Optimization of Cholera Spreading Using Sanitation, Quarantine, Education and Chlorination Control

Optimization of Cholera Spreading using Sanitation, Quarantine, Education and Chlorination Control

Subchan Subchan1, Sentot D. Surjanto1, Irma Fitria2 and Dwita S. Anggraini1 1Departement of mathematics, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia 2Departement of mathematics, Institut Teknologi Kalimantan, Balikpapan, Indonesia

Keywords: Cholera Model, Optimal Control, Pontryagin Minimum Principle.

Abstract: Cholera is a contagious and deadly disease that requires an effective prevention and control actions. In this paper, several efforts are made to prevent the cholera spreading by reconstructing the mathematical model and adding control sanitation, treatment consisted of quarantine and education as well as chlorination on to the bacteria. The Pontryagin Minimum Principle is employed to derive the optimal control solution and solved by Runge-Kutta method. The computational results showed that the control was able to minimize the number of individuals infected by cholera with mild symptoms at the final time as many as 2 individuals and individuals infected by cholera with severe symptoms at the final time as many as 7 individuals as well as minimize the number of bacteria concentrations at the final time as much as 517 cell/ml.

1 INTRODUCTION reached Bangladesh in 1963, India in 1964 and the Soviet Union, Iran and Iraq in 1965-1966 (Setiadi, Cholera is an acute diarrhea infection which is caused 2014). by the consumption of food or water contaminated Cholera is rapidly spreading in densely populated with Vibrio cholerae bacteria (Organization, 2008). areas, poor water sanitation and lack of clean wa- These bacteria secrete enterotoxins in the intestinal ter supply. Therefore, cholera is widely identified in tract which cause diarrhea accompanied with acute poor and developing countries (Subchan et al., 2019). and severe vomiting. Therefore, an individual will So that it does not rule out the possibility of cholera lose a lot of body fluids only in several days and get spreading in Indonesia, for it is important to conduct dehydration. This condition can cause death if not research on controlling the spread of cholera in order handled quickly (Johnson and R, 2006). The spread- to minimize cases of the spread of cholera.Effective ing process of cholera can occur through the mouth, precautions of controlling for cholera depend on pro- when Vibrio Cholerae bacteria successfully entered viding adequate environmental health services, such through the mouth and ingested, then these bacte- as increasing access to clean water, sanitation, avail- ria will be quickly killed when exposed to stomach ability of cholera vaccines, quarantine and treatment acid. However, if Vibrio Cholerae bacteria success- (Organization, 2008). fully passes the stomach acid, the bacteria will de- The mathematical model related to cholera velop in the small intestine (Setiadi, 2014). spreading with its control had been many conducted About 75% of people infected with Vibrio cholera in the previous research. The research about cholera do not experience any symptoms, even though the disease had been examined by Bakhtiar (Bakhtiar, bacteria are in their feces for 7-14 days after infected 2015). He studied optimum control approach of con- (Organization, 2008), but when there is an infec- tagious disease with the control variable was the role tion attack then the diarrhea and vomiting suddenly of education and chlorination. After that, Lemos- occur with serious condition as acute attack (Sack Paiao et al (Lemos-Paiao˜ et al., 2016), concerned on et al., 2004). Since 1917, cholera has been known cholera spreading model by giving control in the form as seven pandemics which spread to Europe. The of treatment done to the population of quarantined Vibrio Cholerae bacteria first appeared in Sulawesi, people. The population infected which was given a Indonesia and caused a cholera epidemic. Cholera treatment would be quarantined so that it obtained the then spread rapidly to other East Asian countries and quarantined population. In addition, Subchan et al

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Subchan, ., Surjanto, S., Fitria, I. and Anggraini, D. Optimization of Cholera Spreading using Sanitation, Quarantine, Education and Chlorination Control. DOI: 10.5220/0009881902360240 In Proceedings of the 2nd International Conference on Applied Science, Engineering and Social Sciences (ICASESS 2019), pages 236-240 ISBN: 978-989-758-452-7 Copyright c 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

Optimization of Cholera Spreading using Sanitation, Quarantine, Education and Chlorination Control

(Subchan et al., 2019). reasearched cholera disease mathematical models of the spread of cholera are as spreading model by giving the optimal control in the follows. forms of medication and intervention through the improvement of sanitation, education and quarantine. ds βB In this research, the problem was reconstructed by = Λ+νR+εE −µS−uaψ−(1−u1) S (1) dt κ + B the mathematical model of the spread of cholera with the control variable in the form of chlorination in bacteria, improved sanitation, education and quarantine. dE = uaψS − εE − µE − (1 − u1)γE (2) Control was given to reduce the number of individuals dt infected with cholera with mild and severe symptoms and reduce proliferation of the Vibrio Cholerae bacte- dIA βB ria. = (1 − u1)p S + (1 − u1)pγE − µIA − α2IA dt κ + B (3)

2 MATHEMATICAL MODEL dI βB s = {(1 − u )(1 − p) S + (1 − u ) dt 1 κ + B 1 (4) In this research, the type of mathematical model of (1 − p)γE − µIs − µsIs = u2δIs} cholera spreading used was the type of SEIQR which was reconstructed by adding chlorine to the bacteria dQ = u2δIs − µQ − µQQ − α1Q (5) (u4). Other optimal controls were based on the re- dt search (Subchan et al., 2019), which are improvement dR of sanitation (u ), control treatment in the form of = α Q + α I − µR − νR (6) 1 dt 1 2 A medication during quarantine for infected individuals (u ) and education for vulnerable individuals (u ). dB 2 3 = ηI + ηI − dB −U B (7) dt A S 4 with the variables and parameters that formed up the system can be seen in Figure 2. It is assumed that S,E,IA,IS,Q,R ≥ 0 and all parameters are positive, which are taken from (Subchan et al., 2019)

3 OPTIMAL CONTROL PROBLEMS

The purpose of this research is to obtain control by minimizing the number of infected human popula- Figure 1: SEIQR Compartment Diagram the Spread of tions, bacterial populations and minimizing the costs Cholera. incurred for controls by considering equations (1)-(7). The objective function can be defined as follows The spread of water-based diseases, especially cholera, can be reduced by the use of chlorine which 1 Z t0 is believed to be effective in reducing bacteria. In ad- J(x,u) = { [C I2(t)] +C I2(t) 2 1 S 2 A dition, sanitation improvements are carried out to re- t f 2 2 2 (8) duce the level of absorption of bacteria caused by in- +C3B (t) +C4u1(t) +C5u2(t) fected individuals. Control treatment is also given to +C u2(t) +C u2(t)dt} people with cholera through quarantine to accelerate 6 3 7 4 healing of infected individuals and prevent spread to with t0 as initial time and t f is final time, and vulnerable individuals. In addition, education is also Ci was the parameter weight or price coefficient is- provided to individuals who are vulnerable to cholera sued at each control, where Ci > 0 for each i = as an effort to prevent the outbreak of the disease. 1,2,3,4,5,6,7. The interpretation of the mathematical model of cholera spreading by giving optimal control to the compartment diagram as shown in the Fig. 1. The

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The ﬁrst step to solve the optimal control problem using the Pontryagin Minimum Principle is to deﬁne (Subchan and Zbikowski, 2007) as follows

1 H = { (C I2(t) +C I2(t) 2 1 S 2 A 2 2 2 +C3B (t) +C4u1(t) +C5u2(t) 2 2 +C6u3(t) +C7u4(t))+ Figure 2: Variables and Parameters on Mathematical Mod- λS(Λ + νR + εE − µS − u3ψS − (1 − u1) els of the Spread of Cholera. βB S) + λE (u3ψS − εE− (κ + B) Equations (10)-(13) were substituted to Equation µE − (1 − u1)γE) + λ(IA)((1 − u1)p (9) so that it had the optimal Hamiltonian function ∗ βB (9) H . The next step was to determine the state equation S + (1 − u1)pγE− (Subchan and Zbikowski, 2007) as follows (κ + B)

µIA − α2IA) + λ IS)((1 − u1)(1 − p) ∗ ( S˙ = {Λ + νR + εE − µS − u∗ΨS − (1 − u∗) βB 3 1 S + (1 − u )(1 − p)γE βB (14) (κ + B) 1 S} (κ + B) − µIS − µSIS − u2δIS) + λQ ˙ ∗ ∗ ∗ (u2δIS − µQ − µQQ − α1Q) E = u3ΨS − εE − µE − (1 − u1)γE (15) + λR(α1Q + α2IA − µR − νR) + λ (ηI + ηI − dB − u B)} ∗ βB B A S 4 I˙ = p S + (1 − u∗)pγE − µI − α I (16) A (κ + B) 1 A 2 A where λi for each i = S,E,IA,IS,Q,R,B was the costate vector or Lagrange multiplier that depended ∗ ∗ ∗ ∗ ∗ βB ∗ on the state. Next, the optimal control value u1,u2,u3 I˙ = {(1 − u )(1 − p) S + (1 − u ) ∗ S 1 (κ + B) 1 (17) and u4 was found as follows ∗ (1 − p)γE − µIS − µSIS − u2δIS} ∗ 1 βB ˙ ∗ ∗ u1 = { ( S(PλIA + λIS (1 − p) − λS) Q = u2δIS − µQ − µQQ − α1Q (18) C4 κ + B (10)

+ γE(λIA P + λIS (1 − p) − λE )} ∗ R˙ = α1Q + α2IA − µR − νR (19)

∗ δIS(λIS − λQ) u2 = (11) ˙ ∗ ∗ C5 B = ηIA + ηIS − dB − u4B (20) And costate equations can be derived as follows ΨS(λ − λ ) u∗ = S E (12) 3 C ∗ ∗ ∗ 6 λS = − (−µλS − u3ψλS − (1 − u1)

∗ λBB βB ∗ ∗ u4 = (13) λS + u3ψλE + (1 − u1)p C7 κ + B B (21) The optimal control u∗ was obtained from ∂H and β ∗ ∂u λIA + (1 − u1)(1 − p) had the following characteristics κ + B βB ∗ ∗ λIS ) u1 = min(u1min,max(uˆ1,u1max)) κ + B ∗ ∗ ∗ ∗ λE = − (ελS − ελE − µλE − (1 − u1) u2 = min(u2min,max(uˆ2,u2max)) ∗ ∗ γλE + (1 − u1)pγλIA + (1 − u1) (22)

∗ ∗ (1 − p)γλIS ) u3 = min(u1min,max(uˆ3,u1max)) ∗ = − (C I − µ − λIA 2 A λIA α2λIA ∗ ∗ (23) u4 = min(u1min,max(uˆ4,u1max)) + α2λR + ηλB)

238 Optimization of Cholera Spreading using Sanitation, Quarantine, Education and Chlorination Control

∗ = − (C I − µ − µ λIS 1 S λIS SλIS ∗ ∗ (24) − u2δλIS + u2δλQ + ηλB)

∗ λQ = − (−µλQ − µQλQ − α1λQ + α1λR) (25)

∗ λR = − (νλS − µλR − νλR) (26) Figure 3: The Change of Rate on the Number of Infected Asymptomatic (IA) population, Infected Symptomatic (IS), βS Bacteria and Susceptible concentration. λ∗ = − (C B + λ (1 − u∗) κ R 3 S 1 (κ + B)2 Furthermore, the infected symptomatic individual ∗ pβSκ (27) also decreased when control is given. The number of + λI (1 − u1) A (κ + B)2 infected symptomatic individuals with the final time ∗ without control was 145 individuals while with con- + λIS (1 − u1) trol, the number of individuals at the end was 7 in- The optimal state and costate then can be deter- dividuals. The number of infected symptomatic indi- mined by considering the boundary condition x(0) viduals without control increased 680 individuals at = x0 and λ(t f ) = 0. t = 15, while with control, the number of infected symptomatic individuals were decreased at the beginning until the end. It means that the level of control effectiveness had an effect of 95.17%. The concentra- 4 COMPUTATIONAL RESULT tion of bacteria decreased with control. The amount of bacterial concentration at the end without control The parameter values are taken from (Subchan et al., was 7,5 ∗ 105 while with control the number of bac- 2019) and used for numerical simulation. The simu- terial concentrations at the end was 517. In this case, lation is solved by using Fordward-Backward Sweep control had an effect of 99.93%. Based on Fig. 3, Runge-Kutta Order 4 method (Lenhart and Work- it was shown that the level of control of individuals man, 2007; Burden et al., 2016; Lindfield and Penny, and bacteria was on maximum value 1 and sanitation 1995). The purpose of numerical simulation was to control was at value 0.4. determine the effectiveness of optimal control in each population. The simulation result on every population can be seen on Fig. 3 and the control can be seen on Fig. 4. Based on Fig. 2, it was known that infected asymptomatic individual decreased. The number of infected asymptomatic individuals with the final time without optimal control was 133 individuals while with control the number of individuals at the end of time was 2 individuals. This was due to the large in- fluence of β parameter, which was the level of con- ∗ ∗ ∗ Figure 4: Sanitation (u1), Quarantine (u2), Education (u3) sumption of bacteria through contaminated sources. ∗ and Chlorination (u4). If the value of β was getting bigger, then the number of individuals without optimal control would be even Based on Fig. 5, it can be seen that the level of less. It caused the individuals with mild symptoms to susceptible population decreased since the beginning. change into severe symptoms if individuals were not This was caused by the number of susceptible popu- aware of the symptoms because of lacking the knowl- lation interacted with cholera bacterial-contaminated edge or neglected the individual education. So, in this environment so the population were infected. Further- case the effectiveness of giving control in the infected more, the population increased at about t = 27. The asymptomatic subpopulation had an effect of 98.50%. educated population was increased from the very first So, the objective to minimize the number of individu- time and it was proportional to susceptible population als with mild symptoms had been reached. level. The quarantine population increased sharply at

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the beginning till day 15, then it decreased until the REFERENCES end. This is caused by infected symptomatic population decreased. Bakhtiar, T. (2015). Peran edukasi dan klorinasi dalam pen- In this case, the recovered population increased gendalian penyakit menular: Sebuah pendekatan kon- from the beginning then it stay in the certain value trol optimum. Semirata, 1. until the end. The asymptomatic and symptomatic in- Burden, R., Faires, J., and Burden, A. (2016). Numerical fected population decreased because bacteria concen- Analysis. Cengage Learning, Boston USA. tration level kept decrease until the end. Johnson and R, L. (2006). Mathematical modeling of Cholera: from bacterial life histories to human epi- demics. s.l.:University of California, Santa Cruz. Lemos-Paiao,˜ S., a., A., Torres, C., and FM, D. (2016). An epidemic model for cholera with optimal control treatment. Journal of Computational and Applied Mathe- matics. Lenhart, S. and Workman, J. T. (2007). Optimal control applied to biological models. CRC Press, Taylor and Figure 5: The Change of Subpopulation Rate and Bacteria Francis Group, London. Concentration with Control. Lindﬁeld, G. and Penny, J. (1995). Numerical Method Us- ing MATLAB. MPG Book Ltd, Botmin Cornwall. Organization, W. H. (2008). Prevention and control of 5 CONCLUSION cholera outbreaks: WHO policy and recommenda- tions. s.n, s.l. In this paper, system of differential equations were Sack, David, A., Sack, R., Nair, G., and Siddique, A. given as the dynamics model of cholera spreading (2004). Cholera. Lancet, 363:223–33. that was divided into human population and bacteria Setiadi, S. (2014). Ilmu Penyakit Dalam. Interna Publish- population classes. The optimal control in the form ing, Jakarta. of sanitation, chlorination, education, and quarantine Subchan, S., A.M., a. I., and F. (2019). An epidemic cholera model with control treatment and intervention. IOP were given as the attempt to control cholera spread- Publishing, s.l. ing. Subchan, S. and Zbikowski, R. (2007). Computational op- The simulation result showed the effect of the timal control of the terminal bunt manoeuvre—part 1: given control. Based on the computational result, Minimum altitude case. Optimal Control Applications controls affected the number of infected population and Methods, 28(5):311–325. and bacteria experienced decline so that cholera en- demic was not quite big problem. This showed that the optimum control strategy in the form of sanitation, chlorination, education, and quarantine gave signiﬁ- cant positive effect to minimize the spread of cholera.

ACKNOWLEDGEMENTS

The authors wish to thank Department of Mathemat- ics, Institut Teknologi Sepuluh Nopember and Institut Teknologi Kalimantan for their support and funding.

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