Numerical computation and series solution for mathematical model of HIV/AIDS

Transcription

1 Journal of Applied Mathematics & Bioinformatics, vol.3, no.4, 3, ISSN: (print, (online Scienpress Ltd, 3 Numerical computation and series solution for mathematical model of HIV/AIDS Deborah O. Odulaja, L.M. Erinle-Ibrahim and Adedapo C. Loyinmi Abstract In this paper, a mathematical model of HIV/AIDS model was examined and of particular interest is the stability of equilibrium solutions. The characteristic equation which gives the Eigen values was examined. By series solution method the behaviour of the viruses and CD4 + Tcells was looked into. It was shown that if the recovery rate is high enough, the healthy CD4 + Tcell may never die out completely. Hence the patient that test HIV positive, may never develop into full-blown AIDS. Keyword: CD4 + Tcell Department of Mathematics, Tai Solarin University of Education, Ogun State. Nigeria. Department of Mathematical Sciences, University of Liverpool, England. a.c.loyinmi@liv.ac.uk Article Info: Received : August 6, 3. Revised : October 6, 3. Published online : December, 3.

2 7 Computation and series solution for mathematical model of HIV/AIDS Introduction AIDS is caused by Human Immunodeficiency Virus (HIV infection and is characterized by a severe reduction in CD4 + Tcells, which means an infected person develops a very weak immune system and becomes vulnerable to contracting life-threatening infection (such as pneumocysticcarinii pneumonia. AIDS (Acquired immunodeficiency syndrome occurs late in HIV disease. The first cares of AIDS were reported in the United States in the spring of 98. By 983 HIV had been isolated. Several mathematicians have proposed models to describe the dynamics of the HIV/AIDS infection of CD4 + Tcells. In particular Ayeni etal [,] proposed the following model: dx dy dz ( d + k V X + Y k T Z µ (. ( d + Y k T Z k V X µ (. + k Y CZ (.3 where: T Population of CD4 + T cells; T i Population of infected CD4 + T cells; V Virus; π Production rate of CD4 + T cells; d National death rate of healthy CD4 + T cells; d Death rate of infected CD4 + T cells; k Viral infection rate of CD4 + T cells; k Viral production rate for CD4 + T cells; C Viral clearance rate. Clearance Td Death of normal CD4 + T cells; Death of infected CD4 + T cells VC Viral clearance rate. Ayeni ( replaced equation (. by T i d dt V π d T + µ T i (.4 + and (. by

3 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 7 dti V d Ti µ Ti + (.5 where α Disease related to death rate. The mathematical model of (. (.3 and (.4 and (.5 have not been fully established in literature. So the research goes on and on this basis we propose the following model: Mathematical Formulation A model of HIV infection similar to (. and (. but using k TV for infection CD4 + T cells is proposed. + Thus the model is dt k TV π dt + µ T, + T( T dti k TV + dv k T CV, V ( V i ( ( dt i µ Ti, Ti Ti i (. 3 Models 3. Method of Solution To obtain the critical point, we set in infected free equilibrium then, dt dt i dv. Equation (. becomes

4 7 Computation and series solution for mathematical model of HIV/AIDS V π dt + µ Ti + V dt i µ Ti + k T CV i (3. when there is no CD4 + T cells infection then V T i. And equation (3. becomes π d T, with So the un- infected equilibrium is π T. d π ( T,, (,,. d The infected equilibrium when there is CD4 + Tcells infections is V,T I. dt dti dv V π d T + µ + T i ( V d Ti µ Ti + k T i CV T T (3.a i ( Ti ( ( Then equation (3.c becomes CV k T i V C i Substituting (3. in (3.b i C dt i µ Ti. i + α C Then T ( d + µ ( C + αk T T (3.b V V (3.c (3. i (3.3 k k Substituting (3. and (3.3 in equation (3.a

5 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 73 ( + µ ( + α d C k i i ( d + µ ( C+ α i kk C π d µ Ti kk i +. + α C Then the equation becomes T i Then ( + ( + + π kk Cd d µ α dk d µ kkd ( + ( + + π kk Cd d µ V α Cd d µ Ck d Now T becomes. ( d + µ C α k( d + µ π kk Cd( d + µ kk kk α dk ( d + µ + kkd T +. Then the infected equilibrium is ( ( ( + (. ( ( d + µ C α k d + µ π kk Cd d + µ π kk Cd d + µ,, kk kk α dk d µ kkd + + α dk d+ µ + kkd ( + ( π kk Cd d µ α Cd d + µ + Ckd 3. Reduction to origin x Let y T i T i z T T T x + T T y + T i i V V V z + V where ( T, Ti, V is the infected equilibrium such that

6 74 Computation and series solution for mathematical model of HIV/AIDS V π dt + µ T V + k T i + i Then dx dt µ T CV i i (3.. dt dy dti dz dv,, (3.. Substituting (3.. and (3.. in (3. k ( x+ T ( z+ V d ( x T y Ti + α ( + ( + ( + d ( y Ti µ ( y Ti + α ( + dx π + + µ ( + z V dy k x T z V z V dv + + ( ( k y+ T C z+ V i Then equation (3..3 becomes, dx dy dz So d kv z + x + y V µ + α + kv k T z x + α V + + nonlinearterms ( d + µ y + + nonlinearterms k y Cz + nonlinearterms dx kv d + µ + + dy kv dz k C Then ( d µ ( nonlinearterms (3..3

7 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 75 dx x dy A y + z dz A λ I where ( nonlinearterms kv d + µ + + kv A ( d + µ + + k C kv d + λ µ + + A λ I kv ( d + µ λ + + k C λ π 5, d d., k k.3, C., µ. and α. Let Substituting the parameters in (3.4. Then ( T, T, V (857.4, 4.76, i So.49 λ..67 A λi.49. λ λ λ λ +.597λ +.55 The eigenvalues of the system are check by MATLAB function and it gives

8 76 Computation and series solution for mathematical model of HIV/AIDS λ -.774, λ -.45, λ Therefore the Eigenvalues of this system are λ -.8, -.4 and., hence this system is asymptotically stable. The conclusion of this system is similar to Theorem (Derrick and Grossman 976 Let ( x, x x V be a lyapunov function the system, x f ( x, x x 3, x f ( x, x x 3, x f ( x, x x Then, 3, V (x x x 3 is negative semi definite the origin is stable V (x x x 3 is negative definite, the origin is asymptotically stable V (x x x 3 is positive definite the origin is unstable 4 Numerical solution dx dy dz Substituting (3.. and (3.. in (.-(.3, it becomes d kv z + x + y V µ + α + kv k T z x + α V + + nonlinearterms ( d + µ y + + nonlinearterms k y Cz + nonlinearterms

9 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 77 ( + ( + + α ( + ( ( + α ( + dx k x T z V z V dy kv x x+ T z+ V z V (4. dz k y cz In order to find the approximate solution to the model, power series solution is used. Let the solution to the system ( be n x( t x + at + at + + at n n y( t y + bt + bt + + bt n n z( t z mt mt mt n Let x( t x + at y ( t y + bt z ( t z + mt x ( t a, y ( t b, z ( t m Substituting (4. and (4.. in (4., the system becomes (4. (4.. ( α α µ ( α α + α ( z + V ( + µ ( + α + α + + α ( z + V d + z + V + kv x + y + z + V z a kv x d z V y z b m k y Cz Then equation (4. can be written as ( α α µ ( α α + α ( z + V d + z + V + kv x + y + z + V z x( t x + t (4..

10 78 Computation and series solution for mathematical model of HIV/AIDS ( µ ( α α + α ( z + V kv x d+ + z + V y+ z y( t y + t ( + ( z t z k y Cz t Let ( ( ( α α µ ( α α + α ( z + V x ( t x d + z + V + kv x + y + z + V z t at + + ( µ ( α α + α ( z + V y ( t y kv x d + + z + V y + z t bt + + z t z + k y Cz t + m t Perturb (4..3 and substitute into (, we have a b ( α α + α ( z + V d( αz α V kv x µ y( αz z + α ( z + V kvx + ( d+ µ ( + αz + y+ z ( ky Cz + α( z + V ( + αz + d + z + V + kv { ( + µ ( } (4..3 { (4..4 ( + αz + d( + αz + α V + kv x + µ y( + αz + z ( + α ( z + V kvx ( d+ µ ( + αz + y+ z ( ky Cz ( d + µ + + α( z + V ( + αz + kv x ( d+ µ ( + αz + y+ z ( + α ( z + V Cz } ( } m { k ( Cky Substituting (4.. into (4..4 ( α α µ + α( z + V ( + αz + d z V kv a ( a b m + b ( a d + b + m ( µ ( + αz + ( + αz +

11 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 79 m ( kb Cm Let x, y, z, T T, T, V. i Substituting case in (4.. a dx, b, m d d a ( dx x, b, m. Then the series become xt ( x + x+ x + + x n n n when x, d.. xt t t ( (. + + when x, d t xt ( ( t+ + Case Let x y z T T Ti V,,,,, Substituting case in (4.. a ( da + µ b m ( + α z b ( d + b + m ( µ ( + α z m ( kb Cm If x, y, z, T 5 and considering other parameter k k 3, d, and c. d We have the following solutions:

12 8 Computation and series solution for mathematical model of HIV/AIDS Figure : Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ / 5 and α / 5 Figure : Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ ½ and α /5

13 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 8 Figure 3: Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ and α / The graph of µ 3 / - -4,5,5,75,,5,5,75,,5,5,75,3,35,35,375,4,45,45,475,5 x y z Figure 4: Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ 3 / and α / 5

14 8 Computation and series solution for mathematical model of HIV/AIDS The graph of µ ,5,5,75,,5,5,75,,5,5,75,3,35,35,375,4,45,45,475,5 x y z Figure 5: Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ and α / 5 The graph of µ ,5,5,75,,5,5,75,,5,5,75,3,35,35,375,4,45,45,475,5 x y z Figure 6: Graph of x (CD4+T cells, y (infected cells, z (virus against time at µ 3 and α / 5

15 D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi 83 5 Conclusion Figure shows that the healthy CD4+T cells are all infected around t.35, when µ. Figure shows that the healthy CD4+T cells are all infected around t.35, when µ.5 Figure 3 shows that the healthy CD4+T cells are all infected around t.365, when µ Figure 4 shows that the healthy CD4+T cells are all infected around t.3875, when µ.5 Figure 5 shows that the healthy CD4+T cells are all infected around t.4, when µ Figure 6 shows that the healthy CD4+T cells are all infected around t.4875, when µ3 The above results show that as µ increases, the duration of time for all the CD4 + T cells to get infected also increases µ is the recovery rate of the CD4 + T cells. This implies that when the recovery rate µ is high, CD4 + T cells take longer time for all to get infected. In this paper, we modified an existing HIV/AIDS model. We investigated the characteristic equation and discussed the stability of equilibrium points by finding the eigenvalues of the model that were previously considered. We solved existing characteristic equations numerically using realistic values for the parameters and we interpreted the graphs that resulted from the numerical solution. The stability criteria showed that HIV may not lead to full blown AIDS since the healthy CD4 + T cells may never die out completely.

16 84 Computation and series solution for mathematical model of HIV/AIDS References [] R.O. Ayeni, T.O. Oluyo and O.O. Ayandokun, Mathematical Analysis of the global dynamics of a model for HIV infection of CD4+ T cells, JNAMP,, (7, 3-. [] R.O. Ayeni, A.O. Popoola and J.K. Ogunmola, Some new results on affinity hemodialysis and T cell recovery, Journal of Bacteriology Research,, (, [3] W.R. Derrick and S. Grossman, Elementary Differential Equations, Addison Wesley, Reading, 976. [4] L.B. Steven and A.S. Peter, Substance Abuse Treatment For Persons with HIV/AIDS, U.S. Department of health and services, (8, 7-9. [5] D.O. Odulaja, Numerical Computation and Series solution for mathematical model of HIV/AIDS, M.ed dissertation, Tai Solarin University of Education, Ogun State, Nigeria, 3.