SVIA HIV model

The SVIA HIV model created by Andrew Josselyn at the University of New Brunswick in 2010, was based on the work done by Elbasha and Gummel in 2005 and describes how the spread of HIV would be affected in a population in which the use of a HIV vaccine is incorporated. Due to the cost and problems associated with drug therapy treatments such as HAART, Josselyn wanted to see how a relatively cheap vaccine could reduce HIV infection mainly in developing African countries where the HIV pandemic is greatest. Josselyn assumed that every member of the population must be vaccinated before they become sexually active and no immigrants can enter the poulation unless they are HIV/AIDS free and vaccinated.
The population is classified into 4 classes; S or susceptibles, people who were unsuccessfully vaccinated or the vaccine wore off and prone to infection, V or vaccinated, people who were successfully vaccinated and have a reduced chance of contracting the HIV virus, I or infected, people infected with the HIV virus, and A or AIDS, people who have AIDS.
Model
dS/dt = Q(1-p) - μS - λcS + γV
dV/dt = pQ - μV - qλcV - γV
dI/dt = λcS + qλcV - θI - μI
dA/dt = θI - μA - ξA
S(0) S0>0, V(0) V0>0, I(0) I0>0, A(0) A0=0
N = S + V + I + A Where N is the total population
The infection rate λ depends on the transmission probability β, and the proportion of people
infected with HIV, I/N. λ βI/N and c the number of sexual partners. The Q value represents the new members into the population through immigration and birth, while the variable p is a composite measure which includes the vaccine take (α) and vaccine coverage (ε). The variable μ represents the death rate due to causes asides from HIV infection, so 1/μ is the lifespan, while the variable γ represents the rate in which the vaccine wanes or wares off, so 1/γ, is the time spent in the vaccinated, V, class. The variable θ represents the rate in which people infected with HIV progress to AIDS or the A class, so 1/θ is the time spent in the infected or I class. The variable ξ represents the death rate due to AIDS, so 1/ξ is the time people remain in the AIDS or I class, and q is the fraction the vaccine reduces susceptibility.
Basic Reproductive Value Rop
To find the basic reproductive value also called the basic reproduction number is to first find the disease free equilibrium (E0 = [S*, V*, 0, 0]) then use the Jacobian matrix to find under what conditions Rop <0.
Rop = βc[μ(1-p)+γ]/(μ+γ)(θ+μ)
If Rop<1, the disease dies out and the disease free equilibrium exsists, however, if Rop>1, the disease is prevalent within the population and the endemic equilibrium exsists which is given by Ee = [S*, V*, I*, A*], where S*, V*, and I* >0, and A* > or = 0.
Limitations
Unfortunately the SVIA HIV model is very limited in its uses. Due to the fact that Josselyn did not incorporate the effects of birth control such as condoms, and the effects of drug therapy such as HAART, the model is only applicable in very limited situations and not in modern western medical systems. Josselyn also assumed that immigrants who are HIV positive or who have not been vaccinated cannot enter into vaccinated populations which is very unreasonable, especially in developing countries like those found in eastern Africa. However, since Josselyn only wanted to see how a relatively cheap vaccine on its own could reduce the prevalence of HIV, especially in east Africa, these limitations are of little importance.

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