Mathematical Models of Control Measures in the Spread of Cholera
Chapter One
Objectives of the study
The objective of this study is to introduce a mathematical model for the spread of cholera disease by providing controls in the form of treatment, sanitation improvement and education about cholera. Giving control is done to reduce the number of population affected by cholera disease and reduce the proliferation of Vibrio Cholarae bacteria.
CHAPTER TWO
LITERATURE REVIEW
This section focuses on the review of literature pertinent to this study under the following sub-headings: conceptual review, empirical review, theoretical framework and summary of literature review.
Conceptual Review
A study carried out in Cameroon indicated a great handicap in information flow during and prior to the cholera outbreak (Njoh, 2010). UNICEF (2012) mentioned responding to cholera emergency through interpersonal communication reaching about 245,000 people and one million were reached with priority behaviour change messages through radio and television programs focusing on disease recognition, treatment seeking practices, and good hygiene practice. UNICEF however, asserted that implementing certain interventions in high-risk groups with poor knowledge of and attitudes toward cholera may not be appropriate. It is therefore important to understand the current levels of knowledge, attitudes, and practices of a given community to implement campaign programs and other preventive measures (UNICEF, 2012).
The World Health Organization (2014), indicated that cultural concepts of illness and how to treat and prevent it cannot be over looked because, it has practical implications for behaviour, public health, and disease control that need to be considered (WHO,2014). According to WHO (2000), health education should continue throughout the year with intensification before the cholera season. As mentioned by WHO (2014), most of the educational messages are technically good, but difficult to implement. WHO (2014), cited that if soap for hand or chemicals for water treatment are not available, alternative solutions should be recommended to ensure basic hygiene practices to limit cholera transmission. Lime juice added to water, beverages or other foods have the ability to inactivate Vibrio Cholerae (WHO, 2014). Also, WHO cited that it would be necessary to organize focus group discussions to identify gaps in knowledge and the kind of reinforcement needed in cholera periods in high risk communities. Checking to see whether soap and chemicals to treat water are available and affordable is another important thing recommended to do during these periods (WHO, 2014).
Causes of Cholera
According to the World Health Organization (2014), cholera is a diarrhoea disease caused by infection of the intestine with the bacterium, either by type 1O or O139 Vibrio Cholerae, which can infect both adults and children. In a retrospective descriptive study by Alam et al. (2006), on 16,379 stool samples cultured in Karachi Pakistan, it was discovered Vibrio Cholerae O1 Ogawa was the most common organism isolated in (32.8%) of the stool cultured. It was also found as the most common enteric pathogen isolated in an urban setting. According to Opare et al. (2012) Vibrio Cholera Serotype Ogawa caused the East Akim Municipality cholera-outbreak affecting many young adult males then females.
Incubation period of cholera . The incubation period of cholera according to World Health Organization (2014), is within two (2) hours to five (5) days; because of this short incubation period, death can occur within two (2) to twenty four (24) hours if medical care or treatment is not promptly sought for. This short incubation period can lead to a large number of cases reported extremely quickly (WHO, 2014). In other words, the short incubation period of two to five days, according to Opare et al. (2012), enhances the potentially explosive pattern of outbreaks.
CHAPTER THREE
MATERIALS AND METHOD
Mathematical Model
We propose models of type SEIQR (Susceptible-Education-Infectious-QuarantinedRecovered) and recognize the bacterial concentration for cholera spread. The dynamics equation splits the population of humans N(t) into six classes based on the disease condition, ie; susceptible population S(t), educated population E(t), asymtomatic population IA(t), symtomatic population IS(t),quarantined population Q(t) dan recovered population R(t). The bacterial aquatic (V. cholerae biomass level) population is B(t) which is the bacterial concentration at t. In this model the infected population does not consider the age and sex factors.
It assumes that a positive recruitment rate of Λ on the susceptible population class S(t) and µ natural mortality rate at t and applies to all human population. The susceptible population may be infected with cholera at the rate of transmission rate. The consumption rate of bacteria through contaminated sources is β > 0, k is half the constant saturation for the population of bacterial, is a possibility that an infected individual has cholera disease due to contact with a contaminated source. Susceptible populations are educated at ψ and some educated populations can stop following preventive measures at , while very small numbers have cholera at γ.
CHAPTER FOUR
COMPUTATIONAL RESULTS
The simulation results of the optimal control problem are given by solving the differential equation (13) – (25) using forward-backward sweep method to obtain optimal state and costate. The state system is completed using the forward order Runge-Kutta 4. Then the costate system is solved using backward runge kutta order-4. The state and costate values are used to update the control values using the characterization of each control and this process keeps repeating until the state, costate and control values have converged. Any parameter values used in this simulation are shown in Table 1. The optimal state is shown in Figure 1 and the optimal control is shown in Figure 2.
CHAPTER FIVE
CONCLUSION AND RECOMMENDATIONS
Conclusion
Optimal control of sanitation, education and quarantine improvements in controlling the spread of cholera diseases has been discussed. Given differential equations as a dynamic system of cholera dispersion models that were divided into classes of human populations and bacterial populations.
In this study, optimal control problems are designed using the Pontryagin Minimum Principle which aims to minimize infected individuals, bacteria and reduce the cost of sanitation, education and quarantine. The simulation results are given at the end of the section to show the effect of the given control. Based on the results, the number of infected individuals and bacteria have increased very large at certain times for the case without control. The controls reduced on the number of infected populations and bacterial population so that the cholera endemic disease could be minimized. This suggests that control strategies in the form of sanitation, education and quarantine improvements can have a good effect on minimizing the spread of cholera.
Recommendation from the study
Based on the findings of this study, the following recommendations were made:
- The health promotion unit and of the Ministry of Health needs to intensify their information and education on behaviour change and communication to help prevent the future occurrence of cholera.
- The community stakeholders should partner to promote water and sanitation practices.
- Government and health authorities need to reposition environmental and sanitation issues as a priority within the national development strategies and plan commitment and financial support which will greatly enhance sanitation programs.
- Lastly, there is a need to raise awareness of people regarding habits which facilitate spread of cholera.
Suggestion for Areas of further research.
Based on the findings the following areas are suggested for further research:
- Future studies should be planned to investigate why knowledge is not effective in the presence of socio-cultural and socio-economic problems with regards to cholera control.
- Further Studies should also be done to investigate how government health policies and health related policies reflect the needs of people.
- Lastly, studies of the same variables should be replicated in similar areas and communities in order to obtain a baseline data for informed decisions.
REFERENCES
- Ali, M., Lopez, A. L., You, Y., Kim, Y. E., Sah, B., Maskery, B., & Clemens, J. (2012). The global burden of cholera. Bulletin of the World Health Organization, 90(3), 209-218.
- Barua, D., (2012).The global epidemiology of cholera in recent years Journal of Royal Society Medicine, (65).423-428.
- Bandura, A. (2000). Exercise of human agency through collective efficacy. Current directions in psychological science, 9(3), 75-78.
- Brown, T. (2018). Design thinking. Harvard business review, 86(6), 84.
- Cumberland, S. (2017). An old enemy returns: the recent cholera outbreak in Zimbabwe highlights failures in the global fight against an old enemy. Bulletin of the World Health Organization, 87(2), 85.
