A MATHEMATICAL ANALYSIS OF THE EFFECTS OF CONTROL OF PLASMODIUM KNOWLESI MALARIA

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1 Pak. J. Statist Vol. 31(5), A MATHEMATICAL ANALYSIS OF THE EFFECTS OF CONTROL OF PLASMODIUM KNOWLESI MALARIA Moaed Baba Abdullai, Yaya Abu Hasan and Fara Aini Abdulla Scool of Mateatical Sciences, Uniersiti Sains Malaysia USM Penang Malaysia ABSTRACT Tis paper considers a ateatical odel for te spread of Plasodiu knowlesi alaria. Te odel incorporates recruitent of uan, osquito and acaque populations troug a constant iigration, wit a fraction of infectie iigrants. Te basic properties of odel and equilibrius for te odels are analyzed rigorously. Te sensitiity analysis of te odel is carried out to recognize te ost sensitie paraeters for te disease transission. Tus, control strategy is presented and studied. Optial control teory is applied to inestigate optial strategies for controlling te spread of te alaria using quarantine, insecticide spray and culling as te control ariables. Te results obtained fro nuerical siulation of te odel sow possible significant reduction of te disease spread using te cobine control strategies. KEY WORDS Plasodiu knowlesi alaria; Control; infectie iigrants; Drug Resistance. 1. INTRODUCTION Malaria is ostly ignored in te Sout-East Asia, altoug it as te igest figure of populace ore ulnerable to te disease [1]. In te Region te disease exibits strange epideiological rareness suc as jungle alaria and alaria due to oeent across intercontinental borders [1]. Oters include rising resistance to drugs and insecticides, detecting and treating Plasodiu knowlesi alaria. Broader callenges around ealt syste capacity also exists wic ipact on te ability of countries to sustain control actiities and work towards eradication of te disease (Malaria 2012) [2]. Its eradication is now a target of a lot of countries in te region and te Western Pacific. Howeer, uge reduction in te disease occurrence as been acieed [3, 4]. Te ajor callenge lingers, and toug te treats of arteisinin resistance ae been te center of uc intercontinental concern, zoonotic alaria species (Plasodiu knowlesi alaria) ae receied less consideration [3, 4]. Plasodiu knowlesi alaria is te ost significant eerging ector-borne infectious disease [4]. Presently, tis alaria species is prealent in Souteast Asia were te osquito and its reseroir ost, te acaque, are abundant [5-7]. In 1931, te parasite was first recognized in India fro an iported long-tailed acaque fro Singapore [8]. Its capability to transit a disease to uans was earliest depicted in 1932, after Knowles and Das Gupta effectiely transitted te parasite to two uan wo offer teseles on free will by blood passages fro infected acaques [8] Pakistan Journal of Statistics 483

2 484 A Mateatical Analysis of te Effects of Control of Plasodiu Te anageent of te disease is still unknown, and te alaria cure guidelines by WHO alaria reports (2012) do not proide any suggestions for its treatent [9, 10]. Te infectiity as been treated effectiely wit efloquine, quinine and cloroquine [10, 11]. Te disease is transitted by Anopeles leucospyrus to uan and acaques, were te acaques sere as te origin and te reseroirs of te parasite [12]. Huan can be infected eerywere witin te distribution restrictions of Anopeles leucospyrus osquitoes if te infected acaques are present [13]. Forest dwelling osquitoes of te Anopeles leucospyrus group, suc as Anopeles latens, Anopeles leucospyrus, and Anopeles dirus are responsible for te disease transission fro uan to uan only [14]. Countries suc as Malaysia, Tailand, Cina, Singapore, Pilippines, Vietna, Myanar and Indonesian ae recorded uan infections of te disease [14]. Traditionally, uan oeents ae added to te stretc of diseases [15, 16]. Failure to reflect on tis reason contributed to decline of success of alaria eradication figts in te 1950s and 1960s [17]. Huans often trael across national borders, and national alaria control efficacy can be coproised by iported alaria [18]. Iported alaria cases transit parasites, as well as resistant strains, een wen at asyptoatic stage [19]. Altoug Plasodiu knowlesi alaria infection aong western traelers returning fro prealent area as been reported interittently for years, tere as been no logical description of te clinical pattern of Plasodiu knowlesi alaria infection aong traelers [5]. Te international transfer of alaria can coe about during te oeent of an infected osquito into a non endeic-disease area; tis is described as airport alaria [15]. Te osquitoes are typically oed into non endeic-disease nations on an intercontinental fligt. For instance, accidental searc of aircrafts at Gatwick Airport (London) establised tat 12 of 67 aircrafts fro tropical nations contained osquitoes [15, 20]. After leaing te plane, osquito igt lie long enoug to take a blood eal and spread te disease, abitually in te neigborood of an airport [15, 20]. Toug te occurrence of tese circustances is low, tey account for te ajority of te disease spread in deeloped countries [5]. A traeler infected wit te disease can aid as a reseroir and seed localized outbreaks or epideics in tose areas, and tus infected traelers becoe actie transitters of infection in low transission areas [21, 22]. Soe alaria odels ae been deeloped for drug resistance and effect of infectie iigrants. Tese include te studies of Aneke, (2002) [23] tat forulated a deterinistic odel wic illustrates te incident of drug resistant alaria in a yper prealent region by a scee of ordinary differential equations odels. Meanwile, no control strategy is incorporated into te odel. Koella & Antia (2003) [24] deried a odel to exaine te spread of anti-alarial resistance. Te autors concluded tat te odels deeloped in teir work are an initial stage in tougtful of te causes of alaria drug resistance and assessing easures to ease te stretc of resistance. Altoug, tis is one of te earliest drug resistance alaria odel, but te precise suggestion for te anageent of resistance were not gien due to unaailability of data. Ciyaka et al., (2009) [25] forulated and analyzed a odel for anageent and stretc of drug resistance in a population wit partly resistant uans. Te odel extends te odel of (Aneke, 2002; Koella & Antia 2003; Bacaër & Sokna 2005) [23, 24, 26] by including discrete tie delays on te dorant stage in bot ectors and uans and fractional

3 Abdullai, Hasan and Abdulla 485 resistance. Tey also reflect on occasions were te treated uans are contagious and wile tey are not transittable. Tey concluded tat te stretc of drug resistance wit anageent as a control easure relies on te fraction of te contagious tie of treated and untreated uans. Also te spread rates fro infectious uans wit resistant and sensitie strains. Te sortcoings of te odel includes: not considering te recoered uans and te return pat fro infected uans to te susceptible. Estea et al., (2009) [27] deried a deterinistic odel for exaining te effect of drug resistance on te spread of alaria in uans. Te odel extends te earlier work of (Koella & Antia 2003; Aneke, 2002) [23, 24] tis is by including te infectious osquitoes as a ariable and also allows treated uans wit te wild strain to adance to te resistance strain. Te ai was to obsere te epideiological ipact of te antialarial drug in dropping te infection saddle in known inabitants. Also to obsere te ipact predisposed by te deelopent of resistance in addition to te suitability of te resistant strain. Te autors concluded tat wile te two strains exist togeter, te fraction of uans wit te resistant strain at equilibriu-state reduces wit growing scale of resistance. Howeer, no control strategies were suggested. Tuwiine et al., (2010) [28] forulated a uan-osquito odel for te transission of alaria tat includes uans recruitent ia a steady igration, wit a portion of infectie igrants. Te odel extended te earlier odel by Tuwiine et al., (2007) [29] were recruitent into te uans was ia birt. Tey establised tat because of iigration of infectie uans, a steady state wit a positie portion of infectiity is constantly present. Terefore, te disease-free steady state of te odel will not exist and only te endeic steady state were te infection perseeres in te population for an extended period. Howeer, te odel does not consider any control strategies and infectie iigrants in te osquito population. Tcuence et al., (2011) [30] forulated and analyzed a deterinistic odel for alaria wit effect of treatent and tey recognized tree classes of resistance in indiiduals. Te odel includes in cooperated responsie and resistant strains of te parasites in uans and te osquitoes. Te autors establised tat te odel exibits backward bifurcation and tat wit specific odel suppositions; wen edication is increased it would ae an inadequate adantage, particularly in setting wit extraordinary transference rate. Te ai of tis work tus, is to build a odel tat would illustrate te essential procedures of te progress and te dependent inestigation of te equilibriu points. Teir conclusions entail tat ore adanced stages of edication will possibly guide to aplify endeic extent and te scope to wic tis appens relies on additional issues, for instance te degree of edication and resistance adanceent. Howeer, efficiency of te anti-alarial edications or edication sensitiity is not included into te edication section of te odel and no control strategy was considered. Makinde & Okosun (2011) [31] deeloped and inestigated a deterinistic odel tat illustrates te dynaics of te disease infectiity wit te conscription of infected igrants, anageent of infectie and te use of insecticides to counter te ectors in te inabitants. Te work aied at deeloping a deterinistic odel wit infected and transittable igrants suc tat tey exaine te function of introduction of infected igrants in te disease spread. Tey applied Pontryagin s Maxiu Principle, to obtain te best possible approaces for te disease anageent. Howeer, tey concluded tat

4 486 A Mateatical Analysis of te Effects of Control of Plasodiu infected igrants ae no robust effect in te spread of te infection, if tere is an efficient anageent syste and te ector control. Altoug, te odel considered te control easures and sensitiity analysis but te odel is ore suitable for Plasodiu falcipariu. Furterore, infected iigrants are soe sources of resistance strain te disease. Okosun & Makinde (2012) [32] deried and analyzed a ateatical odel tat differs fro tose earlier works on drug-resistant anti-alaria; since it includes te copartents of bot uans and osquitoes wit drug-resistant strain parasites. Te study aied at addressing te extent of lack of basic aenities and drug resistance strains anipulating te spread of alaria disease. Te autors exained and establised te likely effect of control easures on te stretc of te disease. Tey establised te conditions necessary for best possible anageent of te disease applying te Pontryagin s axiu principle to establis best possible anageent for te stretc of te infection. Tey concluded tat using best possible anageent actions can lead te disease to a stable disease free equilibriu state (DFE). Altoug, control strategies were suggested and sensitiity analysis ealuated, te odel is ore suitable for Plasodiu falcipariu. Tuwiine et al., (2014) [32] extended te odel of Tuwiine et al., (2007) [29], by adding to uan population drug resistance and drug sensitie uans. Te odel was forulated and analyzed. Te autors concluded tat, odel can elp out in deterining control easures and can present broad assessent of odel assuptions tat anipulate decisions. Howeer, te odel does not suggest any control strategy. Altoug, te osquito population ougt to ae bot drug resistance and drug sensitie due to teir interaction wit uan population. Few studies ae been carried out to quantify te ipact of Plasodiu knowlesi alaria infection in uan and acaques [34, 35]. Fro te work of [34] tey forulated a deterinistic syste of differential equations wit tree classes wit infected uans, osquitoes and acaques to study te ector preference and ost copetence. Te work of [35] use a deterinistic syste of differential equations to study te ipact of treatent, biological and ceical control strategies in controlling te spread of Plasodiu knowlesi alaria in a odel. In tis paper, culling as a control strategy for dislodging osquitoes and it larae as used in [36, 37], and also for acaques control by reoing te acaques fro te endeic areas to a resered area. Ten, te tie dependent control easure using optial control teory is considered. Te teory as been applied wit alaria odel see for exaple [36-40]. Iai et al., (2014) [41] te autors deeloped a new odel tat is te extension of an earlier odel of te disease by Yakob et al., (2010) [35]. Te extension includes addition of tree caracteristic geograpical sites (forest (J), far (F) and illage (V)) in wic exposure to infection and transission can occur. Te odel is aied at ow elpful currently accessible alaria control strategies are against te disease was as well a ital concern tat was explored using te odel. Te autors use te odel to assess te probable ipact of rapid treatent and te use of insecticide-treated bed nets in preenting wider spread of tis rising infectiity. Te autors conclude tat teir outcoe deonstrated tat continued uan-ector or uan transission is unlikely to be taking place at present. As ecological alteration continues, tere is te possible for te occurrence of te disease to aplify and to turn out to be an iportant public ealt crisis. Tese outcoes

5 Abdullai, Hasan and Abdulla 487 also draw attention to te need for continued control and alertness of tis zoonotic alaria ostly as Malaysia enters te pre-eliination pase for oter alaria species. Te ai is to deelop a deterinistic ateatical odel and suggest possible optial control for te disease. Tis is to incorporate te effect of iigrations (uans, osquitoes and acaques) and drug resistance. Te extension to consider te effect of infectie iigrants is otiated by soe earlier work on alaria (see, for instance, Tuwiine et al., 2010; Makinde & Okosun 2011; Okosun & Makinde 2012; Tuwiine et al., 2014) [28, 31-33]. And effect of drug resistance is inspired by te work of (Aneke, 2002; Koella & Anita 2003; Ciyaka et al., 2009; Estee et al., 2009; Tuwiine et al., 2011) [23-25, 27] Te sensitiity analysis of te odel will be carried out to recognize te ost sensitie paraeters for te disease transission. Tis is inspired by te soe recent researc on alaria epideics (see, for instance, Okosun & Makinde 2013; Okosun et al., 2013) [42, 43]. Te paper is organized as follows: Section 2 is dedicated to odel depiction and equilibriu analysis. Sensitiity analysis of te odel is presented in section 3. In section 4, te existence of control proble stated and ten applies te Pontryagin s Maxiu Principle to find te necessary conditions for optial control. Nuerical siulations of te odels are carried out in section 5 and finally conclusion in section MODEL FORMULATION N t, Te odel sub-diides te total uan population at a tie t, denoted by into te following sub populations of susceptible uan S ( t ), uan infected wit parasite sensitie to arteisinin Is( t ), uan infected wit parasite tat is arteisinin resistant strain Ir() t and recoered or iune uan R ( t ). Tus, N () t S I I R s r Te odel sub-diides te total acaque population at a tie t, denoted by N() t, into te following sub populations of susceptible acaque S () t, acaque infected wit parasite sensitie to arteisinin I ( t ), acaque infected wit parasite tat is arteisinin resistant strain tat, s I t and te recoered or iune acaque () r N () t S I I R s r R ( t ). So Te odel sub-diides te total osquito population at a tie t, denoted by N() t, into te following sub populations of susceptible osquitoes S () t osquito infected by te parasites sensitie to arteisinin Is ( t), and osquitoes infected by alaria parasites wit arteisinin resistant strain Ir ( t ). Terefore, osquito populations are gien by; N () t S I I s r

6 488 A Mateatical Analysis of te Effects of Control of Plasodiu It is assued tat uans are born susceptible witout infection at a rate. Te odel wit flow into te population tat is eiter infected wit te parasite tat is sensitie to arteisinin or parasite wit arteisinin resistant strain or susceptible uans is considered. Tis flow is assued to occur troug iigration at constant rate. A fraction s is uan infected wit parasite sensitie to arteisinin, a fraction r is uan infected wit parasite wit arteisinin resistance strain and te reaining fraction is susceptible uan. Te uan population suffers a natural deat at a 1 s r rate and disease induced deat at a rate 0. Susceptible uans infected wit parasites sensitie to arteisinin are oed to te infected uans wit parasite sensitie to arteisinin Is() t class at a rate sci s s, were s is te transission probability per bite, c s is te contact rate of uan wit osquitoes wit parasite sensitie to arteisinin. Susceptible uan infected wit parasites wit arteisinin resistant strain Ir () t oe to infected uans wit arteisinin resistant strain at a rate rci r r, were r is te transission probability per bite, c r is te contact rate of uan wit arteisinin resistant strain osquitoes. A fraction of uans wit parasites sensitie to arteisinin after treatent progress to te recoered/iune class wit at a rate 1 d. Te fraction d of treated infected uans wo deelop arteisinin resistant strains are oed to te infected uan wit arteisinin resistant strain class. Te recoered indiiduals lose teir iunity at a constant rate, and returned to te susceptible class. Te osquito population is recruited at a rate. A fraction s is osquito infected wit parasite sensitie to arteisinin, a fraction is osquito infected wit parasite wit arteisinin resistance strain. Te reaining fraction r 1 s r is susceptible osquito. Mosquito population suffers a natural deat at a rate. Te susceptible osquitoes c I I and rcr Ir Ir were s S oe to infected class at a rate s s s s is te transission probability per bite of eiter te uans infected wit parasite sensitie to arteisinin I s or acaques infected wit te parasites wit sensitie to arteisinin I s. Oter possibility is acaques infected wit parasites resistance to arteisinin I r or infected uan wit arteisinin resistant strain Ir. r is te transission probability per bite of eiter te uans infected wit parasite resistance to arteisinin. Susceptible osquitoes infected wit alaria parasites sensitie to arteisinin are oed to te class of infected osquitoes wit alaria parasite sensitie to arteisinin I s. And tose susceptible osquitoes infected wit parasites wit arteisinin -resistant strain are oed to te class of infected osquitoes wit drugresistant strain I r. It is assued tat te acaque population, increased troug birt or iigration at constant rate. A fraction s is acaque infected wit parasite sensitie to

7 Abdullai, Hasan and Abdulla 489 arteisinin. A fraction r is acaque infected wit parasite wit to arteisinin resistance strain and te reaining fraction 1 s r is a susceptible acaque. Te acaque population suffers a natural deat at a rate and disease induced deat at a rate 1. Susceptible acaque infected wit te parasites sensitie to arteisinin are ten oed to te infected uans wit parasite sensitie to arteisinin Is() t class at a rate sci s s, were s is te transission probability per bite, c s is te contact rate of acaque wit osquitoes wit parasite sensitie to arteisinin. Susceptible acaque infected wit parasites wit arteisinin resistant strain Ir () t oed to infected uans wit arteisinin resistant strain at a rate rci r r, were r is te transission probability per bite, c r is te contract rate wit arteisinin resistant strain. Infected acaque recoers fro te disease by acquiring iunity at a rate. Te recoered acaques lose teir iunity at a constant rate, and return to te susceptible class. Tese assuptions lead to te following coupled syste of ordinary differential equations wic describe te progress of te disease (see Table 1 and 2 for te ariables description and nuerical alues of te paraeters of te odel). ds 1 s r R scsis rcr Ir S dis s scsiss 0 Is dir r rcr Ir S dis r I r dr 1 d Is R ds 1s r scs Is Is rcr Ir Ir S dis s scs Is Is S Is dir r rcr Ir Ir S Ir ds 1 R c I c I S dis s scsiss 1 I s dir r rcr Ir S r Ir dr I R s r s s s r r r (1)

8 490 A Mateatical Analysis of te Effects of Control of Plasodiu were; s r s r s ci s s s ci r r r r c I I s s s s c s r r r r r s s s s s r r r r r I ci c I I refer to ass action force of infection for infected uan wit strain sensitie to arteisinin, infect uan wit arteisinin resistant strain, osquito infected wit sensitie strain to arteisinin, osquito infected wit resistance strain to arteisinin, acaques infected wit sensitie strain to arteisinin and acaques infected wit resistance strain to arteisinin respectiely. 2.1 Basic Properties In tis section, te basic dynaical features of te odel (1) will be explored. Te next is asserted; Lea 1: Te closed set 11 S Is Ir R S Is Ir S Is Ir R,,,,,,,,,, : S Is Ir R ; S Is Ir ; S Is Ir R, is positiely-inariant and attracting wit respect to te basic odel (1). Proof: Adding te first four equations for uans, te next tree equations for osquitoes and te last four equations for te acaques in te odel respectiely yields: and dn dn dn N I I (2) 0 s r r N (3) N I I (4) 1 s r r Assuing Is Ir Is Ir 0 tus, equation odel equation (2) and (4) becoes;

9 Abdullai, Hasan and Abdulla 491 dn dn dn N N N Since dn N, dn N and dn N, dn dn dn it follows tat 0, 0 and 0, if N, N and N, respectiely. Tus, using te standard coparison teore (Laksikanta et al., 1988) [44] it as been sown tat and N t ( t) ( t) ( t) N ( t) N 0 e 1 e, N ( t) N 0 e 1 e ( t) ( t) N( t) N 0 e 1 e. In particular, N, N and N 0 and N 0, Moreoer, if N, N enter in finite tie, or N () t approaces N () t approaces, if N 0, respectiely. Tus, is positiely-inariant. and N, ten eiter te solution, N () t approaces and, and te infected ariables Is, Ir, Is, Ir, Is and I r approaced zero. Hence is attracting (tat is, all solutions in 11 eentually enter ). Tus, in, te odel is well-posed epideiologically and ateatically. Hence it is sufficient to study te dynaics of te odel in Disease Free Equilibriu (DFE) Te Plasodiu Knowlesi alaria odel (1) as a DFE, obtained by setting te rigt-and sides of te equations in te odel (1) to zero, gien by te following;

10 492 A Mateatical Analysis of te Effects of Control of Plasodiu 1 R c I c I s scs Is S 0 Is 0 r r cr Ir S dis r Ir 0 1 d Is R 0 1 s r scs Is Is r cr Ir Ir s scs Is Is S Is 0 r r cr Ir Ir S Ir 0 s scs Is S 1 Is 0 r rcrirs r Ir 0 I R 0 s r s s s r r r S 0 S 0 1 s r R scs Is r cr Ir S 0 Te disease-free equilibriu points (DFE) are equilibriu-state solutions were tere is no disease (Plasodiu knowlesi alaria). Te diseased classes in te uan, osquito and acaque populations are equal to zero. Tus, te (DFE) of te basic odel (1) is gien by, s r s r s r E0 S, I, I, R, S, I, I, S, I, I, R,0,0,0,,0,0,,0,0,0, Tis represents te state were tere exist no infectiity in a counity and it is acknowledged as te disease-free equilibriu point (DFE). Te linear stability of te disease can be establised using te next generation operator etod (Van den Driessce & Watoug 2002) [45] on te odel equation (1), te atrix F and V, for te new infection ters and te reaining transfer ters, are respectiely gien by; (5)

11 Abdullai, Hasan and Abdulla 493 scs rcs d scs scs F rcr rcr scs rc r r V r It follows tat te drug sensitie basic reproduction nuber of te odel equation (1) denoted by. And gien by; s s s s s cs and te drug resistance basic reproduction nuber is; r r r r r r r c r r Furterore, using Teore 2 of (Van den Driessce & Watoug 2002) [45] te following result is establised. Teore Te DFE of te odel (1), gien by s 1, and unstable if 1. s s (6) (7) is locally asyptotically stable (LAS) if 3. SENSITIVITY ANALYSIS OF MODEL PARAMETERS Sensitiity of eac paraeter is exained wit respect to te basic reproduction nuber. In tis way, te paraeters tat are ore sensitie to te disease r

12 494 A Mateatical Analysis of te Effects of Control of Plasodiu transission are identified. And by eiter reducing or increasing suc paraeters will as well reduce or increase te transission of te disease. Sensitiity index of te basic reproduction nuber, r wit respect to eac paraeter is coputed as gien in Table 1 for te odel equation 1 Definition: Te noralized forward sensitiity index of a ariable wit respect to a paraeter is te ratio of te relatie cange in te ariable to te relatie cange in te paraeter. Wen te ariable is a differentiable function of te paraeter, te sensitiity index peraps is on te oter and defined using partial deriaties as: r r r r r r Sensitiity indices of r Tus, te Sensitiity of r to eac of te (14) different paraeters described in table 6.1 are deterined using drug resistance basic reproduction nuber of te basic odel (1) stated; r 2 2 r r r r r r c r r Te Sensitiity indices of r wit respect to, r,, cr for exaple are, 0 0 : 1, cr c : 1, r c r r 1 :, r r :. 0 2 Tese paraeters do not depend on any paraeter alues, wile te reaining paraeter depends of oter paraeters for teir alues. Te paraeters alues are gien on Table 3 (8)

13 Abdullai, Hasan and Abdulla 495 Table 3 Sensitiity Indices of Paraeter Value Sensitiity Index c r r r r r r Table 3 aboe entails tat an increase in te osquito deats as an optiistic ipact in controlling te disease. Te paraeters are prearranged starting fro te ost sensitie to te least. Te ost sensitie paraeters are c r is te contact rate of uan and acaques wit arteisinin resistant strain osquitoes and osquito deat rate. Tis is followed by deat rate of acaques and recruitent rate of osquitoes. Oter iportant paraeters include probability of infection of acaques wit arteisinin resistant strain osquitoes r, iigration rate of acaques and te least sensitie is te birt rate of acaques. Te sensitiity index of r wit respect to probability of osquitoes getting infected wit arteisinin resistant strain r is +0.5, tis iplies tat decreasing (or increasing) te r by 20%, decreases (or increases) r by 10% is te sae wit recruitent rate of osquitoes,. Since 0 1, increasing (or decreasing) cr by 20%, decreases (or increases) te 0 by 20%. Terefore, reducing te nuber of contacts between uan and acaques wit arteisinin resistant strain osquitoes, troug reducing te nuber of osquitoes, will reduce te contact and biting rate wit bot uan and acaques. Tus, te nuber of bits tat uan and acaques will receie will ae a large effect on disease transission. 4. OPTIMAL CONTROL MODEL DESCRIPTION In order to inestigate te optial control easures for te control of te disease, soe control paraeters are considered. Based on te sensitiity analysis result, te r c r

14 496 A Mateatical Analysis of te Effects of Control of Plasodiu following control strategies will be considered; quarantine, spray of insecticides and culling. Te quarantine will be used against uan iigrants and iported acaques, spray of insecticides against te osquitoes and culling against te acaques. Model equation (1) will be extended to include te tree control strategies stated aboe. Tus, te odified odel description is gien below; 1 s r u1 were s and r are te fractions of uans infected wit te parasite sensitie to arteisinin and u 0u 1 is te control efforts of Susceptible uans are recruited at a rate resistance arteisinin respectiely and 1 1 iigrants (quarantine). Susceptible uans get infected following contact wit infected u c I c I u 0u 1 is osquitoes wit bot strains at a rate. And te control on te use of insecticide spray. 1 2 s s s r r r Te osquitoes are recruited at a rate s r u2 were s and r are te fractions of osquitoes infected wit te parasite sensitie to arteisinin and resistance arteisinin respectiely and u2 0u2 1 is te use of insecticide spray. Susceptible osquitoes S get infected following effectie contacts wit eiter fro infected uans or infected acaques wit sensitie to arteisinin and resistance u c I I S u c I I S arteisinin te disease at a rate, respectiely. 1 2 s s s s Te acaques are recruited at a rate 1 2 r r r r 1 s r u1 were s and r are te fractions of acaques infected wit te parasite sensitie to arteisinin and resistance u 0u 1 is te control efforts of iigrants arteisinin respectiely and 1 1 (quarantine). Susceptible acaques get infected following contact wit infected u c I c I u 0u 1 osquitoes wit bot strains at a rate. And is te control on te use of culling. 1 3 s s s r r r 3 3 Terefore, putting te aboe forulations and assuptions togeter gies te following uan osquito - acaque odel, gien by syste of ordinary differential equations below as;

15 Abdullai, Hasan and Abdulla 497 ds 1 s r u1 R 1 u2 scs Is r cr Ir S S dis su1 1 u2scs Is S 0 Is dir ru1 1 u2r cr Ir S dis r Ir dr 1 d Is R ds 1 s r u2 1 u2scs Is Is S 1 u2 r cr Ir Ir S u2 S dis su2 1 u2scs Is Is S Is dir ru2 1 u2r cr Ir Ir S Ir ds 1 u1 R 1 u3 c I c I S dis su1 1u3 scs Is S 1 Is dir ru1 1 u3r cr Ir S r Ir dr I R s r s s s r r r In inestigating te optial control efforts tat would be needed to control te disease, it is necessary to consider an optial control proble wit te objectie (cost) functional gien by; in T u 0 1, u2, u r r 3 J A I A I A u A u A u (10) were Ai, i 1,...,5 are positie weigts. A quadratic cost on te controls is cosen tat is siilar to oters in literature (see, for instance, Okosun & Makinde 2013; Okosun et al., J u, u, u ; tis is aied to iniize 2013) [42, 43]. Wit te gien objectie function te nuber of infected uans wit resistance arteisinin strain to acaques wit resistance arteisinin strain I r (9) I r and infected, wile iniizing te cost of controls u1 ( t), u2( t)and u3( t ). Terefore, an optial control u 1, u 2, u 3 is obtained suc tat;, J u u, u in J u, u, u u, u, u Y (11) were te control set; u1, u , u Y u, u, u u :[0, T] [0,1], Lebesgue easurable i 1, 2,3. (12) i

16 498 A Mateatical Analysis of te Effects of Control of Plasodiu 4.1 Pontryagin s Maxiu Principle Te necessary conditions tat an optial control ust satisfy coe fro te Pontryagin s Maxiu Principle (Pontryagin et al., 1964) [46]. Te principle conerts (9) (10) into a proble of iniizing point wise a Hailtonian H, wit respect to u1, u2, u 3. Using te optial control teory, let i t, be adjoint ariables wit i 1,...,11. Te Hailtonian for te present optial control proble is gien by; H A1 Ir A2 Ir A3 u1 A4 u2 A5 u su1 u3 c I S 10 u I 1 s r u1 R 1 u2 scs Is r cr Ir S S su 1 u scs Is S 0 Is 1 1 r d s 1 d I R ru u r cr I S I r Ir s 1 s r u2 1 u2 scs Is Is r cr Ir Ir 2 su 1 u scs Is Is S u2 Is ru 1 u r cr Ir Ir S u2 Ir 1 s r u1 R S S u S 1 u3 scs Is r cr Ir S 1 s s s 1 1 u c I S r I R s r r r r r I (13) Teore 1 For te optial control triple u1, u2, u3 tere exists adjoint ariables for i 1,...,11 satisfying i tat iniizes,, J u u u oer, ten 1 2 3

17 Abdullai, Hasan and Abdulla Adjoint Syste d1 1 1 u2 scs Is r cr Ir 2 1 u2 scs Is 3r cr Ir d1 d 1 d 1 u c S c S d3 A1 cr S 5 7 u2 1 r 10 r 3 r d4 1 4 d 5 5 1u2 scs Is Is 1 u2r cr Ir Ir u2 6scs Is Is 7 1 u2 r cr Ir I r d6 1 1 u2 scs S 2 1 u2 scs S u3 scs S 9 1 u3 scs S d7 1 1 u2 r cr S 3r cr S u3 r cr S 10 1 u3 r cs r d8 8 1 u3 scs Is r cr Ir 9 1 u3 scs Is 10 1 u3 r cr Ir d9 9 1 s S 1 u2 5 6 cs 11 d 10 A2 5 1 u2 r cr S 7 1 u2 r cr S d s s 6 s s As te optial control just as initial conditions it is necessary to find te transersality conditions, tat corresponds to a terinal condition in te adjoint equation, 4.3 Transersality Conditions 0, for i 1,...,11 (15) i 4.4 Stationary Values Te control u1, u 2 and u 3 satisfy te optiality condition (14)

18 500 A Mateatical Analysis of te Effects of Control of Plasodiu 1s r 8 s r u1 ax 0, in 1, 2A 3 1 1, r Ir Ir cr 1 cs Is Is s S 2A 4 s r 5 u2 ax 0, in crr 7 Ir Ir S1r cr Ir S s Is S 1 2 cs 7 Ir 6 Is S cr Ir 8 10 r scs Is 8 9 u3 ax 0, in 1, 2A5 (16) Proof: Te goerning equations of te adjoints ariables are soled ia differentiation of te Hailtonian function, estiated at te optial control. Ten te adjoint syste can be written as;

19 Abdullai, Hasan and Abdulla 501 d1 H 1 u c I c I 1 u c I c I S d1 H 2 0 3d 4 1 d 5 1 u2scs S 6scs S Is d3 H A1 cr S 5 7 u2 1 r 10 r 3 r Ir d4 H 1 4 R d5 H 5 1 u2 scs Is Is 1 u2 r cr Ir Ir u 2 S 6scs Is Is 7 1 u2 r cr Ir Ir d6 H 1 1 u2 scs S 2 1 u2 scs S 6 2 I 1 2 s s s r r r 2 2 s s s 3 r r r s 8 1 u3 scs S 9 1 u3 scs S d7 H 11 u2r cr S 3r cr S 7 2 I r 1 u c S 1 u c S d8 H 8 1 u3 scs Is r cr Ir S 8 3 r r 10 3 r r 9 1 u3 scs Is 10 1 u3 r cr Ir d9 H 1 S 1 u c I 9 s s 11 s d10 H A2 5 1 u2 r cr S 7 1 u2r cr S Ir d11 H 8 11 R and wit transersality conditions 0, for i 1,...,11 i On te interior of te control set, were 0 1, for i 1,2,3 yields H 0 2Au s r 8 s r u 1 H 0 2A4u 2 r Ir Ir cr 1 cs Is Is s S s r 5 u2 c I I S c I S I S c I I H 0 2A5 u3 8 scsis rcr Ir S 9scsIs S 10rcr Ir S u3 i r r 7 r r 1 r r r s s 1 2 s 7 r 6 s (17) (18)

20 502 A Mateatical Analysis of te Effects of Control of Plasodiu Tis in anoter clear for becoes d1 1 u c I c I 1 u c I c I d d 4 1 d 5 1 u2scs S 6scs S d3 A1 cr S 5 7 u2 1 r 10 r 3 r d4 1 4 d5 5 1u2 scs Is Is 1 u2r cr Ir Ir u2 6scs Is Is 7 1 u2 r cr Ir Ir d6 1 1 u2 scs S 2 1 u2 scs S u3 scs S 9 1 u3 scs S d7 1 1 u2 r cr S 3r cr S u3 r cr S 10 1 u3 r cs r d8 8 1 u3 scs Is r cr Ir 9 1 u3 scs Is 10 1 u3 r cr Ir d9 9 1 s S 1 u2 5 6 cs 11 d10 A2 5 1 u2 r cr S 7 1 u2 r cr S d Tus, it is obtained tat, 1 2 s s s r r r 2 2 s s s 3 r r r 1 s r 8 s r u1 2A3 r Ir Ir cr 1 cs Is Is s S s r 1 u 2 crr 7 Ir Ir S1r cr Ir S 2A 4 s Is S 1 2 cs 7 Ir 6 Is S cr Ir 8 10 r scs Is 8 9 u3 2A5 5

21 Abdullai, Hasan and Abdulla 503 By standard control arguents inoling te bounds on te controls, it can be concluded tat 0 if if 1 u u if if if if 2 0 if 3 0 u3 3 if if 3 1 were 1s r 8 s r 1 2A3 r Ir Ir cr 1 cs Is Is s S s r cr r 7 Ir Ir S 1 r cr Ir S (20) 2A 4 s Is S 1 2 cs 7 Ir 6 I s S cr Ir 8 10 r scs Is A (19) u in 1,, u in 1,, u in 1,. (21) Subsequently, te nuerical solutions of te optiality syste and te corresponding optial control pair, te paraeter coices, and te interpretations fro arious cases are discussed. 5. NUMERICAL ANALYSIS In tis section, te effect of te optial control strategies on te transission of disease is inestigated nuerically. Using te iteratie etod, te optiality syste, consisting of 11 ordinary differential equations fro te state and adjoint equations, coupled wit te tree control caracterizations is soled. Te state differential equations, wit initial estiates for controls and te state are soled using fourt order Runge-Kutta scee. Using te result of state and te gien final tie alues, te adjoint syste is ten soled backward in tie, using fourt order Runge-Kutta scee. Te state and te adjoints syste are used to update te tree control strategies using te caracterizations

22 504 A Mateatical Analysis of te Effects of Control of Plasodiu gien by (16). Te process is repeated and te iteratie process coplete wen te current state, adjoint, and control alues conerge sufficiently (Lenart & Workan 2007) [47]. Ten, te effect of te following optial control strategies on te spread of te disease in a population is inestigates nuerically. Using two controls at a tie wile setting te oter to zero and finally considering te tree controls at sae tie. Strategy A: cobination of use of quarantine and insecticide spray. Strategy B: cobination of use of insecticide spray and culling. Strategy C: cobination of use of quarantine and culling. Strategy D: cobination of use of insecticide spray, quarantine and culling. For te nuerical siulation te following weigt factors are used Ai 5, i 1,...,5 and use te paraeter alues fro table 6.2. Initial states ariables are cosen as S (0) 100, I (0) I (0) 0, R (0) 0, S (0) 1000, I (0) I (0) 0, S (0) 100, s r s r Is (0) Ir (0) 0, R (0) 0. Oter paraeter alues are in table 6.2 to illustrate te effect of different optial. Table 1 Description of Variables Var. Description S Susceptible uan I s Infected uan wit parasite sensitie to arteisinin Ir Huan infected wit parasite tat is arteisinin resistant strain R Recoered/iune uan S Susceptible osquito I s Infected osquito wit parasite sensitie to arteisinin Ir Infected osquito wit parasite tat is arteisinin resistant strain S Susceptible acaque Is Infected acaque wit parasite sensitie to arteisinin I Infected acaque wit parasite tat is arteisinin resistant strain R Recoered/iune acaque

23 Abdullai, Hasan and Abdulla 505 Table 2 Description of Paraeters of Model (1) Par. Est. Value Ref Augusto et al., (2012) [38] Ciyaka et al., (2008) [48] Augusto et al., (2012) [38] 0.2 Assued Assued Niger & Guel, (2008) [49] 0.05 Okosun &Makinde (2012) [32] r 0.05 Hoe-Musekwa, (2008) [50] Assued 0.02 Assued r 0.06 Assued Menac et al., (2005) [51] 0.03 Menac et al., (2005) [51] s 0.17 Ciyaka et al., (2008) [48] r c s 0.6 Citnis et al., (2006) [52] d 0.02 Okosun &Makinde (2012) [32] 0.09 Menac et al., (2005) [51] s 0.83 Ciyaka et al., (2008) [48] r 0.03 Assued s 0.03 Assued r c Assued r 0.5 Assued 0.8 Augusto et al., (2012) [38] 0.04 Ciyaka et al., (2008) [48] Assued Assued s Assued r Assued s Assued r Assued s Assued r

24 Infected Huans wit arteisinin resistant strain 506 A Mateatical Analysis of te Effects of Control of Plasodiu 5.1 Optial Use of te Quarantine ( u 1) and Insecticide Spray ( u 2) Wit tis control strategy, quarantine ( u1 ) and insecticide spray ( u2) are bot used to optiize te objectie functional J, wile te control culling ( u3) is set to zero. In figure 1, te result sows a significant difference in te Ir, Ir and I wit optial control strategy copared to Ir, Ir and I witout control. It was obsered in Figure 1(a) tat te infected uans wit resistance arteisinin strain ( I r ) decrease as a result of control strategies against te increase in te uncontrolled case. In Figures 1(b) and 1(c), siilar situation was also obsered in te case of infected acaques wit resistance arteisinin strain and infected osquitoes. 30 Witout control Optial control Strategy A Tie (days) Figure 1(a)

25 Infected Mosquitoes Infected Macaques wit arteisinin resistant strain Abdullai, Hasan and Abdulla Witout control Optial control Strategy A Tie (days) Figure 1(b) 2.5 Witout control Optial control Strategy A Tie (days) Figure 1(c) Fig. 1: Siulations of te Plasodiu Knowlesi Malaria Model Sowing Effect of Optial Use of Quarantine and Insecticide Spray on te Spread of te Disease 5.2 Optial use of te Insecticide Spray ( u 2) and Culling ( u3) In tis control strategy, use of insecticide spray ( u 2) and culling ( u3) are bot used to optiize te objectie functional J, wile te control biological control ( u2) is set to zero. In figure 2, te result sows a significant difference in te I r and I wit optial

26 Infected Mosquitoes Infected Macaques wit arteisinin resistant strain 508 A Mateatical Analysis of te Effects of Control of Plasodiu control strategy copared to I r and I witout control. It was obsered in Figure 2(a) tat te infected acaques wit resistance arteisinin strain ( I r ) decrease as a result of control strategies against te increase in te uncontrolled case. In Figure 2(b) siilar situation been also obsered in te case of infected osquitoes. 1.2 Witout control Optial control Strategy B Tie (days) Figure 2(a) 2.5 Witout control Optial control Strategy B Tie (days) Figure 2 (b) Fig. 2: Siulations of te Plasodiu Knowlesi Malaria Model Sowing Effect of Optial Use of Insecticide Spray and Culling on te Spread of te Disease

27 Infected Macaques wit arteisinin resistant strain Infected Huans wit arteisinin resistant strain Abdullai, Hasan and Abdulla Optial use of te Quarantine ( u 1) and Culling ( u3) In tis control strategy, quarantine ( u1 ) and culling ( u3) are bot used to optiize te objectie functional J, wile te control insecticide spray ( u2) is set to zero. In figure 3, te result sows a significant difference in te I r and Ir wit optial control strategy copared to I r and Ir witout control. It was obsered in Figure 3(a) infected uans wit resistance arteisinin strain ( I r ) decrease as a result of control strategies against te increase in te uncontrolled case. In Figure 3(b) siilar situations been also obsered in te case of infected acaques wit resistance arteisinin strain. 30 Witout control Optial control Strategy C Tie (days) Figure 3(a) 1.2 Witout control Optial control Strategy C Tie (days) Figure 3(b) Fig. 3: Siulations of te Plasodiu Knowlesi Malaria Model Sowing Effect of Optial Use of Quarantine and Culling on te Spread of te Disease

28 Infected Macaques wit arteisinin resistant strain Infected Huans wit arteisinin resistant strain 510 A Mateatical Analysis of te Effects of Control of Plasodiu 5.4 Optial Use of te Quarantine ( u 1), Insecticide spray ( u 2) and Culling ( u3) Here, all te control strategies ( u1, u2, u 3) are used to optiize te objectie functional J. In figure 4, te result sows a significant difference in te Ir, I and I r wit optial control strategy copared to Ir, I and Ir witout control. It was obsered in figure 4(a) tat te infected uans wit resistance arteisinin strain ( I r ) decrease as a result of control strategies against te increase in te uncontrolled case. In Figures 4(b) and 4(c) siilar situations been also obsered in te case of infected acaques wit resistance arteisinin strain and te infected osquitoes. 30 Witout control Optial control Strategy D Tie (days) Figure 4(a) 1.2 Witout control Optial control Strategy D Tie (days) Figure 4(b)

29 Infected Mosquitoes Abdullai, Hasan and Abdulla Witout control Optial control Strategy D Tie (days) Figure 4(c) Fig. 4: Siulations of te Plasodiu Knowlesi Malaria Model Sowing Effect of Optial Use of Insecticide Spray, Quarantine and Culling on te Spread of te Disease CONCLUSIONS In tis paper, a ateatical odel for te spread of Plasodiu knowlesi alaria is deeloped. Te odel incorporates recruitent rate of te tree populations troug a constant iigration, wit a fraction of infectie iigrants. Te effect of drug resistance is also considered. Te ipact of a control easure, culling against larae and acaques is considered. Te sensitiity index of te odel is inestigated to understand te iportance of eac paraeter to te disease transission. Te condition for optial Plasodiu knowlesi alaria were deried and analyzed wit tie dependent preentie. Te optial control as a ery desirable effect for reducing te Plasodiu knowlesi alaria. Howeer, based on te results of te analysis, tree control strategies were considered, insecticide spray, quarantine and culling. Te nuerical siulation results ae sown tat te best control strategies for control of te disease cobination of te tree control strategies (Strategy D). Howeer, te iplication of using all te controls is tat additional cost will be incurred. Tis is because strategy D and insecticide spray and quarantine (strategy A) as sae effect on tis control of te disease. It can be concluded tat to control te disease, te ost cost-effectie of all te strategies is te use of insecticide spray and quarantine (strategy A) in te presence of infectie iigrations and drug resistance. Public ealt establisents ougt to coose te appropriate control strategy were teir situation lies in te scenarios discussed in te result. ACKNOWLEDGMENT Te present work is supported by sort ter grant (PMATHS/ ) fro RCMO, Uniersiti Sains Malaysia. Te autors would like to express teir sincere tanks to Scool of Mateatical Sciences for proiding adequate facilities for te researc.

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