Journal of Mathematics and Applications

06/41, DOI: 10.7862/rf.2018.6

# On Population Dynamics with Campaign on Contraception as Control Strategy

Virtue U. Ekhosuehi, Fidelis O. Chete

DOI: 10.7862/rf.2018.6

## Abstract

This work considers a population divided into two groups according to the adoption of contraception. The campaign in favour of contraception is modelled as a bounded optimal control problem within the framework of the logistic and the Malthusian models of population dynamics. The control is the fraction of non-adopters successfully educated on contraception. The objective is to maximise the number of non-adopters successfully educated on contraception over time. The optimisation problem is solved using the Pontryagin's maximum principle and the parameters of the model are estimated using the method of least squares.

## References

- S. Anita, V. Arnautu, V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Springer, New York, 2011.
- Q. Ashraf, O. Galor, Dynamics and stagnation in the Malthusian epoch, The American Economic Review 101 (5) (2011) 2003-2041.
- G.S. Becker, E.L. Glaeser, K.M. Murphy, Population and economic growth, The American Economic Review 89 (2) (1999) 145-149.
- P.A. Bourne, A.D.C. Charles, T.V. Crawford, M.D. Kerr-Campbell, C.G. Francis, N. South-Bourne, Current use of contraceptive method among women in a middle-income developing country, Open Access Journal of Contraception 1 (2010) 39-49.
- F. Caliendo, S. Pande, Fixed endpoint optimal control, Economic Theory, 26 (4) (2005) 1007-1012 (Accessed: 02/09/2009 from www.jstor.org).
- A.L. Doontchev, W.W. Hager, The Euler approximation in state constrained optimal control, Mathematics of Computation 70 (233) (2001) 173-203.
- J.E. Eko, K.O. Osonwa, N.C. Osuchukwu, D.A. Offiong, Prevalence of contraceptive use among women of reproductive age in Calabar metropolis, Southern Nigeria, International Journal of Humanities and Social Sciences Intervention 2 (6) (2013) 27-34.
- S. Enke, Leibenstein on the benefits and costs of birth control programmes, Population Studies 24 (1) (1970) 115-116.
- G.T. Ijaiya, U.A. Raheem, A.O. Olatinwo, M.-D.A. Ijaiya, M.A. Ijaiya, Estimating the impact of birth control on fertility rate in sub-Saharan Africa, African Journal of Reproductive Health 13 (4) (2009) 137-146 (Accessed: 22/04/2014 from www.jstor.org).
- A.L. Jensen, Comparison of logistic equations for population growth, Biometrics 31 (4) (1995) 853-862 (Accessed: 15/05/2014 from www.jstor.org).
- Y. Kwon, C.-K. Cho, Second-order accurate difference methods for a one-sex model of population dynamics, SIAM Journal on Numerical Analysis 30 (5) (1993) 1385-1399.
- R. Lande, S. Engen, B.-E. Saether, Optimal harvesting of fluctuating populations with a risk of extinction, The American Naturalist 145 (5) (1995) 728-745 (Accessed: 15/05/2014 from www.jstor.org).
- D. Leach, Re-evaluation of the logistic curve for human populations, Journal of the Royal Statistical Society, Series A (General) 144 (1) (1981) 94-103 (Accessed: 05/05/2014 from www.jstor.org).
- M.L. Lee, D. Loschky, Malthusian population oscillations, The Economic Journal 97 (387) (1987) 727-739.
- Y. Li, Y. Kuang, Periodic solutions in periodic state-dependent delay equations and population models, Proceedings of the American Mathematical Society 130 (5) (2001) 1345-1353 (Accessed: 15/05/2014 from www.jstor.org).
- A.J. Lotka, Some recent results in population analysis, Journal of the American Statistical Association 33 (201) (1938) 164-178 (Accessed: 15/05/2014 from www.jstor.org).
- N. Meade, A modified logistic model applied to human populations, Journal of the Royal Statistical Society, Series A (Statistics in Society) 151 (3) (1988) 491-498.
- R.E. Mickens, Exact solutions to a population model: the logistic equation with advection, SIAM Review 30 (4) (1988) 629-633 (Accessed: 15/05/2014 from www.jstor.org).
- J. Mokyr, Malthusian models and Irish history, The Journal of Economic History 40 (1) (1980) 159-166.
- R.H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Advances in Applied Probability 14 (4) (1982) 687-708.
- I. Nwachukwu, O.O. Obasi, Use of modern birth control methods among rural communities in Imo State, Nigeria, African Journal of Reproductive Health 12 (1) (2008) 101-108.
- F.R. Oliver, Notes on the logistic curve for human populations, Journal of the Royal Statistical Society, Series A (General) 145 (3) (1982) 359-363 (Accessed: 05/05/2014 from www.jstor.org).
- R. Parker, Minimizing cost to maintain a steady-state growth rate in a population, Operations Research 25 (2) (1977) 326-329.
- R. Pearl, L.J. Reed, A further note on the mathematical theory of population growth, Proceedings of the National Academy of Sciences of the United States of America 8 (12) (1922) 365-368 (Accessed: 15/05/2014 from www.jstor.org).
- Prajneshu, Time-dependent solution of the logistic model for population growth in random environment, Journal of Applied Probability 17 (4) (1980) 1083-1086.
- H.B. Presser, M.L.K. Hattori, S. Parashar, S. Raley, Z. Sa, Demographic change and response: social context and the practice of birth control in six countries, Journal of Population Research 23 (2) (2006) 135-163.
- M.S. Ridout, D.J. Cole, B.J.T. Morgan, L.J. Byrne, M.F. Tuite, New approximations to the Malthusian parameter, Biometrics 62 (4) (2006) 1216-1223.
- M.R. Rosenzweig, K.I. Wolpin, Evaluating the effects of optimally distributed public programs: child health and family planning interventions, The American Economic Review 76 (3) (1986) 470-482 (Accessed: 22/04/2014 from www.jstor.org).
- M.C. Runge, F.A. Johnson, The importance of functional form in optimal control solutions of problems in population dynamics, Ecology 83 (5) (2002) 1357-1371 (Accessed: 15/05/2014 from www.jstor.org).
- R.M. Schacht, Two models of population growth, American Anthropologist, New Series 82 (4) (1980) 782-798.
- A.R. Solow, On fitting a population model in the presence of observation error, Ecology 79 (4) (1998) 1463-1466 (Accessed: 15/05/2014 from www.jstor.org).
- T.J. Trussel, Cost versus effectiveness of different birth control methods, Population Studies 28 (1) (1974) 85-106.
- S.L. Tucker, S.O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM Journal on Applied Mathematics 48 (3) (1988) 549-591.
- P.J. Wangersky, Lotka-Volterra population models, Review of Ecology and Systematics 9 (1978) 189-218 (Accessed: 15/05/2014 from www.jstor.org).
- S.A. Weinstein, G. Goebel, The relationship between contraceptive sex role stereotyping and attitudes toward male contraception among males, Journal of Sex Research 15 (3) (1979) 235-242.
- M.K. Welch, Not women's rights: birth control as poverty control in Arkansas, The Arkansas Historical Quarterly 69 (3) (2010) 220-244.
- P. Whittle, J. Horwood, Population extinction and optimal resource management, Philosophical Transactions: Biological Sciences 350 (1332) (1995) 179-188 (Accessed: 05/05/2014 from www.jstor.org).

## About this Article

TITLE:

On Population Dynamics with Campaign on Contraception as Control Strategy

AUTHORS:

Virtue U. Ekhosuehi ^{(1)}

Fidelis O. Chete ^{(2)}

AUTHORS AFFILIATIONS:

^{(1)} Department of Mathematics, University of Benin, NIGERIA^{(2)} Department of Computer Science, University of Benin, NIGERIA

JOURNAL:

Journal of Mathematics and Applications

06/41

KEY WORDS AND PHRASES:

Contraception; Logistic model; Malthusian model; Optimal control model; Population

FULL TEXT:

http://doi.prz.edu.pl/pl/pdf/jma/69

DOI:

10.7862/rf.2018.6

URL:

http://dx.doi.org/10.7862/rf.2018.6

RECEIVED:

2017-10-05

COPYRIGHT:

Publishing House of Rzeszow University of Technology Powstańców Warszawy 12, 35-959 Rzeszow