Optimal Control Analysis of an SIR Epidemic Model with Constant recruitment
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Abstract
A mathematical model of an SIR epidemic model with constant recruitment and two control variables using control terms and a deterministic system of dierential equation is presented and analyzed mathematically and numerically. We intend to control the susceptible and infected individuals with educational campaign and treatment strategies. We analyzed the model by non-dimensionalizing the system of equations of our SIR epidemic model and derived our basic reproduction number.We aim to minimize the total number of infective individuals and the cost associated with the use of educational campaign and treatment on [0; T ]. We used Pontryagin's maximum principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically. The results show that the optimal combination of treatment and educational campaign strategy required to achieve the set objective will depend on the relative cost of each of the control measures. The results from our simulation is discussed.
Keywords: Computational simulations, Disease Free Equilibrium, Optimal control, Pontryagin's Maximum Principle, stability theory.
References
[1] R.M. Anderson, R.M. May. ”Infectious Diseases in Humans Dynamics and Control”. Oxford University Press, Oxford,(1991).
[2] E. A Bakare , Yongzheng Sun. ”On the Optimal Control of Vaccination and treatments for an SIR-Epidemic with infected immigrants”,accepted in Applied Mathematical Sciences, (2014), Hikari Ltd.
[3] E. Beretta, Y.Takeuchi ”Convergence results in SIR epidemic model with varying population sizes”, Nonlinear Anal.,29,(1997), 1909-1921; .
[4] E. Beretta, Y. Takeuchi ”Qualitative properties of Chemostat equations with delayes: Boundedness”, local and global asymptotic stability,Diff.Equs.Dynam.SyS., 2,(1944) 19-40.
[5] E. Beretta, V. Capasso, F. Rinaldi. ”Global stability results for a generalized Lotka- Voltera system with distribution delays”, J. Math.Biol., 26, (1988), 661-688.
View more references (32)
[6] F. Brauer, ”Model for the spread of universally fatal diseases”, J.Math.Biol., 28(4), (1990), 451-462; .
[7] F. Brauer, ”Some simple epidemic models”, Math. Biosci. Engr., 3(1), (2006) 1-15; .
[8] K.L. Cooke ,”Stability analysis for a vector disease model”, Rocky Mount. J.Math., 7, (1979),253-263 .
[9] K.L. Cooke , J.A. Yorke, ”Some equations modelling growth processes and gonorhea epidemics”, Math. Biosci., 16,(1973), 75-101;
[10] K. Fister, J. Donnelly, ”Immunotherapy: An optimal control theory approach”. Mathematical Biosciences and Engineering, (2005), 499-510.
[11] H. Gaff, E. Schaefer. ”Optimal control applied to vaccination and treatment strategies for various epidemiological models”. Mathematical Biosciences and Engineering, (2009), 6: 469-492.
[12] S. Gakkhar, K. Negi ”Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate”, Chaos Solitions fractals, 35, (2008), 626-638;.
[13] R. Granish, C. Gilks, et al. ”Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission:a mathematical model”. The Lancet, 2008. Doi:10.1016/S0140-6736(08)61697-9.
[14] H.W. Hethcote, ”Qualitative analysis of Communicable disease models”, Math.Biosci., 28,(1976) 335-356; .
[15] H.W. Hethcote, ”The mathematics of infectious diseases”, SIAM Review, 42,(2000) 599-653; .
[16] Y. Hsieh, S. Sheu., ”The effect of density-dependent treatment and behaviour change on the dynamics of HIV transmission”. Journal of Mathematical Biology, (2001), 43: 69-80.
[17] H. Joshi, S. Lenhart, et al., ”Modeling the effect of information campaign on the HIV epidemic in Uganda”. Mathematical Biosciences and Engineering, (2008), 5: 757-770.
[18] Y. Jin , W. Wang, S. Xiao, ”An SIRS model with a non-linear incidence rate” , Chaos Solitions fractals, 34,(2007) 1482-1497. 284 International Journal of Applied Mathematical Research (a) (b) (c) Figure 4: Simulation showing the infected with varying parameters and control(u1) with varying parameter
[19] T.K Kar, Ashim Batabyal., ”Stability Analysis and Optimal Control of an SIR Epidemic Model with vaccinations”.BioSystems 104 (2011) 127-135.
[20] M. J. Keeling, P. Rohani., ”Modelling Infectious Diseases in Humans and Animals”. Princeton University Press (2008) 2-3.
[21] M. Kgosimore, E. Lungu., ”The effects of vertical transmission on the spread of HIV/AIDS in the presence of treatment”. Mathematical Biosciences and Engineering, (2006), 3: 297-312.
[22] D. Kirschner, S. Lenhart, S. SerBin., ”Optimal control of the chemotherapy of HIV”. Journal of Mathematical Biology, (1997), 35: 775-792.
[23] A. Korobeinikov , ”Global properties of infectious disease models with non-linear incidence”, Bull. Math.Biol.,69,(2007) 1871-1886.
[24] S. Lenhart, J. Wortman., ”Optimal control applied to biological models”. Taylor and Francis, Boca Raton, (2007).
[25] J. Liu , Y. Zhou ,” Global stability of an SIRS model with transport-related infection”, Chaos Solitions fractals, 40, (2009),145-158; .
[26] J. Mena-Lorca , H.W. Hethcote , ”Dynamical models of infections diseases as regular of populations size,” J. math.Biol., 30,(1992) 693-716.
[27] Z. Mukandavire, W. Garira., ”Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe”. Journal Mathematical Biology, (2007), 54: 669-699.
[28] R. Naresh, A. Tripathi, S. Omar., ”Modelling the spread of aids epidemic with vertical transmission.” Applied Mathematics and Computation, (2006), 178: 262-272.
[29] A. Nwagwo, E.A. Bakare., ”Stability Analysis of an SIR Epidemic model and effect of control strategies with constant recruitment”, accepted in Applied Mathematical Sciences, (2014), Hikari Ltd.
[30] O.S. Obabiyi A. Nwagwo , ”Local asymptotic stability of recurrent diseases”, Int.J.Math.Comp., 8(10),(2010) 1-5. International Journal of Applied Mathematical Research 285
(a) (b) Figure 5: Simulation showing the susceptible with varying parameter and control(u2) with varying parameter
[31] O.S. Obabiyi , A. Nwagwo , I.I.Raji, ”Stability analysis of an SIR epidemic model with constant recruitment”, FJMS, 42(1), (2010) 21-36.
[32] B. Shulgin, L. Stone, Z. Agur, ”Pulse vaccination strategy in the SIR epidemic model”, Bull. Math. Biol., 60,(1998) 1123-1148.
[33] Y.Takeuchi Ma W, Beretta E. ”Global asymptotic properties of SIR epidemic model with finite incubation time”, Nonlinear Anal., 42,(2000), 931-947.
[34] K. Wang, A. Fan, A. Tores., ”Global properties of an improved hepatitis B virus model”. Nonlinear Analysis: Real World Applications, (2009). Doi:10.1016/j.nonrwa.2009.11.008.
[35] T. Yusuf, F. Benyah., ”Optimal Control of Vaccination and Treatment for an SIR epidemiological model”. World Journal of Modelling and Simulation. Vol.8(2012) No 3, pp.194-204
[36] G. Zaman, Y. Kang, I. Jung., ”Stability analysis and optimal vaccination of an sir epidemic model”. BioSystems,(2008), 93: 240-249.