Parameter and State Estimation in a Cholera Model with Threshold Immunology: A Case Study of Senegal

It is often impossible to measure all states affecting spread of a disease. In cholera, asymptomatic and cholera pathogen densities are not practically measurable despite playing a big role in its transmission. They are referred to as inaccessible states of the model and can only be manipulated using the measurable states of the given model. Our interest lies in estimating such states and the parameters catalyzing the spread. A mathematical model for cholera dynamics consisting of five compartments (susceptible, symptomatic, asymptomatic, recovered and bacteria population) with a minimum infection dose (MID) is considered. A method based on observer (from modern control theory) is proposed to estimate the state variables not accessible to measurement and the time-dependent parameters from real data of Senegal. We suppose that the total population of Senegal, the monthly reported cholera-induced deaths and the monthly recovered individuals are known inputs obtainable from real data, and the monthly new cholera cases the system output. An auxiliary system is used, an observer whose solutions converge exponentially to those of an original system and solely utilize known inputs and output of the model. Thus, the estimation of the unmeasured state variables like the pathogen density and the yearly asymptomatic population within the human community playing an important role in the spread of cholera is possible. We derive the expressions for time-dependent infection rate, induced cholera death rate and symptomatic recovery rate and their estimations done using real data. Numerical simulations are then performed for the validation of estimation results. The system together with the observer designed is detectable but is not observable. The observer delivered estimates reflect a close trend already ascertained by other researchers. They indicate the existence of bacteria and asymptomatic individuals in the population of Senegal at any single time for the duration of collection of the data. The ever existence of cholera pathogen explains the endemicity of Cholera in Senegal and other sub-Saharan-African countries owing to role played by the asymptomatic individual in the bacteria density. As such, the heath authorities in Senegal need to educate the general public on hygiene irrespective of observable symptoms to lower the possible number of new infections. We have analytically showed and numerically confirmed the exponential convergence to zero of the estimation errors resulting from the observer model hence the high quality of the estimates.