A compartmental model for vector transmitted diseases: an application to Xylella fastidiosa

Abstract

[eng] In 2016 the presence of the bacterium Xylella fastidiosa was detected in Majorcan almond trees and was regarded as the causal pathogen agent of the mortality of these trees over the last fifteen years. Xylella fastidiosa, being endemic of the Americas, is an insect-transmitted bacterium which is considered one of the major threats to plants worldwide for its large number of hosts, strains and potential vectors. By 2017 the reported incidence of the bacterium over Majorcan almond trees was 79.5% ± 2.0, presenting a great danger to one of the historical crops of the Balearic Islands. then, the aim of this project is to develop a first step on the study of this epidemic by means of a deterministic compartmental (mean-field) model. In particular, we present a model for the vector assisted transmission of the pathogen, performing an analytical and computational analysis. The model incorporates the specific biological and epidemiological considerations of the bacterium Xylella fastidiosa and its interactions with insects and hosts in Majorca Island. In our work the vector (insect) population varies with time, mimicking the field observations. One of the conclusions is that, in general, this temporal variation hinders the theoretical characterization of epidemic thresholds. We show that only in the case that the vector population is all the time in its stationary value, the epidemic threshold can be calculated using the standard techniques, and we numerically characterize the value of this threshold

[cat] El passat 2016 es va detectar la pres`encia del bacteri Xylella fastidiosa als ametllers de les Illes Balears, i es determin`a com la causa de la mortaldat en aquests arbres dels darrers quinze anys. Aquest bacteri, que es transmet per via d’insectes, ´es end`emic del continent Americ`a, i ´es considerat un dels pat`ogens que major risc representa per a les plantes a escala global, a causa del seu gran nombre d’hostes, ceps i vectors potencials. En data de 2017, es report`a una incid`encia del bacteri d’un 79.5% ± 2.0 en els ametllers mallorquins, posant en greu risc un dels cultius hist`orics de les Balears. L’objectiu d’aquest projecte ´es doncs presentar una primera base per estudiar aquesta epid`emia mitjan¸cant models compartimentals deterministes (de camp mitj`a). Concretament, presentam un model matem`atic per a la transmissi´o de l’epid`emia, que analitzem anal´ıticament i computacionalment, on incorporam els trets biol`ogics i epidemiol`ogics caracter´ıstics del bacteri Xylella fastidiosa i la seva interacci´o amb insectes i hostes a l’illa de Mallorca. En el nostre model la poblaci´o del vector (insecte) varia amb el temps, imitant les observacions de camp. Un dels resultats d’aquest treball ´es que aquesta variaci´o temporal obstaculitza la caracteritzaci´o te`orica dels llindars d’epid`emia. Demostram que nom´es en el cas que la poblaci´o de vectors estigui tot el temps en el seu valor estacionari, el llindar de l’epid`emia es pot calcular utilitzant les t`ecniques est`andard, i caracteritzam num`ericament el valor d’aquest llindar.