimates the dynamics of disease increase caused by primary infections, including seed-borne and soil-borne cases, as well as secondary infections using infected scissors. TBC is a disease that can be transmitted through both soil-borne and seed-borne routes [2,3,[7][8][9]. However, it is not clear which infection route has a stronger effect on the onset of TBC. The results of this study showed that PHLID model A, which considers soil-borne infections and uses the infection rate parameter from Kawaguchi et al. [3], showed slightly more disease incidence than model B, which considers seed-borne infections and uses the infection rate from Hadas et al. [23]. However, both models were similar and would produce almost the same results in commercial greenhouses, suggesting that both infection routes have a similar impact on disease incidence. The results also indicate that pruning using infected scissors as a secondary infection is stronger than primary infection in affecting disease incidence. These findings support the previous reports by Kawaguchi et al. [11] and Kawaguchi and Tanina [9], which suggest that primary inoculum primarily serves as a source of secondary spread. Although controlling the primary inoculum is still important, according to PHLID models C, D, and E, the control effect of soil disinfection alone could not be enough to control TBC occurrence during cultivation, but scissors disinfection and combining both treatments could strongly control it, indicating that preventing the secondary spread is effective and efficient for TBC management. Thus, scissors disinfection should be essentially done in commercial greenhouses with TBC outbreaks.
The PHLID model was developed based on the SEIR model suggested by Madden and van den Bosch [16] and modified for botanical epidemic models. Madden and van den Bosch's model included parameters for primary inoculum (P) and primary infection rate (v) [16], with P representing the abundance of pathogen inoculum. When P 0 units of pathogen inoculum were introduced into a disease-free crop at time (t = 0), new infections occurred at a rate of vSP during the growing season [16]. This theory was incorporated into our PHLID model and combined with the parameter c 1 . However, Madden and van den Bosch's model was developed as a theory of plant epidemics and was not specifically designed to fit actual and specific plant diseases [16].
This study involved fitting the PHLID model to observe the incidence of TBC in commercial greenhouses. The PHLID model incorporated parameters c 1 and c 2 , which represented the control effects of interrupting primary and secondary inoculum transmission and was able to simulate both effects. However, the average observed disease incidence in commercial greenhouses, where both soil disinfection and contaminated scissors treatments were carried out was higher than the simulation result of the PHLID model E. This indicates that the simulation result tended to show a higher expected control effect than the actual effect. The parameters c 1 and c 2 , were obtained from well-designed experiments, suggesting that the control effects can be estimated accurately [2,3]. Occasionally, growers may not be able to completely disinfect their scissors before each contact with tomato plants during the cultivation period. To prevent any secondary infections caused by contaminated scissors, growers should dip their scissors into a 70% ethanol solution every time after pruning around 50 to 100 tomato plants for cultural practices. However, continuous disinfection may be difficult to maintain due to other practices such as harvest, pesticides, and/or fungicide spraying. Thus, it is important for users of the PHLID model to understand that the control effects estimated by the model may be slightly higher than the actual effects. Nonetheless, the PHLID model remains a useful tool for estimating not only the disease incidence dynamics but also the control effects before planting.
In general, when developing statistical models, a vast amount of data are used to estimate parameters, including meteorological and disease incidence data obtained from governmental and local governmental offices [26][27][28][29]. However, in the case of developing the previous HLD model for TBC, parameters were assumed based on several actual experiments [3]. In this study, the incubation period of TBC was assumed to be 10 days based on experimental results for the population dynamics of Cmm in latently infected tomato plants. According to Wang et al. [30], Cmm has the ability to migrate both downward and upward in the tomato vascular system, but upward migration via xylem is faster than downward movement. When the stem base was inoculated, Cmm was able to migrate further (up to 6 cm from the inoculation site) and reach a higher population of approximately 10 7 CFU/g plant tissue compared to inoculation at the stem top (which was only 3 cm away from the inoculation site). Our findings that Cmm migrated up to 10 cm from the inoculation site at 10 dpi and reached a population of approximately 10 7 CFU/g plant tissue were consistent with the previous report by Wang et al. [30]. We also found that Cmm migrated up to 20 cm from the inoculation site at 10 dpi and reached a population of approximately 10 6 CFU/g plant tissue in this study. Our previous report [3] indicated that a population of over 1.0 × 10 6 CFU/g of Cmm in plant tissue is sufficient to infect plants and cause TBC symptoms. Based on these findings, we assumed an incubation period of 10 days for TBC. Our findings suggest that Cmm is capable of spreading widely within the plant, which highlights the importance of early detection and effective control measures for preventing the spread of TBC. As mentioned previously, it is necessary to disinfect infected scissors (and perform soil disinfection if possible) in order to prevent the spread of infection.

Conclusions
This is the first study to develop a PHLID model of TBC using biological parameters obtained from precise inoculation experiments, including the incubation period of TBC. Utilizing an epidemiological model such as PHLID can not only enhance our understanding of the relationship between increasing numbers of diseased plants, but also simulate the effectiveness of control measures, either alone or in combination, during cultivation. This could be beneficial for growers in making informed decisions on controlling TBC based on their cultivation timeline.
Funding: This research was supported by Japan Society for the Promotion of Science, KAKENHI Grant 21K05606 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan to A.K.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.