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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies

6.2.3

Example: SIR-UV model

We illustrate this approach for a simple single-strain host-vector SIR-UV model, read-

ing

dS

dt ^{= ε}



µ(N −S)−^β

M ^SV



,

dI

dt ^{= ε}

 β

M ^{SV −(γ +µ)I}



,

dR

dt ^{= ε(γI −µR) ,}

dU

dt ^{= ν(M −U)−ϑ}

N ^{UI ,}

dV

dt ^{= ϑ}

N ^{UI −νV ,}

discussed in [44] and recently in [42]. Assuming the total host and vector populations N,M

to be constant over time, we arrive to the ODE model

dS

dt ^{= ε}



µ(N −S)−^β

M ^SV



,

dI

dt ^{= ε}

 β

M ^{SV −(γ +µ)I}



,

(6.4a)

dV

dt ^{= ϑ}

N ^{(M −V )I −νV .}

(6.4b)

The trajectory in (S,I)-plane of the system (6.4) is shown in Figure 6.1 (full model, blue

curve) where the parameter values are given in Table 6.B.1. In [42] is was shown that with

these parameter values the vector dynamics occurs at a much faster time-scale relative to

the host’s. This means that after a fast transient dynamics the slow dynamics occurs near a

manifold that is a smooth surface in R³ given by the QSSA that reads:

VQSSA =

ϑMI

ϑI +νN ^,

V ^S

QSSA ^{= ϑM}

νN ^{I ,}

(6.5)

where V ^S

QSSA ^{denotes the value of the infected vector population under QSSA when the}

number of infected humans I is small. This plane is named M0 in App. 6.A. Starting with

the same initial conditions as for the full model, the reduced system where VQSSA is sub-

stituted in (6.4a), the trajectory is also shown in Figure 6.1 (left, QSSA green curve). Only

after some transient dynamics this trajectory is close the that of the full model.

Here a mechanistic derivation is obtained for an infection mechanism with a satu-

rated incidence rate where the force of infection is not linear in I but when the expression

V = VQSSA is substituted in the infection rate terms in (6.4a), it is a hyperbolic relation-

ship. In the epidemiological literature it has been used in [15, 21, 36]. In [15] the use of a

saturated incidence rate is motivated because for large I the population may tend to reduce

the number of contacts per unit of time. Below we will, in a host-only model formulation,

use the simplified linearized expression V ^S

QSSA ^{which holds only for small I to get the un-}

saturated incidence rate back. This gives an "ordering" of nested models from a host-vector

model to a host-only model with the QSSA-host-only model inbetween.

In Figure 6.1 also two approximations of V for terms up to O(ε) are shown (left panel,

red curve) and up to O(ε²) (right panel, red curve) where ε = 1. See also Appendix 6.A.

For the first-order approximation the trajectory converges onto a so called spurious, stable

equilibrium. This unexpected spurious equilibrium exists in addition to the expected en-

demic equilibrium in the full model. The spurious equilibrium exists with S < N and I > 0