Mathematical Model for the Transmission of P. Falciparum and P. Vivax Malaria along the Thai-Myanmar Border

Transcription

1 Mathematical Moel for the ransmission of P. alciparum an P. Viax Malaria along the hai-myanmar Borer Puntani Pongsumpun. an I-Ming ang Abstract he most Malaria cases are occur along hai-mynmar borer. Mathematical moel for the transmission of Plasmoium falciparum an Plasmoium iax malaria in a mixe population of hais an migrant Burmese liing along the hai-myanmar Borer is stuie. he population is separate into two groups hai an Burmese. Each population is iie into susceptible infecte ormant an recoere subclasses. he loss of immunity by iniiuals in the infecte class causes them to moe back into the susceptible class. he person who is infecte with Plasmoium iax an is a member of the ormant class can relapse back into the infecte class. A stanar ynamical metho is use to analyze the behaiors of the moel. wo stable equilibrium states a isease-free state an an epiemic state are foun to be possible in each population. A isease-free equilibrium state in the hai population occurs when there are no infecte Burmese entering the community. When infecte Burmese enter the hai community an epiemic state can occur. It is foun that the isease-free state is stable when the threshol number is less than one. he epiemic state is stable when a secon threshol number is greater than one. umerical simulations are use to confirm the results of our moel. Keywors Basic reprouction number Burmese local stability Plasmoium Viax malaria. I. IRODUCIO ALARIA occurs throughout the tropical an subtropical Mregions of the worl. his isease is a mosquito-borne isease cause by the protozoan parasites of the genus Plasmoium. Malaria is ue to four species P. falciparum P. iax P. malariae an P. oale. he two most common malaria infections are cause by the first two: P. falciparum which causes 90% of the malaria in Africa an is the cause of oer 2-3 million (mostly chil cases in the worl (mainly Africa [1]; an P. iax which is the cause of 50% of the malaria outsie of Africa. Less than two percent of the infections are ue to mixe infection by P. iax an P. falciparum together. P. iax an P. oale iffer from the other species [234] in that at the sporizoite stage an after they moe to the lier some of them are transforme into P. Pongsumpun is with the Department of Mathematics an Computer Science aculty of Science King Mongkut s Institute of echnology Lakrabang Chalongkrung roa Lakrabang Bangkok hailan (corresponing author phone: ext. 6196; fax: ext.284; kppuntan@kmitl.ac.th. I. M. ang is with the Department of Physics aculty of Science Mahiol Uniersity Rama 6 roa Bangkok hailan. hypnozoites. Most of these are then transforme into merozoites which inae the re bloo cells where they cause the illness. he remaining hypnozoites lie ormant in the lier for arying lengths of time (up to 3 years. he relapse of malaria occurs when some of these hypnozoites are transforme into schizents an then into merozoites. hey can reinae the bloo stream an cause the illness to recur. Between the relapses of the illness only small number of the merozoites remains in the bloo. P. iax an P. oale selom cause the eath of the human host. Due to the ifferences in economic conitions between hailan an Myanmar temporary migration of Burmese into hailan occurs eery year. More than 60% of the Burmese in some groups (in Mae Sot an Bo Basi two proinces in hailan along the borer are infecte with mefloquineresistant malaria[5]. hese economic migrations from neighboring countries into hailan hae cause problems for the malaria control program in hailan [6]. Especially troubling is multi-rug resistance malaria the presence of which is now seen in the high transmission areas aroun the market centers along the migratory routes. he first cases of malaria resistance were foun along the hai-kampuchean borer another borer where the economic conitions on the two sies are again quite ifferent. It is beliee that the areas where the parasites hae the highest rug resistance are along this latter borer. he meical recors for malaria in hailan [7] inicate that most of the malaria infections in hailan are ue to P. falciparum an P. iax. he most foreigner cases are Burmese. he ata also show that a small number of people are infecte with P. malariae an a small number hae been infecte by both P. falciparum an P. iax. here is no report of an infection ue to P. oale. o reuce the outbreak of Malaria in hailan a new mathematical moel shoul be introuce to anticipate what the response woul be to a plan of action when there are two ifferent forms of malaria in co-circulation in a population. It was long assume that strategies for hanling P. iax coul be extrapolate from those use against P. falciparum. his assumption was challenge at a conference conene by the Multilateral Initiatie on Malaria [8]. he transmission of malaria is usually escribe by the Ross-MacDonal (RM moel [9]. Howeer the RM moel is only suitable for the transmission of P. falciparum malaria since it oes not aress possible relapses of the illness. One of the present authors (IM has introuce a simple mathematical moel [10] to 439

2 escribe the transmission of P. iax malaria. In that moel a ormant class was inclue in which there are no merozoites in the bloo only ormant hypnozoites in the lier. A person becomes ill when the hypnozoites are re-actiate. He oes not hae to be bitten by an infecte mosquito again. In this stuy we formulate a moel in which ifferent mathematical moels are use to escribe the separate transmissions between P. falciparum an P. iax. Ethically there is no place for human experimentation to see what woul happen if new therapies were aopte. Mathematical moeling allows one to simulate what coul occur. Since we are intereste in applying the moel to the situation along the hai-myanmar borer (an to a lesser extent the hai- Kampuchean borer we hae allowe the rates of infections to iffer if the infecting malaria is P. falciparum (enote by f or P. iax (enote by or if the person is a hai (enote by or a Burmese (enote by B. In section 2 we introuce a moification of the moel that woul make it applicable to the transmission of P. falciparum an P. iax between hai an Burmese. In Section 3 we analyze our moel to fin the conitions for the local stability of each equilibrium point. umerical simulations are shown to confirm the local stability of the enemic equilibrium point. II. RASMISSIO MODEL In 1911 Ross formulate the mathematical moel of the epiemiology of malaria (P. falciparum [11] an improe by MacDonal [12]. In the Ross moel an iniiual in the human population is classifie as being in a non-infecte or infecte state. MacDonal propose that the human population shoul instea be iie into three states - noninfecte infecte but without any acute clinical signs infecte with acute clinical signs - to reflect the clinical status of the iniiual better. Others beliee that the population shoul be iie into susceptible infecte but not infectious an infectious. In our moel we consier the transmission cycle between humans in the two populations an in the ector populations. Both human populations (hai an Burmese are iie into susceptible infecte ormant an recoere subclasses. he ector population is separate into susceptible an infecte subclasses [13]. We let S is the number of susceptible hai humans SB is the number of susceptible Burmese humans I is the number of infecte hai humans IB is the number of infecte Burmese humans D is the number of ormant hai humans D B is the number of ormant Burmese humans R is the number of recoere hai humans R B is the number of recoere Burmese humans S is the number of susceptible ectors I is the number of infecte ectors An infectious human can recoer an re-enter the susceptible class. Only the recoere humans who were infecte with P. iax are susceptible to further infections. Howeer an infecte mosquito cannot recoer. We efine as the number of hais entering the susceptible class through birth an r5 I an r I 5 as respectiely the numbers of infecte hais who were infecte with P. falciparum or P. iax malaria but hae recoere. he rate at which susceptible hais are lost by becoming infecte with P. falciparum is h I S an by becoming infecte with P. iax is h I S. A susceptible hai will be infecte by the P. falciparum (P. iax parasite if bitten by a mosquito carrying the particular parasite. o take this into account the infection rates r h an r h shoul be proportional to the fraction of the infecte mosquitoes with the particular type of parasite Aitional increases in the number of people infecte with P. iax malaria occur when the members of the ormant class relapse. he rate of change of the number of susceptible members is equal to the number entering minus the number leaing. his gies us the following ifferential equation for the rate of change of the susceptible hai human population: S r I -hs -( I S h h (r r I r D r R 1 3 I ( I S - (r r I h h 1 - I - (r r I r D h (2 D r I - (r r D h (3 R (r r I - (r R h (4 SB (1PB r1 I -( S B h B -( I S (r r I r D r R B 1 B 3 B 4 B h B h B 1 (5 IB PB ( I SB - (r r Î B hb hb 1 - ( I - (r r I r D (6 h B B B (1 D r I - (r r D B B h B (7 2 3 R (r r I - ( r R B 5 5 B 4 h B (8 where the parameters are efine as follows. h is the eath rate in the human population 440

3 is the percentage of infecte humans in whom some hypnozoites remain ormant in the lier r is the rate at which a person infecte with P. falciparum 1 leaes the infecte class r is the rate at which a person infecte with P. iax leaes 1 the infecte class r is the rate at which the ormant human relapses back to 2 the human infecte by P. iax r is the recoery rate of the ormant human ue to P. 3 iax r is the rate at which the human recoere after P. iax 4 infection relapses back to the susceptible human r is the rate at which the human infecte by P. falciparum 5 recoers r is the rate at which the human infecte by P. iax 5 recoers is the rate at which Burmese moe out the country P is the percentage of Burmese who are infectious when they enter the community B is the constant recruitment rate of Burmese. We assume that P. falciparum an P. iax infections are non-lethal so the eath rates will be the same for all human classes an we will hae S I D R an B SB IB D B R B. he ynamics of the mosquito populations are gien by S A S (( I ( B IB S B I (( I ( B B IB S I (10 At equilibrium the total number of female mosquitoes will be A/. A is the rate at which the mosquitoes are recruite an is the eath rate of the mosquitoes. B an B are the rates at which the mosquitoes become infecte with the parasites (P. falciparum ( an P. iax (V once the mosquito has bitten an infecte human (hai ( an Burmese (B. We also assume S I. he working equations of the moel are obtaine by iiing (1 (2 (3 an (4 by (5 (6 (7 an (8 by B an (9 an (10 by A/. his woul gie us ten equations expresse in terms of the renormalize ariables: S I S D I R D R (9 S B I SB B D IB D B B (B/( h (B/( h (B/( h R R B S B I S I (B/( h (A/ (A/ B A where B. h Conitions S I D R 1 SB IB D B R B 1 an S I 1 lea to only seen of these being inepenent. We choose the seen inepenent equations to be S h(1 S r I-( I S h h (r r1 I r3 D r4 (1 (S I D 1 I ( IS - (r r I - I h h 1 h - (r5 r5 I r2 D D r I - (r r hd 2 3 SB (1 P( h r1 I -( S B h B -( h IS B (r r1 IB h B B 1 r D B r (1 (S B IB D B 3 4 IB P( h ( I S hb h B B -(r r1 IB-( hib 1 -(r5 r5 IB r2 DB DB r1 IB - (r2 r3 h DB an (11 (12 (13 (14 (15 (16 I (( I ( IB(1- I B B (17 I where the new transmission rates are 441

4 I B h ( A / ( / h h A h V V h ( A / h ( A / h B h B VB V B B B B B B B h h (18 he omain of solutions is {(0 S I D R 10 SB IB DB RB 1 0 S I 1} (19 At this point we shoul mention that (14 an (15 contain an explicit epenence on P the percentage of Burmese entering hailan who are infecte with the malaria parasite. hese are the people who will be responsible for malaria epiemics along the hai-myanmar borer. III. AALYICAL RESULS o fin the equilibrium points we set the RHSs of (11 to (17 to zero. his yiels the equilibrium state (S I D S B I B D B I where D r3 I (r1 f (1 r1 r4 r 4 (1 D h S r ( 4 h h I h (20 I r D r r ( h ( h (1 P D B r3 I B (r1 f r1 (1 r4 r4 (1 D B S B ( (h h I h r4 B B (22 I B r D B r r ( h I (h h I (h r 4 (h h I r4 h (h h I ((1 r r4 r 1 r r h 1 (r5f r 5 (h h I r4 h r r (h h I r1 (r3 r4 2 r r 2 3 h ((h h I h r4 (r2 r3 h (24 (h h I (( 1P( h r B B 4 ( hp (h h I h r4 B B (h h I (r1 (1-r r4 B B h r1 r (r5f r5 (h h I h r 4 ( I r (r r r r h h B B (r r h 2 3 (( h h I h r4 ( h r2 r3 B B an I being the solutions of (25 I (1-I ((( ( I ( r ((( I h h h 4 h h ( h h I (r 1 (1r r 4 hr 4( hr1r r 5r5 (h h I h r4 rr (h 2 h I r1 (r3 r 4 h r2 r 3 ( h h I h r4 h r2 r3 ( h B h I ( h ( h P r B 4 B B (( h P ( I hb hb h r4 ( I 1 4 (r r (1 r hb hb h r1 r ( I hb hb h r4 r r r r h r2 r3 (26 ( I r (r r hb hb 3 4 ( ( I h r4 ( h r2 r3 hb hb he solution to (26 will be physically meaningless if it is negatie since the normalize infectious mosquito population must be a non-negatie real number. So we nee to fin all possible conitions for I to be real an positie. P is in the range [0 1]. We consier two cases: P = 0 an 0 < P 1. or P = 0 (26 becomes I (1-I (((h h ( ( hr 4 ( (( h h I h r 4 ( h h I (r 1 (1r r 4 (hr1r ( h h I h r4 rr r 2 5 r5 h r2 r3 442

5 ( h h I r (r 3 r 4 ( h h I h r 4 h r 2 r 3 ( h h ( h r4 ( B B B B ( ( h B h I B h r4 ( h r1 r1 ( I (r1 r (1 r hb hb 4 ( h B h I B h r4 r r r 2 5 r5 h r2 r3 ( h I r (r r B hb 3 4 (27 ( ( I h r4 ( h r2 r3 hb hb One of the solutions of (27 is I = 0. he other solutions are the solutions of a quaratic equation. he numerical alues of these two solutions will epen on the numerical alues of the parameters in the moel. hese are often unknown. Using stanar ynamical analysis (base on the Hopf Bifurcation heory [14] we can establish the conitions for the stability of the isease-free state. We fin the conition is R 0 < 1 where R 0 = R + R B (28 with ( ( ( r r h h h 2 3 R 2 ( (r1 r5 r5 (r2 r3 r1 (r2 (1 r3 h h (r1 r1 r2 r3 r5 r5 an (h h ( ( r r h 2 3 B B B R B B 2 2 ( (r1 r5 r5 (r2 r3 r1 (r2 (1 r3 h ( h (r1 r1 r2 r3 r5 r5 (29 Determining whether the numerical alues of the parameters satisfy (28 is not of irect concern to us in this paper. he important thing to remember is that the iseasefree state is one of the equilibrium states. his means that in the absence of any infectious Burmese entering hailan malaria will not become epiemic in hailan proie that the alues of the parameters lea to the conitions gien by (28. or 0 < P 1 the equilibrium state will not be the iseasefree state since the ifference between (26 an (27 is the term ( ( h P in (26. If we substitute I B B = 0 into (26 all the terms except ( ( h P woul anish leaing only that B B term present. Since the term is non zero I = 0 can not be a solution to Eqn. (26.. or this case the equilibrium state will be the epiemic state E 1 (S I D S B I B D B I. It remains to be etermine if this state is stable. Performing an analysis similar to the one use to establish the conitions uner which the isease-free state is stable we fin that the epiemic state will be stable if R E 1 (30 R E where R 1 E R E 2 with R E ( ( h h i h r4 1 B B ( ( B i B B ( i ( h h i h r1 r 1 r2 r3 r4 r5 r5 R E (1 r2 ( h h h h i 2 B B (r1 (1 r 1 r 4 ( h h B B ( B (1- i B s B ( h h ( s (31 he numerical alues of the equilibrium epiemic will again epen on the numerical alues of the parameters. Stability analysis of the eigenalues of ynamical systems will place limits on the alues of the parameters that woul lea the epiemic state to be stable. Again what these alues are is of no irect concern in this paper. What is known is that the equilibrium state will not be the isease-free state but will instea be the epiemic state. Without infectious Burmese entering the community there will be no infecte population proie the numerical alues of the parameters in the moel are such that the conitions gien by (28 are satisfie. IV. UMERICAL RESULS In this section we present the results of our numerical simulations for the case of P = 0 in ig. 1a. he alues of the parameters are taken from real life obserations. We hae set h per ay which correspons to the real life expectancy of 70 years for human an 1/ 30 which correspons to the life expectancy of 30 ays for the Anopheline mosquito. he alues r =1/20 per ay 1 r 1 =1/14 per ay correspon to the time it takes people who are infecte with P. falciparum an P. iax to leae the infecte class an become susceptible again i.e. 20 ays for P. falciparum an 14 ays for P. iax. he alues r =1/365 per ay r =1/(2365 per ay correspon to 2 3 the time it takes people who are infecte with P. iax to leae the ormant class i.e. 1 year to enter the infecte class an 2 443

6 years to enter the susceptible class. he alue r 4 =1/(3365 per ay correspons to 3 years for the people who are infecte with P. iax to relapse. he alues r 5 =1/30 per ay r =1/25 per ay correspon to the time it takes 5 people who are infecte with P. falciparum an P. iax to recoer i.e. 30 ays for P. falciparum an 25 ays for P. iax. 1 / is the aerage time a Burmese stays in hailan an we take this to be per ay. o hae the isease-free state as the stable equilibrium state we set P = 0. o hae the stable equilibrium state as the epiemic state we set P = 0.6 [5]. he transmission rates h h h h B B are arbitrarily chosen. 1 a 1b ig. 1 1a ime series of S I D SB I B DB an i. he parameters for the transmission rate are as follows: h h hb hb B B he other parameters are gien in the text an R 0 = b he solution trajectories of our moel. he parameters are similar to fig.1a. As we see in ig.1a the seen populations go to ( as t meaning that the equilibrium state is the isease-free state. he numerical alues of the parameters lea to a threshol number R 0 = 0.9. he trajectories of the solutions in the 2D: D I plane D B I plane IB D plane an DB IB plane are shown in ig. 1b. he arrows in these planes show the irections of the trajectories as t which are towars the isease-free state. he numerical simulation is therefore in agreement with the behaior preicte when R 0 < 1. We now change the alues of the parameters an set P = 0.6. he alues are gien in the caption of ig.2. hese alues gie R E 98. his is the conition for the epiemic state E 1 (S I D S B I B D B I to be the stable equilibrium state. his is inee seen in ig. 2a. he trajectories of the solutions in the 2D: D I plane DB I plane IB D plane an DB IB plane are shown in ig. 2b. As t the trajectories ten to the limiting alues inicate on ig.2a. 2a 2b ig. 2 2a ime series of S I D SB I B DB an I. he parameters for the transmission rate are as follows: h h hb hb B B he other parameters are gien in the text an R E

7 2b he solution trajectories of our moel. he parameters are similar to fig.2a. before they return to Myanmar. he time eolutions of the three populations shown in ig. 2a are those when the Burmese stay a long time. he present behaiors are for the case when the Burmese stay 1/5 ay 50 ays an 5000 ays. ig. 3 shows that a higher number of hais will be infecte if the Burmese stay in hailan for shorter perios. If the Burmese stay for longer perios the number of hais who are infectious at a gien time will be lower. he reason for this is that initially the Burmese hae a higher incience rate of actie malaria infection. hey woul be able to pass the illness to the hais at the beginning. If they stay longer they woul eelop the same incience rate as the hais an are less likely to pass on the malaria. ig. 3.ime series of I I B an I for the ifferent alues of. he alues of the other parameters are similar to fig. 2. In ig. 3 we plot the time eolution of the three infecte population ( I I B an I for ifferent alues of the reciprocal of the time the Burmese stay in hailan V. DISCUSSIO In this stuy we hae analyze a mathematical moel of malaria that coul escribe the situation along the hai- Myanmar borer. Along this borer there are two major types of malaria in circulation P.falciparum an P. iax. here is a migration of Burmese into hailan. We fin that there are two equilibrium states a isease-free state an an epiemic state. We establish the threshol conitions neee for each of the equilibrium states to exist. he numerical results confirm our analytical results (see ig. 1 an 2. When R 0 is less than one the normalize iniiual populations ten to the iseasefree state. he normalize iniiual population tens to the epiemic state when RE is greater than one. ACKOWLEDGME his work is supporte by the Commission on Higher Eucation an the hailan Research un accoring to contract number MRG REERECES [1] WHO: Worl Malaria Situation in 1994: Weekly Epiemiological Recor. Genea 1997 [2] M. Mayxay S. Pukrittayakamee P.. ewton an. J. White Mixe-Species malaria infections in humans rens in Parasitology ol. 20 pp [3] PC. C. Garnhan Malaria parasites of man: life-cycles an morphology (excluing unltrastructure. In Malaria eite by Wernsorfer WH McGregor I. Einburgh: Churchill Liingstone; [4] R.. Price. Emiliana C. A. Guerra S. Yeung. J. White an. M. Anstey Viax Malaria: eglecte an ot Benign Am J rop Me Hyg 2007 ol. 77 pp [5] C. Wongsrichanalai J. Sirichaisinthop J. J. Karwacki K. Congpuong S. R. Miller L. P. himasarn an K. himasarn Drug Resistant Malaria on the hai-myanmar an hai-camboian Borers. SEA J rop Me Pub Health 2001 ol. 32 pp [6] S. Pinichpongse he Current Situation of the Anti-Malaria Programme in hailan. Preceeing of the Asia an Pacific Conference on Malaria Honolulu Hawaii 1985 pp [7] Annual Epiemiological Sureillance Report. Diision of Epiemiology Ministry of Public Health Royal hai Goernment; [8] B. Sina ocus on Plasmoium Viax. rens in Parasitology 2002 ol. 18 pp [9] R. M. Anerson an R. M. May Infectious Disease of Humans Dynamics an Control. Oxfor: Oxfor Uniersity Press;

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