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<h1 id="page.title">Forest Disease Dynamics and Management - Outline</h1>
<p><em>Noam Ross</em></p>
<p>12-12-12 12:00:06</p>
<hr />
</div>
<div class="contenttext">
<h2 id="overall-questions">Overall Questions</h2>
<ul>
<li>How does forest disease spread through forest systems?</li>
<li>What aspects of forest structure (species composition, age and spatial strucuture) are important in predicting the spread and impact of disease?</li>
<li>What forest composition minimizes the risk of outbreak while maintaining desirable populations of at-risk species?</li>
<li>What is the most cost-effective strategy for maintaining populations of tree species?</li>
</ul>
<h2 id="study-system---sudden-oak-death-in-redwoodtanoakbay-forests">Study System - Sudden Oak Death in Redwood/Tanoak/Bay Forests</h2>
<ul>
<li>Caused by water mold <em>Phytophthora ramorum</em> <span class="citation">(Rizzo et al. 2002)</span></li>
<li>Spreads via wind-blown raid and fog, as well as human vectors (short- and long-distance dispersal) <span class="citation">(Meentemeyer et al. 2011)</span></li>
<li>Dynamics dependent on forest composition:
<ul>
<li>Wide host range, but lethality and transmissivity vary by host</li>
<li>Lethal in tanak, spreads most widely from Bay Laurel, little effect or spread in Redwood and other species <span class="citation">(DiLeo et al. 2009)</span></li>
<li>Lethality varies by tree size <span class="citation">(Cobb et al. 2012)</span></li>
</ul></li>
</ul>
<h2 id="general-apparoch">General Apparoch</h2>
<ul>
<li>Extend mechanistic model from <span class="citation">Cobb et al. (2012)</span> that incorporates disease and stand dynamics.</li>
<li>Build series of nested models of varying complexity that incorporate
<ul>
<li>Species differences in demographic and epidemiological parameters</li>
<li>Size structure in demographic and epidemiological parameters</li>
<li>Density dependence on recruitment, growth, and mortality</li>
<li>Shape of spore dispersal kernel</li>
</ul></li>
<li>Compare models to determine best predictor of disease outbreak</li>
<li>Determine optimal combination of host- (harvesting) and pathogen-centric (spraying) controls over management periods, given constraints, to minimize probability of disease outbreak</li>
</ul>
<h2 id="model">Model</h2>
<p>SIS model with a structured population, implemented in discrete time and space.</p>
<p>Forest stand is represented as a lattice, with populations of trees in each grid cell. Populations in each cell are divided into classes with the following attributes:</p>
<ul>
<li>Species</li>
<li>Size class</li>
<li>Disease status (S or I)</li>
</ul>
<p>Each class has its own demographic and epidemiological parameters:</p>
<table>
<caption>Class-specific demographic and epidemiological parameters</caption>
<col width="44%" />
<col width="52%" />
<thead>
<tr class="header">
<th align="center">Parameter Symbol</th>
<th align="left">Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">\(\boldsymbol m\)</td>
<td align="left">Probability of death per year.</td>
</tr>
<tr class="even">
<td align="center">\(\boldsymbol b\)</td>
<td align="left">Fecundity per individual per year.</td>
</tr>
<tr class="odd">
<td align="center">\(\boldsymbol g\)</td>
<td align="left">Probability of transition to the next size class.</td>
</tr>
<tr class="even">
<td align="center">\(\boldsymbol R'\)</td>
<td align="left">Probability of recovery from disease.</td>
</tr>
<tr class="odd">
<td align="center">\(\boldsymbol r\)</td>
<td align="left">Probability of resprouting after death by disease.</td>
</tr>
<tr class="even">
<td align="center">\(\boldsymbol w\)</td>
<td align="left">Relative space taken up by an individual tree of the class, or the competitive coefficient.</td>
</tr>
<tr class="odd">
<td align="center">\(\boldsymbol\theta\)</td>
<td align="left">Spore production and dispersal kernal parameters</td>
</tr>
<tr class="even">
<td align="center">\(\boldsymbol E\)</td>
<td align="left">Effect of density on \(m\),\(b\),and \(g\).</td>
</tr>
</tbody>
</table>
<p>In different variations of the nested model, these parameters will either be held at zero, be held constant across classes or allowed to vary across classes.</p>
<p>At each time step, populations transition between classes and new individuals enter and exit each cell via recruitment and mortality. Transition between S and I depends on the class-specific force of infection \((\boldsymbol\Lambda)\). \(\boldsymbol\Lambda\) on each class depends on both the burden of spores arriving in the cell and the class’s susceptibility to those spores. Spores arriving in the cell are calculated by summing the dispersal of spores for each infected class in the lattice, to get a vector of spore burden from each class for a location. This is multiplied by a “Who Acquires Infection From Whom” matrix (WAIFW, <span class="citation">Anderson and May (1985)</span>) to get \(\boldsymbol \Lambda\). There will be an addition component of \(\Lambda\) representing the force of infection from arrival of spores from outside the stand.</p>
<p>Stochasticty will be incorporated into the model. Spore production and transmission will be subjec to environmental stochasticity, and recruitment, demographic transitions, and infection will be subject to demographic stochasticity.</p>
<h3 id="essential-topics-and-papers-related-to-disease-modeling">Essential topics and papers related to disease modeling</h3>
<ul>
<li>SIR Models: <span class="citation">Kermack and Mckendrick (1927)</span>, <span class="citation">Kermack and McKendrick (1932)</span>, <span class="citation">Kermack and McKendrick (1933)</span>, <span class="citation">Anderson (1991)</span></li>
<li>Host heterogeneity: <span class="citation">Anderson and May (1985)</span></li>
<li>Multispecies transmission: <span class="citation">Dobson (2004)</span>, <span class="citation">Craft and Hawthorne (2008)</span></li>
<li>Age-structured models: <span class="citation">Metcalf et al. (2012)</span>, <span class="citation">Klepac et al. (2009)</span></li>
<li>Plant-specific spatial models: <span class="citation">Ferrari et al. (2006)</span>, <span class="citation">Bolker et al. (2009)</span>, <span class="citation">Gilligan and van den Bosch (2008)</span></li>
<li>Stochastic spatial spread: <span class="citation">Lewis and Pacala (2000)</span></li>
<li>Kernels for spatial spread on a discrete grid: <span class="citation">Chesson and Lee (2005)</span></li>
</ul>
<h2 id="the-model-selection-process">The model selection process</h2>
<p><strong>Goal</strong>: Indentify the most parsimonious model structure for predicting the spread of disease and mortality through the forest stand</p>
<h3 id="model-fitting">Model fitting</h3>
<ul>
<li>Model data as a <em>hidden Markov process</em> <span class="citation">(Gimenez et al. 2012)</span>
<ul>
<li>Smallest size classes (seeds, seedlings) unobserved</li>
<li>Disease observations imperfect - binomial process with underlying probability dependent on tree class and disease status</li>
</ul></li>
<li>With likelihoods for the mechanistic model intractable, likelihoods can be calculated using particle filtering <span class="citation">(Arulampalam et al. 2002)</span></li>
<li>Maximum likelihood estimates of parameters calculated using <em>iterated filtering</em> <span class="citation">(Ionides et al. 2006, He et al. 2010)</span></li>
</ul>
<h3 id="model-selection">Model selection</h3>
<ul>
<li>Fit all possible combinations of models:
<ul>
<li>Including or excluding size structure of each species (by fixing parameters across classes)</li>
<li>Including or excluding density dependence on reproduction, growth, and mortality</li>
<li>Including or excluding disease effects on reproduction, growth, and mortality</li>
<li>Normal, fat-tailed, or uniform (no spatial structure) dispersal</li>
</ul></li>
<li>Compare models via AICc. Eliminate models with AICc weights < 4x the lowest AICc weight (following <span class="citation">Maunder and Deriso (2011)</span>)</li>
</ul>
<h3 id="other-essential-topics-and-papers-related-to-model-fitting">Other Essential topics and papers related to model fitting</h3>
<ul>
<li><span class="citation">Hartig et al. (2011)</span> and <span class="citation">Wood (2010)</span> describe a summary statistic approach (Approximate Bayesian Computing), also implemented in the <code>pomp</code> package in R [@King2010]. An example in <span class="citation">Martínez et al. (2011)</span></li>
<li>More theory about the approach in <span class="citation">Andrieu et al. (2010)</span></li>
<li>If only distinguishing between a couple of models for hypothesis testing, might follow <span class="citation">Boettiger and Hastings (2012)</span></li>
<li>General approach for forecasting, example from dynamic range models: <span class="citation">Pagel and Schurr (2012)</span></li>
</ul>
<h2 id="optimization">Optimization</h2>
<p><strong>Goal</strong>: Determine the optimal course of action for minimizing probability of disease outbreak over a management period, given budget constraints and and species conservation goals</p>
<ul>
<li><p>The best-fit model provides us with estimates of the probability over disease outbreaks in a forest stand over a management period. This probability is contingent on both forest composition and spatial structure, and the burden of spores from other areas, and changes over time as forest dynamics change the host population.</p></li>
<li><p>Both composition and the arrival of disease can be modified by controls. Select cutting may modify the composition and managment actions such as the cleaning of logging equipment can reduce the arrival of spores. Pesticides can also reduce the probability of spores successfully infecting healthy trees. All of these management actions (including logging) have costs, and are limited by budget constraints of management agencies and/or landowners</p></li>
<li><p>The solution to this optimal control problem will be an optimal <em>treatment rotation</em> - a schedule of harvest and spraying actions that minimize risk over the treatment period.</p></li>
<li><p>An alternative useful formulation of the problem is to determine the minimum-cost method given constraints of desired tree population and <em>maximum acceptable probability of outbreak</em>. An advantage of this approach is to provide managers with an method of implicit valuation of the tanoak population by illustrating the trade-off between costs and acceptable tree populations and risks.</p></li>
<li><p>An extension of the analysis above will include the parameter uncertainty of the best-fit model, and an optimization under the scenario that parameter uncertainties decrease with time</p></li>
</ul>
<h3 id="topics-and-papers-relevant-to-optimization">Topics and papers relevant to optimization</h3>
<ul>
<li>Optimal rotation under stochastic risk: <span class="citation">Reed (1984)</span></li>
<li>Forestry with multiple age classes: <span class="citation">Tahvonen (2004)</span></li>
<li>Optimal Management with risk of crossing uncertain thresholds: <span class="citation">Polasky et al. (2011)</span>, <span class="citation">Chen et al. (2012)</span></li>
<li>Risk minimization over time</li>
</ul>
<h2 id="references">References</h2>
<p>Anderson, R. M. 1991. Discussion: the Kermack-McKendrick epidemic threshold theorem. Bulletin of mathematical biology 53:121–134.</p>
<p>Anderson, R. M., and R. M. May. 1985. Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes. J. Hyg. Camb.</p>
<p>Andrieu, C., A. Doucet, and R. Holenstein. 2010. Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 72:269–342.</p>
<p>Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp. 2002. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing 50:174–188.</p>
<p>Boettiger, C., and A. Hastings. 2012. Quantifying limits to detection of early warning for critical transitions.. Journal of the Royal Society, Interface / the Royal Society 9:2527–39.</p>
<p>Bolker, B. M., M. E. Brooks, C. J. Clark, S. W. Geange, J. R. Poulsen, M. H. H. Stevens, and J.-S. S. White. 2009. Generalized linear mixed models: a practical guide for ecology and evolution.. Trends in ecology & evolution 24:127–35.</p>
<p>Chen, Y., C. Jayaprakash, and E. Irwin. 2012. Threshold management in a coupled economic–ecological system. Journal of Environmental Economics and Management 64:442–455.</p>
<p>Chesson, P., and C. T. Lee. 2005. Families of discrete kernels for modeling dispersal.. Theoretical population biology 67:241–56.</p>
<p>Cobb, R. C., J. A. N. Filipe, R. K. Meentemeyer, C. A. Gilligan, and D. M. Rizzo. 2012. Ecosystem transformation by emerging infectious disease: loss of large tanoak from California forests. Journal of Ecology.</p>
<p>Craft, M. E., and P. L. Hawthorne. 2008. Dynamics of a multihost pathogen in a carnivore community. Journal of Animal …:1257–1264.</p>
<p>DiLeo, M. V., R. M. Bostock, and D. M. Rizzo. 2009. Phytophthora ramorum does not cause physiologically significant systemic injury to California bay laurel, its primary reservoir host.. Phytopathology 99:1307–11.</p>
<p>Dobson, A. 2004. Population dynamics of pathogens with multiple host species.. The American naturalist 164 Suppl :64.</p>
<p>Ferrari, M. J., O. N. Bjø rnstad, J. L. Partain, and J. Antonovics. 2006. A Gravity Model for the Spread of a Pollinator-Borne Plant Pathogen. The American Naturalist 168.</p>
<p>Gilligan, C. A., and F. van den Bosch. 2008. Epidemiological models for invasion and persistence of pathogens.. Annual review of phytopathology 46:385–418.</p>
<p>Gimenez, O., J. D. Lebreton, J. M. Gaillard, R. Choquet, and R. Pradel. 2012. Estimating demographic parameters using hidden process dynamic models. Theoretical Population Biology 82:307–316.</p>
<p>Hartig, F., J. M. Calabrese, B. Reineking, T. Wiegand, and A. Huth. 2011. Statistical inference for stochastic simulation models–theory and application.. Ecology letters 14:816–27.</p>
<p>He, D., E. L. Ionides, and A. a King. 2010. Plug-and-play inference for disease dynamics: measles in large and small populations as a case study.. Journal of the Royal Society, Interface / the Royal Society 7:271–83.</p>
<p>Ionides, E. L., C. Bretó, and a a King. 2006. Inference for nonlinear dynamical systems.. Proceedings of the National Academy of Sciences of the United States of America 103:18438–43.</p>
<p>Kermack, W. O., and A. G. McKendrick. 1932. Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proceedings of the Royal society of London. Series A 138:55.</p>
<p>Kermack, W. O., and A. G. McKendrick. 1933. Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proceedings of the Royal Society of London. Series A 141:94.</p>
<p>Kermack, W. O., and A. G. Mckendrick. 1927. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 115:700–721.</p>
<p>Klepac, P., L. W. Pomeroy, O. N. Bjø rnstad, T. Kuiken, A. D. M. E. Osterhaus, and J. M. Rijks. 2009. Stage-structured transmission of phocine distemper virus in the Dutch 2002 outbreak.. Proceedings. Biological sciences / The Royal Society 276:2469–76.</p>
<p>Lewis, M. A., and S. Pacala. 2000. Modeling and analysis of stochastic invasion processes. Journal of Mathematical Biology 41:387–429.</p>
<p>Martínez, I., T. Wiegand, J. J. Camarero, E. Batllori, and E. Gutiérrez. 2011. Disentangling the formation of contrasting tree-line physiognomies combining model selection and Bayesian parameterization for simulation models.. The American naturalist 177:136.</p>
<p>Maunder, M. N., and R. B. Deriso. 2011. A state – space multistage life cycle model to evaluate population impacts in the presence of density dependence : illustrated with application to delta smelt ( Hyposmesus transpacificus ). Canadian Journal of Fisheries and Aquatic Science 1306:1285–1306.</p>
<p>Meentemeyer, R. K., N. J. Cunniffe, A. R. Cook, J. a. N. Filipe, R. D. Hunter, D. M. Rizzo, and C. a Gilligan. 2011. Epidemiological modeling of invasion in heterogeneous landscapes: spread of sudden oak death in California (1990–2030). Ecosphere 2:1–24.</p>
<p>Metcalf, C. J. E., J. Lessler, P. Klepac, A. Morice, B. T. Grenfell, and O. N. Bjø rnstad. 2012. Structured models of infectious disease: Inference with discrete data.. Theoretical population biology 82:275–282.</p>
<p>Pagel, J., and F. M. Schurr. 2012. Forecasting species ranges by statistical estimation of ecological niches and spatial population dynamics. Global Ecology and Biogeography 21:293–304.</p>
<p>Polasky, S., A. de Zeeuw, and F. Wagener. 2011. Optimal management with potential regime shifts. Journal of Environmental Economics and Management 62:229–240.</p>
<p>Reed, W. J. 1984. The effects of the risk of fire on the optimal rotation of a forest. Journal of Environmental Economics and Management 11:180–190.</p>
<p>Rizzo, D. M., M. Garbelotto, J. M. Davidson, G. W. Slaughter, and S. T. Koike. 2002. Phytophthora ramorum and sudden oak death in California: I. Host relationships. Pages 733–740 <em>in</em> Fifth Symposium on Oak Woodlands: Oaks in California’s Changing Landscapes. . USDA Forest Service, San Diego, CA.</p>
<p>Tahvonen, O. 2004. OPTIMAL HARVESTING OF FOREST AGE CLASSES: A SURVEY OF SOME RECENT RESULTS. Mathematical Population Studies 11:205–232.</p>
<p>Wood, S. N. 2010. Statistical inference for noisy nonlinear ecological dynamic systems.. Nature 466:1102–4.</p>
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