PLP 6404 Epidemiology of Plant Diseases Spring Lecture 17: Disease progress in space: dispersal and disease gradients

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1 PLP 6404 Epidemiology of Plant Diseases Spring 2015 Lecture 17: Disease progress in space: dispersal and disease gradients Dispersal processes Dispersal - the movement of propagative units of a pathogen from the place where they are formed Aerial dispersal of fungal spores requires 3 interdependent events (repeat): 1. liberation - the process by which propagules are removed from where they developed and released into the dispersal medium. 2. transport - the process by which liberated propagative units are dispersed through a turbulent medium, such as air or water. 3. deposition - the placement of a propagule on a surface Dispersal and disease gradients Dispersal gradient - change in inoculum concentration with distance from a source Disease gradient - change in disease or infections with distance from source of inoculum May be caused by: 1. gradient in deposition 2. gradient in environment 3. gradient in susceptibility of host Sources of gradients: Primary vs. secondary gradients: Primary: initial entry of inoculum or vectors (virus diseases) into a field from outside or from surviving propagules in soil or plant debris Secondary: gradient in secondary inoculum produced in the course of the epidemic or in secondary spread by vectors (offspring from primary vectors)

2 Gregory's 5 principles for dispersal gradients (1968): 1. A dispersal gradient implies a local source of inoculum 2. A dispersal gradient requires a population of susceptible hosts 3. Secondary spread (dispersal) flattens a primary gradient 4. Background contamination also flattens a primary gradient 5. Use multiple-infection-transformation (same as monomolecular) if using incidence data Disease gradients are affected by the size, shape and height of the inoculum source Above-ground vs. ground-level source? Wind-dispersed vs. splash-dispersed propagule? Modeling dispersal or disease gradients Model: disease intensity (Y) = function of distance (s; in other handouts called d) Empirical dispersal models 1. Gregory or Power law model: - based on probability of spore landing in a unit distance is inversely proportional to distance from source s s 2. Exponential model: - based on assumption that probability of a spore landing in a unit distance is independent of the distance from source s s 3. Gaussian (normal) distribution model: Which model is best? Fitt et al. (1987) concluded: 1. Neither power law nor exponential was consistently better than the other 2. Splash-dispersed pathogens better described by the exponential model 3. Small propagules (<10 :m) better described by power law model Aylor (1985; 1987) - dispersal is determined by multiplicative effect of 2 factors: dilution deposition

3 Physical models of dispersal - developed for air pollutants 1. Gaussian plume model: atmospheric dispersion model - models concentration of propagules in 3 dimensions downwind from a source - spore/propagule release is independent of wind 2. Gradient transfer theory [don t learn]: - used in models that describe the dispersal of particles and gases in the atmosphere - analogous to diffusion of molecules, propagules move from higher to lower concentrations at a rate that is proportional to the concentration gradient - we will not work with these models 3. Random walk models [don t learn]: - simulate trajectories of individual propagules in packages of air - can account for intermittent spore release and deposition processes - potentially the most realistic biologically of the 3 physical dispersal models - we will not work with these models Disease gradient models (or disease spread models; only in space) Models of disease gradients are modifications of the exponential and power law models and can be divided into 2 categories: 1. Those that do not correct for host number or size 2. Those that do correct for host number or size Models that do not correct for host number or size: Model 1 (exponential): dy/ds = -by Rate is independent of distance Integrated form: y = a e -bs Linearized form: ln(y) = ln(a) b s Model 2 (Gregory or Power): dy/ds = -by/s dy/ds is directly proportional to y, but inversely proportional to s Integrated form: y = a s -b Linearized form: ln(y) = ln(a) b ln(s)

4 Models that do correct for host number or size: Model 3: dy/ds = -b (1-y) Analogous to monomolecular model Does not depend on distance from source (not very realistic) Integrated form: y = 1 a e bs Linearized form: ln[1/(1-y)] = -ln(a) b s Model 4: dy/ds = -by (1-y) Analogous to logistic model Rate depends on healthy and diseased tissue, but not s Integrated form: Linearized form: ln[y/(1-y)] = -ln(a) b s Model 5: dy/ds = -b(1-y)/s Rate is inversely proportional to distance, s But does not depend on diseased tissue, y Log-transformation is version of the power law Integrated form: y = 1 a s b Linearized form: ln[1/(1-y)] = -ln(a) b ln(s) Model 6: dy/ds = -by(1-y)/s Rate is directly proportional to y and inversely proportional to s Most appropriate for a polycyclic epidemic when there is a large number of propagules or large amount infectious diseased tissue at the source compared to the rest of the host area Integrated form: Linearized form: ln[y/(1-y)] = -ln(a) b ln(s) Epidemic development in both space and time Relationship between development of disease in time and space, arising from a single focus or foci Jeger developed several 3-dimensional models to express the progress of disease in time and space. Here are just two of them [don t learn by heart]. 1) y = 1 a exp (bs r M t) (monomolecular) 2) y = 1/[1 + a exp(bs r L t)] (logistic).

5 Isopaths: contours of equal disease levels Example: Isopath map of anthracnose severity In an avocado orchard in Michoacan, Mexico, In (Teliz-Ortiz et al Proc. World Avocado Congress. pp ) Rate of isopath movement velocity of focus expansion If velocity is constant traveling wave For example: when you throw a stone in water the waves will move at constant speed. The same is true for an epidemic starting from a focus, when you assume that the dispersal gradient is Gaussian; daughter foci will not be formed and the margin of the focus is expanding at a constant rate. If velocity is increasing (accelerating) dispersive wave In reality, the dispersal gradient often has a longer tail than the Gaussian distribution, and a focus would be expanding an increasing speed.