JavaScript is disabled for your browser. Some features of this site may not work without it.
Please be advised that the site will be down for maintenance on Sunday, September 1, 2024, from 08:00 to 18:00, and again on Monday, September 2, 2024, from 08:00 to 09:00. We apologize for any inconvenience this may cause.
Mathematical models of the epidemiological dynamics of soil-borne pathogens
Despite the increase in agricultural crop yield over the last century, the world's
food supply is in grave danger, as an estimated 16% of global yield is lost to
various pathogens annually. As a result, mathematical epidemiology is regularly
used to study the mechanisms of transmission, and to determine possible control
strategies. A basic analysis of the SEIR model with linear di usion on the infective
compartment published in Gilligan (1995) is carried out rst. When the
population size is constant the temporal model admits a disease free equilibrium,
which is asymptotically stable when R0 1, and locally asymptotically stable
when R0 > 1, as well as a locally asymptotically stable endemic equilibrium which
only exists when R0 > 1. Numerical investigations con rm the existence of
travelling wave solutions. Next an SEIR model with non-linear di usion on the
infective compartment is investigated numerically. The behaviour of the two
models is consistent, although non-linear di usion with a small di usion constant
results in travelling waves with signi cantly lower speed.
The host-pathogen model was developed to circumvent the underlying issues of
placing a di usion operator directly onto the infective compartment. This model
consists of susceptible and infected hosts, and free and attached pathogen.
Although R0 < 1 for all parameter values, the model admits either only the
pathogen free equilibrium PFE, or the PFE and two endemic equilibria. The
PFE is always locally asymptotically stable and the global asymptotic stability is
proven using two methods: the application of LaSalle's Invariance Principle, and
the construction of a monotone system that approximates the model from above.
These methods lead to two sets of su cient conditions for the global stability of
the PFE. The parameter values satisfying these conditions have some overlap.
However there are values that satisfy one set and not the other.
Although the stability properties of the endemic equilibria have not been proven,
numerical simulations indicate that the equilibrium with the higher level for free
pathogen is asymptotically stable on R4
+, and the other is unstable, with the
possibility of being a saddle point. Conditions for the persistence of the pathogen,
and thus the infection, were derived. A local sensitivity analysis is completed, and
from this possible control methods have been suggested.
The model was extended to include a spatial component, by the addition of
di usion on the free pathogen sub-population. This inclusion did not result in
solutions deviating from the behaviour that had been proven for the temporal
model. Indeed, under the conditions for persistence, solutions initiated at the level
of the stable endemic equilibrium result in a travelling infection front that joins
this equilibrium to the PFE. The wave speed was calculated for di usion
constants 2 [107; 101], and an equation of the form c( ) = a b was tted to
data. The obtained value of b, namely b = 0:4189 is close to the expected value of
0:5 as for FKPP equations.