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JOURNAL OF RESEARCH IN NATIONAL DEVELOPMENT VOLUME 6 NO 2, DECEMBER, 2008 A FOUR DIMENSIONAL, DETERMINISTIC, COMPATMENTAL MATHEMATICAL MODEL OF HIV/AIDS DISEASE PANDEMIC Abdullah Idris Enagi
Abstract Keywords: Susceptible; Removed; latent; infected Introduction In this work, we proposed a deterministic mathematical model which is a system of ordinary differential equations. The population is partitioned into four compartments of the susceptible class S(t), this is the class in which members are virus free but are prone to infection by interaction with the latent and the infected classes; The second class is the removed class R(t), this is the class of those not susceptible to infection, possibly due to their yielding to warnings or changed behavior as a result of public awareness campaign or enlightenment; The third class is the Latent L(t), this is the class of those that have contracted the virus, but have no symptom of the AIDS disease, the members of this class are still active in the population both sexually and economically. The fourth class is the Infected I(t); this is the class of those that have the manifestation of the symptoms of infection; this class is assumed to be generally weak and inactive. It is assumed that while the new birth of S(t) and R(t) are born into S(t), the new birth of I(t) are born into I(t), the offspring of L(t) are divided between S(t) and L(t) in the proportion θ and (1θ) respectively, i.e. a proportion (1θ) of the offspring of are born with the virus while the remaining proportion θ are free from the virus. The three classes have a natural death modulus, while the infected class has additional death modulus arising from the weight of infection. Members of the class move into the class due to change in the behavior or/and as a result of effective public campaign at a rate . Members of the class move into the class at the rate Members of the class moved into the class at the rate α by interacting with or The model equations are presented in section one with the definition of parameters. We obtained the equilibrium states and the corresponding characteristic equations of the
model in section two. We analyse the zero and the non zero equilibrium states for stability in section three. The result of the work is presented in section four in the form of concluding remarks. The Model Equation with the parameters given by Equilibrium State of the Model The Characteristic Equation βµαy – αz  β θβαw αw µ  0 0 0 0 β  µ  δ 
Expanding the determinant gives Hence the characteristic equation is given by Stability of the Equilibrium State The zero equilibrium state We note that 2 < 0, we have that the system will be stable at the origin if β< µ; i.e when the death modulus is higher than the birth modulus. The non zero equilibrium state (w,x,y,z) = , , H(ip)=p4ip3{[µ(βµαyαz)] [(βµδ)+((1θ)βµ+αw)]} Resolving into real and imaginary parts, G(p) = p3{[µ(βµαyαz)] [(βµδ)+((1θ)βµ+αw)]}+p{[µ(βµαyαz)][((1θ)βµ+αw)(βµδ)αw]+[µ(βµαyαz)+β][(βµδ)+((1θ)βµ+αw)]+[α(y+z)(θβαw)(βµδ)µα(y+z)(θβαw)+α2w(y+z)]} (3.7) Differentiating with respect to p we have that = 3p2{[µ(βµαyαz)] [(βµδ)+((1θ)βµ+αw)]}+ [µ(βµαyαz)][((1θ)βµ+αw)(βµδ)αw]+[µ(βµαyαz)+β][(βµδ)+((1θ)βµ+αw)]+[α(y+z)(θβαw)(βµδ)µα(y+z)(θβαw)+α2w(y+z)]} (3.9) F(0) = αµ(y+z)(θβαw)(βµδ)+ α2µw(y+z) [µ(βµα(y+z)+β][((1θ)βµ+αw)(βµδ)αw] (3.10) The condition for stability according to the Bellman and Cooke theorem [3] is given by (3.16) and from (2.8) and (2.9) (y+z) =+ = Substituting (3.16) and (3.17) in to F(0) we obtain
+ α β θ µ δ F(0) GI (0) REMARK .01 .02 .5 .1 .015 .01 .025 1.073065E07 3.182288E05 UNSTABLE .009 .02 .5 .2 .015 .01 .025 3.441995E07 1.195487E05 UNSTABLE .008 .02 .5 .3 .015 .01 .025 4.71596E07 3.472887E06 UNSTABLE .007 .02 .5 .4 .015 .01 .025 5.724663E07 2.055574E06 STABLE .006 .02 .5 .5 .015 .01 .025 6.628787E07 6.384833E06 STABLE .005 .02 .5 .6 .015 .01 .025 7.481069E07 1.010945E05 STABLE .004 .02 .5 .7 .015 .01 .025 8.303894E07 1.348669E05 STABLE .003 .02 .5 .8 .015 .01 .025 9.108385E07 1.664601E05 STABLE .002 .02 .5 .9 .015 .01 .025 9.900689E07 1.965962E05 STABLE .001 .02 .5 1 .015 .01 .025 1.068448E06 2.257105E05 STABLE
From the analysis in the previous section, we observe that the origin is stable when the birth rate (β) is less than the death rate (µ), this consequently leads to the extinction of the population. The analysis of the nonzero state resulting from Bellman and Cooke’s theorem as shown in the table above shows that the population will be unstable when the measure of effectiveness of public awareness/campaign ( ) is low consequently leading to high contraction rate (α), and stable when ( ) is high consequently leading to low contraction rate (α), The work hence gives credence to high level awareness campaign as an effective measure of steaming the scourge of the pandemic. References Akinwade. N.I. (2003). A Mathematical model Akinwande, N.I. and Sirajo, A.(2005). On Bellman, R. and Cooke K.L. (1963): Kinbir. A. (2005) On a twosex mathematical Pictet, G. et al. (1998). Contribution of AIDS

