Multi-Strain Host-Vector Dengue Modeling: Dynamics and Control

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epidemiology occurs at a much faster time scale due to the vector’s much shorter life

cycle relative to the host. The vector dynamics is, therefore, slaved by the host dynamics.

This is an approach which allows model reduction, whereby the complexity of the model

is reduced by considering the dynamics of some variables to be in a quasi-steady state,

and transforming the original system of ordinary differential equations into a system of

algebraic-differential equations.

Singular perturbation theory deals with systems whose solutions evolve on different

time scales with a ratio characterized by a small parameter 0 < ε ≪1

dx

dt ^{= f(x,y),}

dy

dt ^{= ϵg(x,y),}

x ∈R^m,y ∈Rⁿ .

(6.1)

In the context of vector-borne diseases, system (6.1) has the following interpretation: x

describes the unknowns in the vector compartments (variables U,Vi denoting respectively,

the susceptible and infected by DENV-i vectors), and y in the host compartments (vari-

ables Si,Ii,Ri). The dynamics of the vector is given by f, and the dynamics of the host by

g.

A change of time scale τ = εt gives the slow system in the time scale of the host

population:

ε^dx

dτ ^{= f(x,y),}

dy

dτ ^{= g(x,y) .}

(6.2)

Systems (6.1) and (6.2) are equivalent if ε , 0. Letting ε = 0 in (6.2) leads to the reduced

system:

0 = f(x,y),

dy

dτ ^{= g(x,y) .}

(6.3)

The differential-algebraic system (6.3) describes the evolution of the slow variable y(τ)

constrained to the set {(x,y) | f(x,y) = 0}.

With a good Ansatz the relation f(x,y) = 0 can be rewritten as x = q(y), resulting in

the quasi-steady state approximation (QSSA):

dy

dτ ^{= g(q(y),y) .}

The QSSA assumption is the approach followed by the host-only models for vector-borne

diseases that take all fast variables representing classes of vector population to react instan-

taneously to changes in the slow variables, and hence assume them to be in a quasi-steady

state.

In order to get a better approximation for 0 < ε ≪1, we follow the geometric singular

perturbation technique. This approach has been exploited for single strain dengue [50, 44,

42] and for multi-strain dengue models [41]. In particular, it provides a rigorous framework

for reducing the model complexity and deriving robust reduced-order approximations of

the full host-vector model to a host-only model.