Page 8 - IJOCTA-15-1

P. Kumar / IJOCTA, Vol.15, No.1, pp.1-13 (2025)

imaging has been given in ref. 12 The identifica- to describe the transmission dynamics of the cit-
tion of citrus canker by hyperspectral reflectance rus canker disease. In this study, we propose a
imaging with spectral information divergence has novel mathematical model for the citrus canker
been analysed in ref. 13 Some predictions on the epidemic by using integer and fractional deriva-
role of environmental factors on citrus canker via tives. This research study contains the first non-
multiple regression have been given in. 14 Mod- linear mathematical model of citrus canker trans-
elling for the progress of Asiatic citrus canker on mission along with supporting analyses, which is
Tahiti lime in relation to temperature and leaf the main novelty of our work. Firstly, in Section 2,
wetness has been done in. 15 we propose an integer-order non-linear model by
considering five differential equations along with
the equilibrium points with stability. After that,
we propose a fractional-order model to include
memory effects in the system. Hysteresis is a
general sort of memory that we capture in epi-
demic models. In hysteresis, the present stage
of the model is influenced by both current and
previous stages, i.e., past happenings impact the
current stage of the disease. In Section 3, we
generalise the proposed integer-order model into
fractional-order sense by using Caputo fractional
derivative. As a result of the reasons presented,
the justification for the Caputo-type formation
of the given classical model is evident. In Sec-
Figure 1. Citrus canker spread in tion 4, the fractional-order model is numerically
worldwide. 1 solved by using the Chebyshev spectral colloca-
tion scheme along with the graphical simulations.
In the end, in Section 5, we give significant con-
To date, a number of deadly diseases have been
modelled using mathematical models. In the clusions along with the future scope of the study.
field of modelling, the applications of fractional
derivatives 16 can also be seen in a significant 2. Model dynamics
amount. There are several human diseases that
have been modelled using fractional-order math- The mathematical models for the understanding
ematical models. 17–23 Recently, some plant epi- of bacterial disease dynamics had already been
demics have been successfully modelled in terms proposed by a number of researchers for instance,
of fractional-order mathematical models. In, 24 a see. 31–34 On the basis of the above-given intro-
plant epidemic model in terms of Caputo frac- duction about the dynamics of citrus canker dis-
tional derivative has been given. In, 25 the au- ease, we know that this is a bacterial epidemic
thors derived an optimal control problem for mo- and spreads very quickly due to many factors.
saic diseases by using the Caputo fractional-order This disease has a huge potential to infect or dam-
model. In, 26 a model of the maize streak virus age all the surrounding citrus-growing fields. In
has been solved by applying a very recent ver- this regard, we introduce a nonlinear mathemati-
sion of the predictor-corrector scheme in the Ca- cal model with immigration. Here, the diffusion of
puto form. A fractional-order nonlinear model the epidemic is considered by the flow of bacteria
to explore the impact of fungicides on plant epi- in the environment along with the direct trans-
demics has been derived in ref. 27 Kumar et al. mission of the infective plant to the susceptible
in 28 have established a fractional model to define plant. Let us assume that the total plant popula-
the transmission dynamics of huanglongbing in tion size is denoted by P(t) and divided into sus-
the population of citrus trees. A mathematical ceptible class S(t) and infective class I(t). If Λ is
model of the mosaic epidemic using well-known the recruitment rate of susceptible plants caused
Atangana-Baleanu and Caputo fractional deriva- by the plantation of citrus farming, δ is the rate
tives has been derived in. 29 In, 30 the authors pro- of natural deaths of plants, β is the rate of canker
posed piecewise fractional analysis of the migra- transmission from infected plants, λ is the rate
tion effect in plant-pathogen-herbivore interac- of canker transmission from the bacteria popula-
tions. tion, α is the plant’s death rate cause of canker,
After a careful literature survey, we noticed that, and ν is the recovery rate of infected plants, then
to date, there is no mathematical model available the differential equation representing the change
2