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becomes

˙S = −^β

M ^{SV +µ((1−pv)N −S) ,}

˙I = ^β

M ^{SV −(γ +µ)I ,}

˙R = γ(I +Iv)−µR ,

(6.10a)

˙Sv = −^(1−αv)β

M

SvV +µ(pvN −Sv) ,

˙Iv = ^(1−αv)β

M

SvV −(γ +µ)Iv

(6.10b)

˙U = −^ϑ

N ^{U(I +Iv)+ν(M −U) ,}

˙V = ^ϑ

N ^{U(I +Iv)−νV ,}

(6.10c)

N = S(t)+Sv(t)+I(t)+Iv(t)+R(t) ,

M = U(t)+V (t) .

The human N and mosquito M population sizes are constant over time and therefore,

the system can be reduced to a five dimensional system in the remaining state variables

S(t),I(t),V (t),Sv(t),Iv(t), t ≥0.

6.5.1.1

Analysis of the SIRvUV model

When pv = 0 and αv = 0 the vaccinated individuals behave just like the non-vaccinated

and therefore the dynamics is the same as if no individuals were vaccinated. This is de-

scribed by the SIR-UV model with ε = 1 in (6.4). If S vanishes in (6.10), the first equation

implies either N = 0, which is of course senseless, or pv = 1. In the extreme case pv = 1

the system is effectively the SISUV-model [42] and becomes

˙Sv = −^(1−αv)β

M

SvV +µ(N −Sv) ,

˙Iv = ^(1−αv)β

M

SvV −(γ +µ)Iv ,

˙R = γIv −µR ,

(6.11a)

˙U = −^ϑ

N ^{UIv +ν(M −U) ,}

˙V = ^ϑ

N ^{UIv −νV ,}

(6.11b)

N = Sv(t)+Iv(t)+R(t) ,

M = U(t)+V (t) .

In [42] system (6.11) is studied with αv = 0. Using a reduction to a two-dimensional sys-

tem, for the parameter values given in Table 6.B.1, the disease-free equilibrium is unstable

and the endemic equilibrium is stable.

Note that in our case it is possible to introduce in (6.11) a compound parameter β^∗=

(1−αv)β. This means that the analysis of (6.11) is the same as the one performed in [42]

where only the infection rate β is multiplied by a factor 1 −αv. Thus we can now safely

assume in (6.10) that S , 0. There are two equilibria for system (6.10): the trivial, disease-

free equilibrium E0 and the interior, endemic equilibrium E^∗. The trivial equilibrium equals

E0 = (S0 = N(1−pv), I0 = 0, Sv0 = pvN, Iv0 = 0, U0 = M, V0 = 0) ,

(6.12)