Several epidemiological, evolutionary and economic outputs can be
generated by landsepi and represented in text files
(writeTXT=TRUE), graphics (graphic=TRUE) and a
video (videoMP4=TRUE).
See equation below, as well as ?epid_output and
?evol_output for details on the different output
variables.
Equations
In the following equations, \(H_{i,v,t}\), \(L_{i,v,p,t}\), \(I_{i,v,p,t}\) and \(R_{i,v,p,t}\) respectively denote the
number of healthy, latent, infectious and removed host individuals,
respectively, in field \(i\) (\(i=1,…,J\)), for cultivar \(v\) (\(v=1,…,V\)) at time step \(t\) (\(t=1,…,T
\times Y\) with \(Y\) the total
number of cropping seasons and \(T\)
the number of time-steps per season).
\(AUDPC_{v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J} \sum_{p=1}^{P}
\{I_{i,v,p,t}+R_{i,v,p,t}\}}{T \times \sum_{i=1}^{J} A_i}\)
\(AUDPC^r_{v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J} \sum_{p=1}^{P}
\{I_{i,v,p,t}+R_{i,v,p,t}\}}{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J}
\left(H_{i,v,t}+\sum_{p=1}^{P}
\{L_{i,v,p,t}+I_{i,v,p,t}+R_{i,v,p,t}\}\right)}\)
\(GLA_{v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J} H_{i,v,t}}{T \times
\sum_{i=1}^{J} A_i}\)
\(GLA^r_{v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J}
H_{i,v,t}}{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J}
\left(H_{i,v,t}+\sum_{p=1}^{P}
\{L_{i,v,p,t}+I_{i,v,p,t}+R_{i,v,p,t}\}\right)}\)
\(Yield_{v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J} \left( y_{H,v} \times
H_{i,v,t} + \sum_{p=1}^{P} \{y_{L,v} \times L_{i,v,p,t} + y_{I,v} \times
I_{i,v,p,t} + y_{R,v} \times R_{i,v,p,t}\} \right)}
{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J} H^*_{i,v,t}}\) with
\(y_{H,v}\), \(y_{L,v}\), \(y_{I,v}\) and \(y_{R,v}\) the theoretical yield of cultivar
v in pure crop (in weight or volume unit/ha/season) associated with the
sanitary statuses ‘H’, ‘L’, ‘I’ and ‘R’, respectively. \(H^*_{i,v,t}\) is the number of healthy
hosts in a pure crop and in absence of disease.
\(Operational\_cost_{v,y}=planting\_cost_v
\times \frac{\sum_{i \in \Omega_{v,y}}(area_i \times prop_{v,i})}
{\sum_{i \in \Omega_{v,y}}area_i} + treatment\_cost \times \frac{\sum_{i
\in \Omega_{v,y}}(TFI_{i,v,y} \times area_i \times prop_{v,i})} {\sum_{i
\in \Omega_{v,y}}area_i}\) with \(\Omega_{v,y}\) the set of polygon indices
where cultivar \(v\) is cultivated at
year \(y\), and \(prop_{v,i}\) the proportion of cultivar
\(v\) in polygon \(i\) (for mixtures). \(TFI\) stands for the Treatment Frequency
Indicator (number of treatment applications per ha).
\(Product_{v,y}=yield_{v,y} \times
market\_value\)
\(Margin_{v,y} = Product_{v,y} -
Operational\_Cost_{v,y}\)
\(Contrib_{p,v,y} =
\frac{\sum_{t=t^0(y)}^{t^f(y)} \sum_{i=1}^{J}
\{L_{i,v,p,t}+I_{i,v,p,t}+R_{i,v,p,t}\}} {\sum_{t=t^0(y)}^{t^f(y)}
\sum_{i=1}^{J} \sum_{p=1}^{P}
\{L_{i,v,p,t}+I_{i,v,p,t}+R_{i,v,p,t}\}}\)
\(freq(I)_{p,t} = \frac{\sum_{i=1}^{J}
\sum_{v=1}^{V} I_{i,v,p,t}} {\sum_{p=1}^{P} \sum_{i=1}^{J}
\sum_{v=1}^{V} I_{i,v,p,t}}\)
With respect to evolutionary outputs, for each pathogen genotype
(evol_patho) or phenotype (evol_aggr, note
that different pathogen genotypes may lead to the same phenotype on a
resistant host, i.e. level of aggressiveness), several computations are
performed:
- appearance: time to first appearance (as propagule);
- R_infection: time to first true infection of a resistant host;
- R_invasion: time to invasion, when the number of infections of
resistant hosts reaches thres_breakdown, above which the
genotype or phenotype is unlikely to go extinct.
The value Nyears + 1 time step is used if the genotype or
phenotype never appeared/infected/invaded.
Durability is defined as the time to invasion of completely adapted
pathogen individuals.
