### A dynamic model of typhoid fever with optimal control analysis

#### Abstract

In this study, a deterministic mathematical model of Typhoid fever dynamics with control strategies; vaccination, hygiene practice, sterilization and screening is studied. The model is first analyzed for stability in terms of the control reproduction number, Rc, with constant controls. The disease-free equilibrium and endemic equilibrium of the model exist and are shown to be stable whenever Rc<1 and Rc>1 respectively. The model by investigation shows a forward bifurcation and the sensitivity analysis conducted revealed the most biological parameters to be targeted by policy health makers for curtailing the spread of the disease. The optimal control problem is obtained through the application of the Pontryagin maximum principle with respect to the above-mentioned control strategies. Simulations of the optimal control system and sensitivity of the constant control system confirm that hygiene practice with sterilization could be the best strategy in controlling the disease.

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WHO. Typhoid. www.whoints/news-room/fact-sheets/detail/typhoid, 2020. Accessed on 10th May, 2021.

M. M. Gibani, M. Voysey, C. Jin, C. Jones, H. Thomaides-Brears, …, A. J. Pollard. The impact of vaccination and prior exposure on stool shedding of salmonella typhi and salmonella paratyphi in 6 controlled human infection studies. Clinical Infectious Diseases, 68(8): 1265-1273, 2019.

CDC. Typhoid fever. www.nc.cdc.gov/travel/diseases/typhoid, 2020. Accessed on 11th September, 2021.

A. C. Bradley and S. Eli. Typhoid and paratyphoid fever in travellers. Lancet Infect Dis., 5(10):623-8, 2005.

S. Cairncross, C. Hunt, S. Boisson, K. Bostoen, V. Curtis, I. C. H. Fung and W. Schmidt. Water, sanitation and hygiene interventions and the prevention of Diarrhoea. Int J Epidemiol, Suppl 1(Suppl 1):193-205, 2010.

I. Bakach, M. R. Just, M. Gambhir and I. C. Fung. Typhoid transmission: A historical perspective on mathematical model development. Trans R Soc Med Hyg., 109(11): 679-89, 2015.

S. Mushayabasa. A simple epidemiological model for typhoid with saturated incidence rate and treatment effect. International Journal Mathematical and Computational Sciences, 6(6): 688-695, 2012.

H. Abboubakar and R. Racke. Mathematical modelling and optimal control of typhoid fever. Konstanzer Schriften in Mathematik, Universitat Konstanz, Konstanzer Online-Publikatins-System (KOPS), Nr. 384, Dezember 2019.

J. W. Karunditu, G. Kimathi and S. Osman. Mathematical modeling of typhoid fever disease incorporating unprotected humans in the spread dynamics. Journal of Advances in Mathematical and Computer Science, 32(3):1-11, 2019.

O. J. Peter, M. O. Ibrahim, A. Oluwaseun and R. Musa. Mathematical model for the control of typhoid fever. IOSR Journal of Mathematics,13(4): 60-66, 2017.

N. Nyerere., S. C. Mpeshe and S. Edward. Modeling the impact of screening and treatment on the dynamics of typhoid fever. World Journal of Modelling and Simulation. 14(4): 298-306, 2018.

O. J. Peter, A. Afolabi, F. A. Oguntolu, C. Y. Ishola and A. A. Victor. Solution of a deterministic mathematical model of typhoid fever by variational iteration method. Science World Journal, 13(2): 64-68, 2018.

S. Edward and N. Nyerere. A Mathematical model for the dynamics of cholera with control measure. Applied and Computational Mathematics, 4(2): 53-63, 2015.

M. Kgosimore and G. R. Kelatlehegile. Mathematical analysis of typhoid infection with treatment. Journal of Mathematical Sciences: Advancse and Applications, 40: 75-91, 2016.

B. S. Aji, D. Aldila and B. D. Handari. Modeling the impact of limited treatment resources in the success of typhoid intervention. AIP Conference Proceedings, International Conference on Science and Applied Science, 2202(1):020040, 2019. Doi:10.1063/1.5141653

G. T. Tilahum. O. D. Makinde and D. Malonza. Modelling and optimal control of typhoid fever disease with cost-effective strategies. Computational and Mathematical Methods in Medicine, Article ID 2324518, 2017.

P. N. Okolo and O. Abu. On optimal control and cost-effectiveness analysis for typhoid fever model, FUDMA Journal of Sciences (FJS), 4(3): 437 – 445, 2020.

O. J. Peter, M. O. Ibrahim, H. O. Edogbanya, F. A. Oguntolu, K. Oshinubi, A. A. Ibrahim, T. A. Ayoola and J. O. Lawal. Direct and indirect transmission of typhoid fever model with optimal control. Results in Physics, 27, 104463, 2021.

H. Abboubakar and R. Racke. Mathematical modelling forecasting and optimal control of typhoid fever transmission dynamics. Chaos, Solitons and Fractals, 149, 111074, 2021.

T. D. Awoke. Optimal control strategy for the transmission dynamics of typhoid fever. American Journal of Applied Mathematics, 7(2):37 – 49, 2019.

J. H. Jones. Notes on R_0. Department of Anthropological Sciences, Standford University, May 1, 2007.

J. M. Heffernan, R. J. Smith and L. M. Wahl. Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface. 2(4): 281-293, 2005.

C. Castillo-Chavez and B. Song. Dynamical model of tuberculosis and their applications. Mathematical Bioscience and Engineering, 1(2): 361-404, 2004.

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Salsana and S. Tarantola. Global sensitivity analysis - the primer. Wiley, Chichester, 2008.

H. S. Rodrigues, M T. T. Monteiro and D. F. M. Torres. Sensitivity analysis in a dengue epidemiological model. Conference Papers in Mathematics, 2013: Articles ID: 721406, 2013. doi: 10.1155/2013/721406

D. T. Lauria, B. Maskery, C. Poulous and D. Whittington. An optimization model for reducing typhoid cases in developing countries without increasing public spending. Vaccine, 27(10): 1609-1621, 2009.

S. Mushayabasa. Impact of vaccine on controlling typhoid fever in

Kassena-Nankana district of Upper East Region of Ghana: Insights from a

mathematical model. Journal of Modern Mathematics and Statistics. 5(2): 54-59, 2011.

I. A. Adetunde. Mathematical models for the dynamics of typhoid fever in Kassena-Nankana district of Upper East Region of Ghana. Journal of Modern Mathematics and Statistics. 2(2): 45-49, 2008.

M. Gosh, P. Chandra, P. Sinha. and J. B. Shukla. Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population. Nonlinear Analysis Real World Applications, 7(3): 341–363, 2006.

J. Mushanyu, F. Nyabadza, G. Muchatibaya, P. Mafuta and G. Nhawu. Assessing the potential impact of limited public health resources on the spread and control of typhoid. Journal of Mathematical Biology, 77(3):647-670, 2018.

J. M. Mutual, F. B. Wang and N. K. Vaidya. Modeling malaria and typhoid co-infection dynamics. Mathematical Bioscience, 264(1): 128-144, 2015.

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko. The mathematical theory of optimal processes, John Wiley &Sons, Lonon, UK, 1962.

DOI: http://dx.doi.org/10.23755/rm.v41i0.657

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