Simulation of Dengue Disease with Control

Transcription

1 nternational Journal of Scientific and nnovative atematical Researc (JSR) Volume 5, ssue 7, 7, PP 4-8 SSN 47-7X (Print) & SSN 47-4 (Online) DO: ttp://dx.doi.org/.4/ Simulation of Dengue Disease wit Control Nita. Sa, Foram A. Takkar, Bijal. Yeolekar Professor, Department of atematics, Gujarat University, Amedabad, ndia Researc Scolar, Department of atematics, Gujarat University, Amedabad, ndia Researc Scolar, Department of atematics, Gujarat University, Amedabad, ndia Corresponding Autor: Nita. Sa, Professor, Department of atematics, Gujarat University, Amedabad, ndia Abstract: Tese Dengue is a disease wic is transmitted to umans by a female mosquito named Aedes Aegypti. n tis paper, a matematical model as been developed to study te spread of dengue disease. Population of uman, mosquitoes and teir eggs are taken into consideration. To study te dynamics of te disease a compartmental model involving non-linear ordinary differential equations as been formulated for uman, mosquitoes and egg population. Stability analysis as been carried out at dengue free and endemic equilibrium points. Control in terms of preventive measures, spraying insecticides, and killing teir eggs wic is food for guppy fis are to be taken/applied to umans, mosquitoes and eggs respectively. Numerical Simulation as been carried out to sow te impact of control on different compartments. Keywords: atematical model, System of non-linear ordinary differential equations, Dengue, Stability, Control.. NTRODUCTON Dengue is a painful devastating mosquito borne disease caused by te dengue virus name Aedes Aegypti. Aedes Aegypti also known as yellow fever mosquito is a mosquito wic can spread dengue, cikungunya, Zika etc. and many oter diseases. Tis mosquito as a wite marking on its legs and as marking on upper surface of torax in te form of lyre. Tis mosquito wic was originated first in Africa is now found in all te tropical and subtropical regions over te world []. Of tese only female mosquitoes bites wic se needs to mature er eggs. osquitoes are attracted to te cemical compounds like ammonia, carbon dioxide, lactic acid etc. wic are emitted by mammals to find a ost. On taking blood meal tis mosquito can produce an average of to eggs per batc. During a life time tey can produce up to five batces. Tese number of eggs depends on te size of blood meal. Te eggs of tis mosquito are smoot, long, oval and about millimeter long. nitially, wen laid tey appear to be wite but in few minutes it turns out to be siny black. Tey prefer to lay eggs in tree oles, pots, tanks, water cooler, temporary flood areas and a lot at places were rain water is collected. Development of eggs depends upon time. n warmer climate tey take about two days, wile in cooler climate tey may take upon a week to develop. n dry regions, laid eggs can survive more tan years. But once submerged in water, tey atc immediately. Tese viruses are terefore difficult to control. nfected eggs atc and becomes an infected mosquito. Wit regards to control te attack of te diseases, a metod to reduce a population of mosquito as been prepared in wic a guppy fis is placed in water wic eats te mosquito larvae []. Wen a mosquito born wit dengue virus bites a person, te virus enters into te skin along wit its saliva. t enters into te wite blood cells and wile moving trougout te body tey reproduce inside te cells. Tis production of virus in te body arms many organs of te body suc as liver, bone marrow etc. Normally, tese mosquito bites at dawn and dusk time and tus can spread infection at any time of te day. Wen it bites a person wit dengue virus in teir blood te mosquito becomes infected []. t is not a contagious disease. Te symptoms of dengue include fever, eadace, joint and muscular pain, nausea, vomiting etc. Similar symptoms as tose of common cold, vomiting and diarrea are found in cildren, toug tey prove to be mild initially including ig fever but can increase te risk of severity. Particularly, tere is no vaccine for tis disease [4]. Tus, to protect oneself from te bites of infected mosquitoes make use of mosquito repellents and making use of nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 4

2 spraying insecticides as to be done to kill tem. Preventive measures sould be taken care off by individuals. Asmaidi et al. [5] as done teir researc entitled A SR atematical odel of Dengue transmission and its simulation in wic e as incorporated temperature as one of te parameter. Tis paper as been modelled by considering eggs, mosquitoes and uman population as all te tree plays an important role in dengue disease. n tis paper, uman population as been divided into susceptible umans, infected umans, ospitalized umans, recovered umans. osquito population as been divided into susceptible mosquitoes and infected mosquitoes. gg population as been divided into susceptible eggs and infected eggs. atematical model for te transmission of dengue disease as been described in Section. Section, 4 and 5 includes stability, control and simulation for te compartments. Conclusion is described in Section 6.. ATATCAL ODL ere, we formulate a matematical model for uman-mosquito-egg population. Te notations wit its description for eac parameter is given below Table. Table. Notations and its Parametric Values Notations Description Parametric Values N Sample Size of uman population N Sample Size of osquitoes population N Sample Size of mosquitoes ggs population S t Number of Susceptible umans at some instant of time t 5 t Number of nfected umans at some instant of time t 5 t Number of Susceptible umans at some instant of time t 8 R t Number of Recovered umans at some instant of time t 5 S t Number of Susceptible osquitoes at some instant of time t 4 t Number of nfected osquitoes at some instant of time t 5 S t Number of Susceptible ggs at some instant of time t 6 t Number of nfected ggs at some instant of time t 4 New Recruitment Rate.4 umans Deat Rate.5 osquitoes Deat Rate.5 ggs Deat Rate.4 Disease nduced Deat Rate of umans. Transmission rate of umans from susceptible to infection caused by.4 infected mosquitoes Transmission rate of umans from infected to ospitalized.6 Transmission rate of umans from ospitalized to recovered. Transmission rate of mosquitoes from susceptible to infection caused by.6 infected umans nfected ggs atcing Rate.5 Proportion of infected eggs laid by an infected mosquito. C Climatic Factor.7 Oviposition Rate. u Control rates in terms of preventive measures, u Control rates in terms of spraying insecticides, u Control rate in terms of guppy fis, nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 5

3 Te transmission diagram of dengue from different compartments is sown in Figure. Fig. nfected mosquito can infect susceptible egg S and susceptible uman being S. Also, infected uman can infect susceptible mosquito S. So, te disease can spread by two ways: infected mosquito and infected uman. Susceptible uman infects uman at te rate and so total individuals in te infected uman compartment are S and te rate of infected umans are N seeking ospitalization so total individuals in te ospitalized compartment are. Te treatment given in compartment makes individual free of dengue disease at te rate and so total individuals recovered R from it are. Disease induced deat occurs in two uman compartment namely infected umans and ospitalized umans at te rate. nfected umans coming in contact wit a susceptible mosquito makes tem infected by te rate. Total infected mosquito population is S. Susceptible eggs and infected eggs atces teir infected N eggs at te rate depending upon te climatic factor C. Oviposition rate is described by and te proportion of infected eggs laid by an infected mosquito is described wit te rate. and denotes te deat of eggs or mosquitoes due to unfavorable conditions may be ig temperature or ig umidity index, wile denotes natural deat rate of umans. Control u in terms of preventive measures at uman population, u in terms of spraying insecticides for mosquito population, u to curtail number of infected eggs by use of guppy fis wic is an aquatic creature. So, from te above figure, we ave te following set of nonlinear ordinary differential equations describing te causes and cureness of dengue from one compartment to oter. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 6

4 ds N N S K N d S N d dr ds d ds d R S C S S N S C N N S C S K N C K () wit S R N, S N, S N and S, S, S ;,, R,,. Simplification of equation () as been carried out by comparing eac population to a total population i.e. S S, N N R S S,, R, Sm, m, Se, e () N N N N N N To obtain a new set of equations mainly of five dimensions equation () is substituted in equation () namely as rror! Bookmark not defined. () ds N z m S K d z ms d dm C z d m N C z K m m wit S R, S, S m m e e were z N N and z N Te feasible region for te above set of equations () is nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 7

5 N K S m e / S m e ; S,,, m, e N K Tus, Dengue free equilibrium point = S,,, m, e =,,,,. Now, our actual interest lies in calculating te basic reproduction number wic is calculated using te next generation matrix metod wic is defined as FV [6] were F and V bot are Jacobian matrices of and v evaluated wit respect to infected umans, mosquitoes, eggs and ospitalized umans at te point. dx Let X,, m, e, S. ( X ) v( X ) were ( X ) denotes te rate of newly recruited and vx ( ) denotes te rate of derived recruited wic is given as z ms ( X ) m C ze m and v X m N C e z K N z m S K Now, te derivative of and v evaluated at a dengue free equilibrium point and V of order 55wic is defined as gives matrices F i( ) vi( ) F V X j X j for i, j,,,4,5 z N K K So, F and C z V N C z K z N K K nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 8

6 were V is non-singular matrix. Tus, te basic reproduction number R wic is te spectral radius of matrix R FV is given as z K N C K K C N K CK K On equating set of equations () to zero, an endemic equilibrium point follows: S,,,, were S m e K N K z m, z S m z S, m K m C z e. STABLTY ANALYSS, m e zk C K N, is obtained wic is as n tis section, te local and global stability at and using te linearization metod and matrix analysis are to be studied.. Local Stability Teorem.. (stability at ) f R ten a system is locally asymptotically stable at dengue free equilibrium point. f R ten it is unstable. Proof: Jacobian atrix of te system evaluated at point is J z N K K z N K K C z N C z K Te eigenvalues of te caracteristic equation are satisfies te equation a a C a a a a were, and, 4, 5 nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 9

7 C C K N K z K N a is obvious. a KK wic C K C N C K C K K K R So, if R ten a. ence, a, a, a and aa a if R. Tus by Rout urwitz criteria [7] all te conditions for te stability of n are satisfied. Terefore, te system is locally asymptotically stable at. Teorem.. (stability at K ) f QU RT asymptotically stable at an endemic equilibrium point Proof: Jacobian atrix of te system evaluated at point J were P T P Q T S R U Xz V X and X ten a system is locally. m m is P z, Q, R, S, T z S, U, V N, X C z K Te eigenvalues of te caracteristic equation are S and,, 4, 5 satisfies te equation a a a a a were 4 4 a a X U P Q a P Q U Q U X PQ PU PX QU QX UX VXz RT wic is also obvious. a Q U PQ QX UX VXz P Q PU X QU RT X PQU PQX So, if QU RT and X ten a and VXz P Q compared to oter terms and consist of a very negligible value. a PQU PQUX QU RT X VQXz P 4 is very small term as nternational Journal of Scientific and nnovative atematical Researc (JSR) Page

8 But as VQXz P is very small term. So, if QU RT ten a4. Tus, a, a, a, a4 and a a a a a a if QU RT. 4 ence, all te conditions of Rout urwitz criteria for te stability are satisfied for n 4 so te system is locally asymptotically stable at.. Global Stability Teorem.. (stability at ) f asymptotically stable at dengue free equilibrium point. Proof: Consider a Lyapunov function z N K N ten a system is globally K z K m e L t t t t t L t C C z e m N m zs z K N So, if zs tat is z K ten Lt. And if m e ten L t ence, Lt is. z N K N K z K. as and is non negative So, by LaSalle s nvariance Principle [8], every solution of system (), wit initial conditions in, approaces as t. Tus, a system is globally asymptotically stable at point. Teorem.. (stability at ) Te system is globally asymptotically stable at endemic equilibrium point. Proof: Consider a Lyapunov function Lt S t S t t t t t m t m t e t e t Lt S t S t t t t t m t m t e t e t S m e On using S we ave m e S t S t t t t t m t m t e t e t S t S t t t t t m t m t Lt m N e t e t m Ce z K N S m e K ence, te system is globally asymptotically stable at. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page

9 4. OPTAL CONTROL ODL n tis section a control function as been implemented to decrease te spread of dengue disease in uman population. Te objective function along wit te optimal control variable is given by T A S A A A4 R A5 S A6 J ui, A7 S A8 wu wu wu (4) were denotes te set of all compartmental variables, A, A, A, A4, A5, A6, A7, A 8 denotes nonnegative weigt constants for te compartments S,,, R, S,, S, respectively and w, w, w are te weigt constant for te control variable u, u, u respectively. Te weigts w, w and w wic are constant parameters for u, uand u will standardized te optimal control condition. Now, we will calculate te value of control variables u, u and u from t to tt suc tat i J u t, u t, u t min J u, / u, u, u were = smoot function on te interval,. Using, Fleming and Risel results [9], te optimal control denoted by u i is obtained by collecting all te integrands of te objective function () using te lower bounds and upper bounds of te bot te control variables respectively. Using Pontrygin s principle [], we construct a Lagrangian function consisting of state equation and adjoint variables A (,,,,,,, ) wic is as follows: V S R S S L, A A S A A A R A S A A S A w u w u w u nternational Journal of Scientific and nnovative atematical Researc (JSR) Page V N N S u S S K N S u S N R R S S C S u S N S C u S N S K N C u S K N S C S u S Now, te partial derivative of te Lagrangian function wit respect to eac variable of te compartment gives us te adjoint equation suc tat

10 S R S S L S A S u L S S N S A S N L 4 A L R AR L S R R A5S u S S S S N L S N N A 6 S S S N K K L S A S C u 7 S S S S A C 8 L Te necessary conditions for Lagrangian function L to be optimal for control are L wu S S u L wu S S u L wu S S u On solving equation (5), (6) and (7) we get, S S u, S u S and u w w ence, te required optimal control condition is obtained as u max a,min b, S S w S S w (5) (6) (7) nternational Journal of Scientific and nnovative atematical Researc (JSR) Page

11 u max a,min b, S S w max,min, S u a b S w were a, a, a = lower bounds and b, b, b = upper bounds of te control variables u, uand u respectively. 5. NURCAL SULATON Using te data given in Table, it is observed tat 5% population gets infected by dengue disease, owever approximately only % disease is transmitted from dengue free equilibrium point to te existence. n tis section, we will analyze and study te effect of control on eac compartment numerically. Figure. Control in terms of preventive measures on compartment Fig. depicts tat wen control is not applied infected umans increases from 5 to 8 but wen control in terms of preventive measures are taken by tem te individual decrease to 7 approximately witin a day. Figure. Control in terms of preventive measures on compartment From Fig. it can be seen tat wen an individual gets medical treatment along wit preventive measures, tey recover from te diseases faster comparatively. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 4

12 Figure 4. Control in terms of preventive measures on R compartment Fig. 4 sows tat individual affected from dengue recovers soon if te ways to control are adopted by tem. n fact, it seems to be more fruitful ass it increases te recovered individual from 9 to 4 in te same duration of days. Figure 5. Control in terms of spraying insecticides on S compartment Fig. 5 sows tat a mosquito responsible for dengue disease are removed from te community in a small duration of time wen insecticides are used. Figure 6. Control in terms of spraying insecticides on compartment nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 5

13 Fig. 6 depicts tat in te initial stage itself infected mosquitoes decreases in comparison to wen control in terms of spraying insecticides is not done. But approximately after days, no effect of control is observed wic means tat te mosquitoes ave lost teir life span and as died out. Figure 7. Control in terms of guppy fis on S compartment Fig. 7 sows tat te eggs laid by a susceptible mosquito proves to be susceptible wic can be decreased as tis is te food of guppy fis. f tis eggs are eaten by te fises, we can ave our surrounding free from mosquitoes. Figure 8. Control in terms of guppy fis on compartment Fig. 8 depicts tat te susceptible eggs wic are not eaten by guppy fis will surely become infective and will give birt to an infected mosquito wic is armful for te individuals in te society. But it can be decreased by te control wic is observed from te figure. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 6

14 Figure 9. Control Variables versus Time (in days) Fig. 9 sows tat eac control as a vital role in protecting umans from dengue disease. t is seen tat % preventive measures, 8% use of insecticides, 8% Guppy fis are required in te initial days to stop oneself from becoming te victim of tis disease. 6. CONCLUSON n tis paper, a matematical model as been constructed to study te transmission of dengue disease. t as been establised tat tis number of dengue affected individual can be reduced by using control on tem as well on mosquitoes and teir eggs. Two equilibrium points ave been found: Dengue free equilibrium point and endemic equilibrium point. At bot tese points system proves to be locally asymptotically stable and globally asymptotically stable. Results for different compartments ave been calculated numerically wic interprets tat to reduce overall severity of te dengue disease, increase in use of guppy fis at initial stage of mosquito cycle, increase spray of insecticides for te last stage of mosquito cycle and preventive measures at te end of umans is required. ACKNOWLDGNTS Te autor tanks DST-FST file # S-97 for tecnical support to te Department of atematics. [] ttps://en.wikipedia.org/wiki/aedes Aegypti RFRNCS [] ttp:// [] ttps://en.wikipedia.org/wiki/dengue_fever [4] ttps:// [5] Asmaidi, A., Sianturi, P., and Nugraani,.., A SR atematical odel of Dengue Transmission and its Simulation, ndonesian Journal of lectrical ngineering and Computer Science. (), , (4). [6] Diekmann, O., eesterback, J.A.P., Roberts,.G., Te construction of next generation matrices for compartmental epidemic models, Journal of te Royal Society nterface, 7(47), , (). [7] ttp://web.abo.fi/fak/mnf/mate/kurser/dynsyst/9/r-criteria.pdf [8] LaSalle, J. P., Te Stability of Dynamical Systems, Society for ndustrial and Applied atematics, Piladelpia, Pa, (976). ttps://doi.org/.7/ [9] Fleming, W.., Risel, R. W., Deterministic and Stocastic optimal control (). Springer Science and Business edia, (). [] Pontriagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., iscenko,. F., (986). Te atematical Teory of Optimal Process, Gordon and Breac Science Publisers, NY, USA, 4-5. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 7

15 AUTORS BOGRAPY Nita. Sa, is a Professor in te Department of atematics, Gujarat University, Amedabad. Se as 5 years of researc experience in inventory management, forecasting and information tecnology and information systems. Se as publised 75+ articles in international journals, including APJOR (Singapore), nternational Journal of Production conomics, OGA, CCRO (Belgium), CP (Romania), easuring, Control & Simulation (France), JOS (ndia), JOS (ndia), ndustrial ngineering (ndia), uropean Journal of Operational Researc, JR, JOR, JOR, JBPSC, etc. Foram Takkar, is researc scolar in department of matematics, Gujarat University, ndia. Se is working in dynamical systems of socio-medico problems. er researc interest is applications of matematics in various disciplines. Se as publised 4- papers in JOBAR, AR, DCDS-B, ADSA. Dr. Bijal Yeolekar, received P.D. recently in matematics from Gujarat University, ndia. Se worked in dynamical systems of social issues. er researc interest is applications of matematics in various disciplines. Se as publised 8- papers in JOBAR, AR, DCDS-B, ADSA, AC, A, JC, JCCSR, etc. Citation: N.. Sa et al., " Simulation of Dengue Disease wit Control ", nternational Journal of Scientific and nnovative atematical Researc, vol. 5, no. 7, p. 4-8, 7., ttp://dx.doi.org/.4/ Copyrigt: 7 Autors. Tis is an open-access article distributed under te terms of te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original autor and source are credited. nternational Journal of Scientific and nnovative atematical Researc (JSR) Page 8