# A Draft Dissertation Proposal: Forest Disease Dynamics and Management

by Noam Ross, 27 January 2013

A first draft, just out to my committee. Feedback welcome! A modifiable souce version can be found on github here

## Abstract

Forest disease spreads through plant communities structured by species composition, age distribution, and spatial arrangement. I propose to examine the consequences of the interaction of these components of population structure Phytophthora ramorum invasion of California redwood forests. First, I will compare the dynamic behavior of a series of epidemiological models that include different combinations of population structure. Then I will fit these models to time-series data from a network of disease monitoring plots to determine what components of forest population structure are most important for prediction of disease spread. Using the most parsimoniuous models, I will determine optimal schedules of treatment to minimize the probability of precipitous decline of host populations.

## Introduction: Modeling Disease in Complex Populations for Management

Forest diseases can radically transform ecosystems. In North America, chestnut blight, Dutch elm disease and beech bark disease have caused precipitous declines in their hosts, leading to changes in the structure and function of forests. Changes from such diseases may be the dominant force changing the face of some forests in coming decades, overwhelming other forms of rapid environmental change such as climate change (Lovett et al. 2006).

Even simple host-disease systems exhibit complex dynamics (Kermack and Mckendrick 1927). Forest disease dynamics include additional complexity driven by pathogen life cycles, variation in pathogen and host populations, spatial strucure, and demographic and environmental stochasticity (Hansen and Goheen 2000, Holdenrieder et al. 2004). These factors yield greater complexity in disease dynamics. This complexity, combined with limited monitoring data and the uncertainty of the parameters of emergent diseases, makes prediction of disease dynamics difficult. Yet such prediction is needed to allocate limited resources for treatment and prevention.

One source of complexity in forest disease dynamics is structured variation in host populations. Host tree populations may consist of multiple species or genotypes, and consist of multiple sizes, each of which may interact with disease or each other differently (Gilbert and Hubbell 1996). All forest tree populations are spatially structured; non-motile individuals can not be “well-mixed” (Filipe and Maule 2003). Each of these forms of population structure have consequences for population dynamics that have been examined in theoretical models (Park et al. 2001, 2002, Dobson 2004, Klepac and Caswell 2010). However, dynamics resulting from these components of population structure interacting are not well-characterized.

Understanding modeled dynamics of disease in structured populations may provide insights useful for management. However, the ability to use such models for management planning depends requires confidence in their predictive power, particularly under novel management treatments. Recent developments in particle filter techniques allow estimation of the likelihoods of complex dynamic models (Arulampalam et al. 2002, Ionides et al. 2006, Knape and de Valpine 2012), and subsequent comparison of models in out-of-sample predictive performance (Vehtari and Ojanen 2012). Fitting dynamic models direclty provides more confidence in the dynamic results than deriving madel parameters individually from data, because allows tests confidence in models’ emergent dynamics rather than separate components.

Sudden oak death (SOD) is an emerging forest disease in California and Oregon that threatens populations of tanoak (Notholithocarpus densiflorus) (Rizzo and Garbelotto 2003), has the potential to modify community structure (Metz et al. 2012), and cause significant economic damage (Kovacs et al. 2011). Silvicultural, chemical, and other control techniques can modify the progression of disease (Swiecki and Bernhardt 2013), but eradication of SOD is unlikely [@Cobb2013] even at local scales. Long and short- and long-term solutions to limit the damage of disease will require continuous treatment and monitoring regimes. Under budget constraints, planning for such treatment can be informed by dynamics models via optimal control techniques.

I aim to answer the following overall questions:

1. How do interactions between community, size, and spatial structure affect disease dynamics in forests?
2. What aspects of population structure are most important in determining the probability and size of forest disease outbreaks in theoretical models?
3. What aspects of population structure are most important in predicting the probability and size of SOD outbreaks?
4. What control strategy minimizes the probability of SOD outbreaks at least cost?

## Background

### Sudden Oak Death

Sudden Oak Death (SOD) is an emerging forest disease in California and Oregon that poses risks to forests across North America and Europe. First observed in California in the mid-1990s, SOD is caused by the water mold Phytophthora ramorum (Rizzo et al. 2002). P. ramorum often kills tanoak (Notholithocarpus densiflorus), which provide habitat and food to many vertebrate species, and are the primary host of ectomycorrhizal fungi in redwood forests (Rizzo and Garbelotto 2003). Loss of this species may have cascading effects on other species. The disease has also caused significant economic damage through removal costs and property value reduction (Kovacs et al. 2011). It has the potential to spread to species in other regions, such as northern red oak (Quercus rubra), one of the most important eastern timber species (Rizzo et al. 2002).

### Population Structure: Multiple Host Species

Phytophthora ramorum has over 100 host species in 40 genera, which fall into several functional types (Swiecki and Bernhardt 2013). In canker hosts, which are all members of Fagaceae in California, pathogen produces cankers on the trunk. Few if any spores are produced from the cankers, and these hosts are generally dead ends. SOD is fatal to many canker costs. In foliar hosts, P. ramorum resides and reproduces in leaves and twigs. Foliar hosts vary greatly in susceptibility to disease and spore production from diseased individuals. Some hosts are dead ends; others are major drivers of disease spread and survival.

In Redwood forests in Northern California, the hosts of importance are tanoak and bay laurel (Umbellularia californica). Tanoak has the dubious distinction of being the only species that is both a canker and foliar host, and the disease is generally lethal in these trees. Bay laurel is a foliar host in which P. ramorum produces prolifically, and bay suffers no harm from the disease (DiLeo et al. 2009). Redwood and other species are unsusceptible or dead-end hosts where P. ramorum has little detrimental effect. In forests of the redwood-tanoak-bay complex, bay Laurel acts as the primary disease reservoir [@Davidso2008]. Tanoak infection and mortality is dependent on bay density (Cobb et al. 2012), and removal of bay is a commonly suggested treatment to reduce risk of SOD outbreaks (@Filipe2013; Swiecki and Bernhardt 2013).

Dobson (2004) examined a model of disease in multiple host species based on an the SIR (Kermack and Mckendrick 1927) framework. In it, species had density dependence but interacted only though disease transmission. Importantly, transmission between species was always equal or less than transmission between species. Dobson found that in when transmission was density-dependent, species had a complementary effect on the disease’s rate of reproduction $$(R_0)$$; that is, a mix of both species led to faster outbreaks. However, in the frequency-dependent case, $$R_0$$ was greatest when one species dominated. Transmission between species dampens and synchronizes disease oscillations in each species, and at very high contact rates between species the most susceptible species are more like to go extinct. Craft and Hawthorne (2008) implemented a stochastic version of this model in a system of mamallian carnivores, finding that species with low intra-species contact rates acted as sinks for disease, and their contact with more social species increased spread disease within these low-contact species.

SOD dynamics differ from these systems in a few important ways. First, the contact structure among species may not be symmetrical. Contact rates among trees are driven by differences in species transmissivity (amount of spore produced) and susceptibility (probability of infection per spore) than social grouping. Where one species with high transmissivity coexists with a species of high susceptibility, the probability of transmission may be higher between species than within. Also, the spatial structure of species, and the dispersal pattern of spores, may result in a hybrid between density and frequency-dependent disease transmissions.

### Population Structure: Size/Stage Classes

Within the tanoak population, epidemiological characteristics vary with tree size. Trees of larger size classes are more likely to be infected and die more quickly than smaller tree (Cobb et al. 2012), Possibly due to the vulnerability of cracked bark and the amount of bark tissue available for invasion (Swiecki and Bernhardt 2005). There is no evidence currently for variation in spore production or other physiological effects across tree sizes [@Davidson2008].

When disease dynamics occur much faster than demographic processes, we can treat size classes similarly to different species, and examine the progression of disease in a constant population structure. However, when disease progression and growth occur at similar rates, these processes can interact to produce complex dynamics (Klepac and Caswell 2010). This is the case with SOD; infectious periods can last many years, during which trees may continue to grow.

The effects of age-based population structure and contact rates have been studied extensively in the context of human disease and vaccination programs (Anderson and May 1985, Metcalf et al. 2012). Reducing in-class transmission rates at young ages increases the average age of infection, independent of the exact contact structure. When only the susceptibility, and not transmissivity, of each age class varies. Klepac and Caswell (2010) found that increases in within-class transmission increases both the infected and recovered population of those classes, and classes with high transissivity have higher equilibrium populations relative to low-transmissivity classes.

### Population Structure: Space

P. Ramorum dispersal interacts with host population structure at many scales. The pathogen spreads between trees via wind-blown rain and splash, limiting most spores to spread within 15m of host plants (Davidson et al. 2005). However, occaisional weather events, such as fog, can transport spores up to 3 km (Rizzo et al. 2005). Over longer distances, P. ramorum can be transported in streams or spread via human-mediated vectors such as nursery plant trade [@Osterbauer2004]. Meentemeyer et al. (2011) found that the best fit kernel for a statewide model of P. ramorum spread was the sum of two Cauchy kernels, one on the scale of tens of meters and another on the scale of tens of kilometers.

Despite the ability to spread long distance, strong meter-scale gradients mean that individual trees in stands can not be considered “well mixed” in terms of contact rates between trees. Rather, contact patterns across space arise from the interaction of spatial clustering of trees and the dispersal kernel of the disease. Neither frequency- nor density-dependent transmission characterizes this arrangement.

The effect of such spatial structure on development of epidemics has been studied with continuous populations in space [@Bolker1999], metapopulation models (Park et al. (2001); Park et al. (2002)), models of individuals on a lattice (Filipe and Maule 2003, 2004) and discrete individuals in continuous space (Brown and Bolker (2004)). These approaches have reached similar conclusions of the effect of spatial structure in non-mobile populations. In all cases, the threshold of disease growth for a global outbreak $$(R_0)$$ is greater than the threshold for a local epidemic $$(R_L)$$. For instance, in metapopulations on a lattice, $$R_0 = R_L (1 + z\varepsilon)$$, where $$z$$ is the average number of connections between patches and $$\varepsilon$$ is the inter-patch contact rates (Park et al. 2001). In models of discrete individuals in space, $$R_0 = \lambda(1 + \bar{\mathcal{C}} SI)$$, where $$\lambda$$ would be the pathogen growth rate in a well-mixed population, and $$\bar{\mathcal{C}}$$ is the dispersal kernel-weighted spatial correlation between susceptible and infected individuals. $$\bar{\mathcal{C}}$$ evolves over time but reaches a minimum that represents the threshold required for a global epidemic (Brown and Bolker 2004). Also, both clustering and anti-clustering (oversdispersal) reduce the rate of increase of disease, except when clustering occurs at the scale of dispersal, in which case it can accellerate spread. Fat-tailed dispersal kernels accellerate spread, as well.

## Study Approach

I will examine the comparative effects of the above components of population structure on SOD epidemics by (1) characterizing the dynamic behavior of models that include each type of structure and their combinations and (2) identifying the model structure that best predicts data of disease spread over time in redwood-tanoak-bay forests. Using the best-fit model, I determine optimal control plans for stand protection efforts to minimize the risk of loss of host plants.

### Determining the Consequences of Population Structure via Comparative Dynamics

1. How do interactions between community, size, and spatial structure affect disease dynamics in forests?

2. What aspects of population structure are most important in determining the probability and size of forest disease outbreaks in theoretical models?

To characterize the effects of the population structure on disease dynamics, I will compare four models. All are extensions of the epidemiological model of Cobb et al. (2012).

Model A is the simplest of the four models, only representing structure in the form of differences between species of trees. It is described by this system of stochastic difference equations:

\begin{aligned} \boldsymbol{S}_{t+1} &\sim \boldsymbol{S_t} + \overbrace{ \text{Pois}\left[\boldsymbol{b(S_t+I_t)}\left(1-\sum_{i=1}^n w_i (S_{it} + I_{it}) \right) + \boldsymbol{rm_I I_t}\right]}^{\text{new recruits}} - \overbrace{\text{Binom}_1\left((\underbrace{1 - e^{-\boldsymbol{\beta I_t} - \boldsymbol{\lambda_{ex}}}}_{\text{force of infection}})\boldsymbol{S_t} \right)}^{\text{infections}} -\overbrace{\text{Binom}_2(\boldsymbol{m_S S_t})}^{\text{mortality}} \\ \boldsymbol{I}_{t+1} &\sim \boldsymbol{I_t} + \text{Binom}_1\left((1 - e^{-\boldsymbol{\beta I_t} - \boldsymbol{\lambda_{ex}}})\boldsymbol{S_t} \right) - \text{Binom}_3(\boldsymbol{m_I} \boldsymbol{S_t}) \end{aligned}

Here $$\boldsymbol{S}_t$$ and $$\boldsymbol{I}_t$$ are vectors of the populations of susceptible and infected individuals of each species, and $$\boldsymbol{\beta}$$ is the matrix of contact rates. Other parameters are listed below in Table 1.

Model A Parameters
Parameter Vector Symbol Description
$$\boldsymbol m_{S,I}$$ Probability of death per year, for both susceptible and infectious trees
$$\boldsymbol b$$ Fecundity per individual per year.
$$\boldsymbol r$$ Probability of resprouting after death by disease.
$$\boldsymbol w$$ Competitive coefficient (relative contribution to density-dependent recruitment)
$$\boldsymbol \lambda_{ex}$$ Rate of contact of each species with pathogen spores from outside the site

New susceptible trees enter the system via density-dependent seedling recruitment and resprouting from recently killed trees. Susceptible trees become infected at rates proportional to contact with infected trees (density-dependent), and pathogen migrating into the system. Both susceptible and infected trees die at constant, but different, rates.

The model is modified from Cobb et al. (2012) in several ways: (1) inclusion of demographic stochasticity, (2) conversion from continuous to discrete time, (3) the inclusion of $$\lambda_{ex}$$, the force of infection from areas outside the system, and (4) the exclusion of recovery of infected trees, which has not been observed in the field. Finally, the only population structure in Model A is the difference in species parameters. It excludes age structure and spatial structure, assuming a well-mixed population and frequency-dependent transmission (Following Diekmann et al. (1990))

Model B adds stage structure to Model A. In this case, the vectors $$\boldsymbol{S_t}$$ and $$\boldsymbol{I_t}$$ represent the population divided by species and size class as follows

\begin{aligned} S'_t &\sim S_t - \text{Binom}_1\left((1 - e^{-\boldsymbol{\beta I_t} - \boldsymbol{\lambda_{ex}}})\boldsymbol{S_t} \right) \\ I'_t &\sim I_t + \text{Binom}_1\left((1 - e^{-\boldsymbol{\beta I_t} - \boldsymbol{\lambda_{ex}}})\boldsymbol{S_t} \right) \\ S_{t+1} &\sim \text{Multinom}(\boldsymbol{A_S(S'_t,I'_t)S'_t}) + \text{Pois}\left[\boldsymbol{b(S'_t+I'_t)}\left(1-\sum_{i=1}^n w_i (S'_{it} + I'_{it}) \right) + \boldsymbol{rm_I I'_t}\right] \\ I_{t+1} &\sim \text{Multinom}(\boldsymbol{A_I(S'_t,I'_t)I'_t}) \end{aligned}

Here $$\boldsymbol{A}$$ is the matrix of demographic rates specifying transitions between classes and mortality. $$\boldsymbol{A}$$ is a block-diagonal matrix of size transition matrices, with no transitions between species classes.

Disease transitions are separated in time from demmographic transitions (Klepac and Caswell 2010). In California, P. Ramorum reproduces and spreads during the winter rainy season while most tree growth occurs in the spring and fall.

Models 3 and 4 modify Models 1 and 2 by adding spatial structure in the form of a lattice metapopulation. $$\boldsymbol{S_t}$$ and $$\boldsymbol{I_t}$$ become $$\boldsymbol{S_{jt}}$$ and $$\boldsymbol{I_{jt}}$$, matrices of each species and class at each location. Furthermore, the force of infection $$\left((1 - e^{-\boldsymbol{\beta I_t} - \boldsymbol{\lambda_{ex}}})\boldsymbol{S_t} \right)$$ is replaced with an overall force of infection $$\Lambda_{jt}$$ at each location

$\Lambda_{jt} = \boldsymbol\beta \sum_{x=1}^J \phi (\boldsymbol I_{xt}, j, \boldsymbol{\theta}) + \lambda_{ex}$

Here $$j$$ is the location, $$\phi$$ is the dispersal kernel of P. ramorum from all locations $$x$$ to $$s$$, and $$\theta$$ is a vector of parameters of the dispersal kernel. $$\boldsymbol{\beta}$$ remains the contact matrix, representing physiological and vertical dispersal components of spore dispersal. I will test the models using normal, exponential, and power-law disperal kernels.

Using these four models, I will determine:

• How global epidemic growth rate $$R_0$$, probability of global epidemic, the epidemic size, and time to extinction of both hosts and pathogens vary between the models
• How the presence of each form of population structure affect these values when global species densities and mean parameter values remain identical
• Whether the effects of population structure are additive or if they interact in a more complex way.

The model will be parameterized with three species: tanoak, bay, and redwood, (the latter representing all non-host species). Parameters and initial conditions will be drawn from Cobb et al. (2012) and @Filipe2013.

### Selecting the Model that Best Repesents Observed Behavior

3. What aspects of population structure are most important in predicting the probability and size of SOD outbreaks?

To apply the understanding developed in exploring model structure to management, I must determine which model best represents observed dynamics of SOD in forests. I will fit the four models above to data of SOD spread in redwood-tanoak-bay forests in California and compare their ability to predict disease development

#### Data

Data to fit the models comes from a collaboration with the Rizzo lab’s disease monitoring plot network [@Ref; @Ref; 2012]. The data consist of observations tree size, alive/dead status, and disease status for trees greater than 1 cm diameter from the years 2002-2007. Trees were observed in 14 sites along the California coast from X (36.16°N) to Y (38.35°N), each of which contains an average of 8 500 m2, seperated by minimium distances of 100m and arranged in a random linear sequence. All plots are in forests dominated by redwood, tanoak, and bay laurel.

Size, health, and disease status of all trees were measured in 2002 and 2007, as well as for a random sample of 5 previously infected and 5 previously uninfected trees of each species per plot in the years 2003-2006.

Since the time scale of the are not suffiently long to data to determine transition between size classes, the parameters of transition matrices will be drawn from other data from other sites and included as strong priors on these parameters.

#### The Observation Model

The process generating the data above may be considered hidden Markov process (Gimenez et al. 2012), that is, processes where the state of the system is dependent on its previous state, but where states are partially or imperfectly observed. Imperfect observation in this case is due to incomplete counting of trees and error in determination of tree disease status. Tree disease is determined by subjective examination of symptoms in the field, and then confirmed in the lab for those individuals with sufficiently suggestive symptoms. There are both false positives and false negatives:

\begin{aligned} p_{OI} &= p(\text{Observing that tree is diseased}|\text{tree is diseased}) \\ p_{OS} &= p(\text{Observing that tree is healty}|\text{tree is healthy}) \end{aligned}

Then, for each sub-population (by species, size class, or location) observed in each time period, then, the observed number of susceptible and infected trees are:

\begin{aligned} S_{obs} \sim \text{Binom}\left(p_{OS} S + (1-p_{OI}) I \right) \\ N_{obs} \sim \text{Binom}\left(p_{OI} I + (1-p_{OS}) S \right) \end{aligned}

OMITTED HERE: OBSERVATION MODEL FOR PARTIAL CENSUS YEARS THAT I REALIZED WAS WRONG

Finally, for models with size-class structure, the observations omit the smallest size class (seeds and seedlings < 1 cm), as these were not measured.

#### Model Fitting

Particle filtering determines the likelihood of a hidden Markov process by taking advantage of the fact that, at each time step, the likelihood of a system state is dependent on the previous state $$(X_{t+1})$$, and the observation $$(Y_t)$$.

$p(X_t) = p(X_t | X_{t-1}, Y_t)$

While $$X_{1:t}$$ is hidden, $$p(X_t|X_{t-1})$$ may be approximated by by simulation of the model process. Repeated simulations (particles) provide a distribution of outcomes from which average likelihood are determined from the observation model $$(p(X_t|Y_t))$$. For each time step, the particles are re-sampled according to their individual likelihoods, in order to prevent particle degeneracy - most particles approaching zero likelihood. The product of all steps’ averaged particle likelihoods is an unbiased estimate of the model likeihood.(Arulampalam et al. 2002)

With an estimate of likelihood available, one needs an approach to determine the maximum-likelihood estimate of the model parameters $$(\theta)$$. Several methods are available. Iterated filtering (IF) estimates $$\theta$$ by replacing constant parameter values with a random walk $$\theta_t$$ of decreasing variance over time, and determining the “states” of this walk as the variance approaches zero. IF is computationally efficient but requires long time series.

Alternatively, Particle Filter Markov Chain Monte Carlo (PFMCMC) uses a Markov Chain sampler (e.g., Metropolis-Hastings), calculating the likelihood at each iteration using a particle filter. While computationally expensive, it does not require long time series and may be used with multiple time series. PFMCMC also integrates particle filtering into a Bayesian context, allowing the use of informative priors. I will used PFMCMC to determine maximum-likelihood parameterizations for all four models.

When fitting models 1 and 2 to the data, populations from each plot in each site will be aggregated. For models 3 and 4, each plot will represent a single sub-population on a lattice spanning the site. Initial density, species, age and disease distributions for other sub-populations will be randomly generated from other points on the lattice using the proportions from the measured plots. Epidemiological parameters will remain constant across sites except for the external force of disease $$\lambda_{ex}$$, which will be represented as a noramlly distributed random variable to account for site effects. varies.

#### Model selection

Model selection criteria have multiple purposes: to determine which model best represents “true” processes, and to estimate the utility of models for the purposes of prediction. In this case, I am interested in both - determining which components population structure matter, and also providing an estimate of predictive stength of the best model. For these purposes, I will use the deviance information criterion (Spiegelhalter et al. 2002, DIC). DIC has several advantages. Since it estimates the model complexity, or effective number of parameters, directly from the likelohood distribution, it makes it easy to incorporate random parameters such as $$\lambda_{ex}$$ that have non-integer contributions to model complexity. Secondly, it estimates the how priors on parameter values effect their contribution to model complexity.

DIC is not suitable for model-averaging. However, model-averaged results will not likely be useful for the optimization analysis below. Using the best-DIC model for prediction is a compromise between predictive ability and tractable analysis of the model used for prediction. Use of DIC assumes that there is a “best” or “real” model within the comparison. While no models are “true” (Box 1976), this excercise is selecting a “best” or “most useful” model for scenario exploration.

### Using Dynamic Optimization to Determine the Best Control Strategy

4. What control strategy minimizes the probability of SOD outbreaks at least cost?

The best-fit model that captures essential aspects of the disease dynamics will be used to develop an optimal control solution to minimization of disease risk over time.

While forest disease management can occur under multiple, conflicting and uncertain goals and priorities, here I consider the case where conservation of tanoak and its ecosystem services are the goal. This is approximately the case in Humboldt County, California, where large landowners have agreed to work in tandem to control the disease. In this situation, managers goals can be simplified to three components: (1) The minimum acceptable density of the species of interest $$(N_{min})$$, (2) the cumulative acceptable probability that the species will fall below that density $$(p_{acc})$$ during the management period, and (3) the cost of disease control $$C$$. Using acceptable probabilities avoids the challenge of defining and valuing the flows of ecosystem services as a function of the forest composition (Hartman 1976).

Tools for SOD control can be categorized into two types. Silvicultural (“host-centric”) treatments change forest composition. “Pathogen-centric” treatments, such as quarantine, equipment cleaning and spraying, reduce the probability of spore arrival (Swiecki and Bernhardt 2013).

I represent these two types of treatment with two control functions. Cutting is a periodic control treatment that can be imposed in any year. I define the cut as a reduction in species density to the levels that would minimize the probability of species densities falling below the acceptable level over a management period without further controls. This will be a value that balances stochastic risk of extinction with risk of diseae outbreak, bounded by the minimum acceptable species density. As tanoak and bay laurel trees have little timber value, each cut has a cost $$(c_1)$$. For the purpose of this model $$c_1$$ is independent of the extent of density reduction (that is, the variable costs of cutting are negligible. Thus, the annual cost of cutting is $$c_1 E_1$$, where $$E_1$$ is a zero-one control variable

Spraying is an ongoing control treatment that represents the combined annual effort to reduce the arrival of new spores at the site. This modifies the external force of infection

$$\lambda_{ex} \equiv \lambda_0 \times (1 - e^{-a E_2 c_2})$$

where $$a$$ is the effectiveness of treatment, $$E_2$$ is the effort spent at spraying and $$c_2$$ is the annual cost per unit effort.

The dynamic optimization problem is to determine the combination of treatment effort over time $$(E_1t, E_2t)$$ that minimizes the cost of treatment while maintaining the probability of species loss at an acceptable level. That is,

$\min \sum_{t=1}^T c_1(E_1)+ c_2(E_2) \text{ s.t. } p(N_t < N_{\min}) < p_{acc}$

An alternative formulation, also useful to managers, is to minimize the probability of outbreak given budget constraints $$(B)$$. Budget constraints could be forumulated as total budgets for the entire mangement period, or annual limits on spending. That is

$\min \sum_{t=1}^T p_{(N_t < N_{\min})} \text{ s.t. } \sum_0^T c_1 (E_2) + c_2(E_2) \leq B_T$

or

$\min \sum_{t=1}^T p_{(N_t < N_{\min})} \text{ s.t. } c_1(E_1) + c_2(E_2) \leq B_{ann} \forall \, T$

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