Pentatomoids as Vectors of Plant Pathogens 629
change over time and as a function of the model parameters, which do not vary with time. Susceptible hosts
become infected at an inoculation rate β, and non-infectious vectors acquire the pathogen at an acquisition
rate α, both in a frequency-dependent manner (Wonham et al. 2006). Infected hosts lose infection through
recovery at a recovery rate a, and infectious vectors lose infection at a recovery rate μ. Host and vector recov-
ery could be caused by different processes, including death and birth of new susceptible individuals, acquired
immunity/resistance, or simple loss of infection (e.g., vectors of non-persistently transmitted pathogens). The
host population is assumed to be closed (i.e., total population size does not change). The vector population
dynamics, on the other hand, are governed by logistic growth—with a carrying capacity K, constant birth rate
bi, and constant death rate di, where i = v or u to represent rates of infectious or non-infectious vectors, respec-
tively (Gaff and Gross 2007). Thus both birth and death rates can vary according to vector infection status.
Different transmission modes should influence epidemiology through (a) vector movement, (b) vector
recovery, and (c) vector population dynamics (Madden et al. 2000). Pentatomid species exhibit similar
levels of movement among hosts, at least compared to Hemiptera overall, so our model ignores vector
movement. Rather, we focus on parameters specific to the vector-pathogen relationship—vector recovery,
μ, and birth rate of infectious vectors, bv. Although we are unaware of any heteropteran vectors with
transovarial transmission, resulting in bv > 0, this process is known for numerous persistent-propagative
pathogens such as Candidatus Liberibacter spp. transmitted by psyllid and triozid vectors (Haapalainen
2014). Furthermore, vertical transmission of endosymbionts occurs in several families of true bugs includ-
ing pentatomids (Prado et al. 2006). Thus, some form of vertical pathogen transmission in pentatomids
may be possible. Vectors of non-persistently transmitted pathogens by definition have high recovery rates,
whereas those of persistently transmitted pathogens have low recovery rates. Likewise, persistent-
propagative, vertically transmitted pathogens can be differentiated from other transmission modes by
setting bv > 0. We explored the effects of vector recovery and infectious vector birth by running numerical
simulations of model 13.1 by varying μ and bv while keeping all other parameters constant. Numerical
simulations were run in R 3.2.1 with the deSolve package (Soetaert et al. 2010, R Core Team 2015).
As vector recovery rate increases, the percent of infected vectors in the population decreases linearly
(Figure 13.2). Interestingly, the percent of infected hosts is greatest at intermediate vector recovery
100 A500100500
0 0.5 1BParameter valuePercentage InfectedFIGURE 13.2 Percent infected hosts (I, solid line) and vectors (V, dashed line) from an epidemic model of pentatomid-
borne symbiotic pathogens (13.1) at varying values of (A) infectious vector birth rate, bv, and (B) vector recovery rate, μ.
Values are taken as equilibrial (i.e., long-term) values from numerical simulations after 2,000 time steps. Preliminary simu-
lations using 10,000 time steps verified these results. Other parameters were held constant: α = β = 0.4, a = 0.1, bu = 0.3,
du = dv = 0.2, K = 250, S + I = 100; μ = 0 when varying bv and vice-versa. R code for models and simulations are available
at https://github.com/arzeilinger/pentatomid_vector_epidemic_model.