Spatial dynamic on foot and mouth disease in bovines: a mathematical model

Author(s)

Fernando Ferreira

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Faculdade de Saúde Pública (FSP/CIR)
• Defense date: 2000-08-11

Examining board members

Eduardo Massad; Raymundo Soares de Azevedo Neto; Marcelo Nascimento Burattini; José Soares Ferreira Neto; Maria do Rosário Dias de Oliveira Latorre

• Advisor: Eduardo Massad

Abstract

Objectives. It proposes mathematical model which simulate the spatial spread ofthe foot and mouth disease in bovines and allows to estimate the minimum velocity of its dissemination, as well as the extension of a barrier against the foot and mouth disease achieved by vaccination of susceptibles in areas ahead of the advancing wave. The model was validated by comparing the results of the simulation with the data of a real epidemy of foot and mouth disease which occurred in 1990 in the State of Santa Catarina. Methods. A system of differential equations which represent the natural history of the disease in bovines, incorporating the spatial dimension through the use of a diffusion equation, was used. The determination of the birth rate, the discard rate, the diffusion coefficient, the transmission coefficient and the environmental carrying capacity were made based on the populational data of the bovine herd of the State. The immunity loss rate, the mortality rate of foot and mouth disease and the recovery rate were obtained from literature. Results. For the transmission coefficient the value of Β=6,62x \'10 POT. -2\' kmX\'day POT. -1\' was obtained. The diffusion coefficient was estimated as being equal to D=3,19x\'10 POT. -1\' \'km POT. 2\'x\'day POT. -1\'. For the birth rate and discard rate the following values were obtained a=4,71x\'10 POT. -5\' \'day POT. -1\' e b=3,91x\'10 POT. -5\' \'day POT. -1\', respectively. The environmental carrying capacity was estimated as being equal to K=22,93 animalsX\'km POT. -2\'. The value of the constants made it possible to determine the value of \'R IND. 0\' for the foot and mouth disease as being equal to 15. The mathematical treatment of the model made it possible to estimate that the minimum velocity of propagation of the epidemy was equal to V=490,74 kmX\'year POT. -1\' and that a barrier with 20 km of extension, in which the proportion of protected animais is 97,82%, would be sufficient to hinder the propagation of the epidemy, taking into consideration that the probability of an infected animal meeting a susceptible, in this case, is only 0,0001. The model was validated through comparing the results of the simulation in two dimensions with actual data of a foot and mouth epidemy that occurred in the State of Santa Catarina in 1990. Conclusions. The width of a foot and mouth disease barrier depends on the density of susceptibles present in this area and can be determined based on the probability that an infected animal comes into contact with a susceptible animal, which means assuming a known risk that this will occur. Considering that the presence of susceptibles in this area is only 2,28%, the probability of contact between susceptible and infected animais, for a value of p = 0,60 animals/km, varies from 0,025 to 0,0001 for an extension varying from 10 to 20 km, respectively. The model was able to simulate adequately the spatial diffusion of the epidemy of foot and mouth disease that occurred in the State of Santa Catarina in 1990. (AU)