Host-vector interaction in dengue: a simple mathematical model

Transcription

1 Host-vector interaction in engue: a simple mathematical moel K Tennakone, L Ajith De Silva (Inex wors: engue, engue moel, engue Sri Lanka, enemic equilibrium, engue virus iversity) Abstract Introuction Dengue is a mosquito-borne viral infection enemic in tropical an subtropical regions,now spreaing at epiemic proportions causing a major health issue in Sri Lanka an elsewhere. No effective vaccine or a curative antiviral rug is available to prevent or treat the isease. The only way of mitigating engue at present, is through mosquito eraication an eucating the public on preventive measures which can minimize the cycle of transfer. Objectives A theoretical moel of engue with simple mathematics is presente to gain a quantitative unerstaning of the pattern of engue outbreaks in Sri Lanka an suggest control measures. Metho The statistics on incience of the isease reporte by the Epiemiology Unit is analyze using the moel. Despite simplicity, the moel possesses explanatory an preictive capacity, enabling etermination of crucial parameters. The moel shows that the infectives increase exponentially in an outbreak, provie the number of vectors per human excees a threshol, illustrating not only vector eraication but measures which minimize their biting frequency an preventing prolonge survival are effective safeguars. Results In a population consisting of 75% who are susceptible, the threshol is estimate to be 0 mosquitos per person. Conclusions The moel showe that the enemic equilibrium of the system can occur at any level. As emographic changes escalate mosquito breeing, they infect more an more susceptible people. The consequent increase in virus replication inuces new strains broaening the genetic iversity of the virus an helping it to overcome the human immune response. The increasing enemicity of engue ue to this is emonstrate by the moel. Ceylon Meical Journal 08; 63: DOI: Introuction The evolving pattern of engue transmission leaing to sporaic epiemics is a major health issue in Sri Lanka an many other tropical an subtropical countries. In the absence of an effective vaccine or a curative antiviral rug, the only way of mitigating the sprea of this isease is mosquito eraication an aoption of precautionary measures to minimize the cyclical transfer of the virus [,]. Such programs, carefully esigne, base on the epiemiology of infection have been successful in combatting the sprea of mosquito borne iseases [3-5]. In this context, mathematical moels that quantitatively epict the epiemiology of the system, insightfully assist the optimization of control strategies [6-8]. Unfortunately, most moels of engue reporte in literature are too mathematical an cannot be use by non-specialist health care workers, meical professionals, stuents or eucate laymen [8-4]. Mathematical epiemiological moels are intellectually appealing, have a high peagogic value, motivate eeper stuy an suggest control strategies. It is important to recollect the contributions by Ross an Maconal in malaria eraication. They moele the epiemic mathematically following the finings of Manson an these three malaria pioneers exhaustively stuie malaria epiemics in Sri Lanka [6, 5,6]. This paper introuces a simple moel of engue that requires only a high school backgroun in mathematics. The ata on istribution of engue cases reporte by the Epiemiology Unit, Ministry of Health, Sri Lanka is use to test the moel preictions an suggest isease control strategies base on the analysis [7]. The mathematical moel Dengue fever is cause by a virus transmitte by the Aees aegypti mosquito which is mainly active uring ay time. However, mosquitoes which bite at night may also act as engue vectors. Increase human population ensity, effect of climate on mosquito breeing an presence of several stereotypes of the virus an human reservoirs with marginal to severe clinical manifestations 55 Amberville R, North Anover, MA 0845, Unite States, Department of Physics, University of West Georgia, Carrollton, GA 308, Unite States. Corresponence: KT, <ktenna@yahoo.co.uk>. Receive 06 March 08 an revise version accepte 9 April 08. This is an open-access article istribute uner the terms of the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original author an source are creite. 58 Ceylon Meical Journal

2 complicate the epiemiology of the isease [8-0]. The simple moel presente here examines the evelopment of engue infection in an environment of constant human an mosquito populations. Nevertheless, the contribution of these parameters on temporal progression of the isease can be inferre from the analysis. When a mosquito x bites a engue infecte human subject X, there is a high probability of the virus being transmitte to the mosquito an it acquiring the infection after an incubation perio of 4-0 ays, to reach the stage of an infecte mosquito enote x. During its life time, x is capable of transferring the virus to the human subjects it bites. The host X bitten by x coul evelop the infection after 4-4 ays of incubation. Some humans o not get infecte even if they are bitten by virus carrying mosquitoes but those who are susceptible acquire the infection. If S an S enote the population ensities of healthy an infecte human susceptibles in a population of ensity N an n an n the ensities of healthy an infecte mosquitoes, the time evolution of the infection can be escribe by the following rate equations, S an ( S / N ) ks n b( S / N) n hn Vol. 63, No., June 08 () () where k, h, a an b are constants. The first term in () accounts for the interaction between healthy humans an infecte mosquitoes in transmitting the infection. The rate of biting of infecte mosquitoes escribe by the first term of () is assume to be proportionate to the ensity of infecte mosquitoes an the fraction of human susceptibles in the population. The constant a measures the rate of biting per unit time per infecte mosquito with a weightage for the probability of virus transmission. Similarly, the first term of () accounts for the interaction between infecte humans an healthy mosquitoes in transferring the infection to the mosquitoes an b is the biting rate of infecte persons by healthy mosquitoes per unit time, per mosquito with weightage for mosquito acquiring the infection. The secon term in () accounts for recovery of infecte humans an the parameter k - which has the imension of time, measures the meian recovery time. Analogously, infecte mosquitoes are eliminate ue to their senescence, an the inverse of h in the equation () is their mean life time. Even in the case of an intense engue outbreak, S << S an the eath rate is relatively low. Unlike in more etaile moels of engue, approximate but reasonable assumption S << S avois the necessity of writing a separate equation for the time evolution of S. The moel assumes these conitions an oes not explicitly consier the immunities originating from presence of more than one stereotype of the virus or ifferent strains of the same stereotype. The moel oes not explicitly take into account inputs from control programs such as fogging, elimination of stagnant water or aoption measures to reuce the mosquito biting rate, instea the relative importance of these are assesse. The system escribe by the equations () an () has an equilibrium when S = S 0, n = n 0, corresponing to the conition. an ( S / N) ks 0 b( S / N) n hn n N S kh ab ( N / S ) 0 0 Clearly (3) imply that the equilibrium occurs when, S ( S ) n n ( n ) 0 S a( n )( S / N) k( S ) n b[( S ) / N] n h( n ) Equations (6) an (7) can be easily solve as follows: Differentiate (6) with respect t an eliminate n n an using (7) to obtain the equation (8) given below. S ( k h) S [ kh ab( n/ N)( S / N)]( S ) 0 ( k h) [ kh ab ( n / N)( S / N)] 0 t t S S0 Ae Be t t n n0 Ce De ( ) ( ) k h k h 4[ kh ab( n / N)( S / N)] () ( (4) To examine the behavior of the system near the equilibrium, we set, where S an n the eviations of S an n from the equilibrium values. Inserting (5) in () an () subject to the conition (3), we obtain the equations, ( ) ) k h k h 4[ kh ab( n / N )( S / N )] 0 (5) Solutions of (8) have terms of the form, S= Ke t where K is constant an the possible values of is are being given by the quaratic equation, (6) (7) Thus the solution of () an () can be expresse as, (0) () where A, B, C, D are arbitrary constants an (9) (3) (8) () 59

3 The quantity is essentially real an negative, implying that the secon term in (0) an () vanishes as t, whereas can be positive or negative, epening on whether (n/n) is greater or less than a threshol value (n/n) c = (kh/ab) (N/S). The former conition is inicative of the onset of an epiemic an in the latter case any suen increase in number of infectives above the equilibrium, ecays off without expaning into epiemic proportions. The parameter can be approximate an written as, ab n kh [ ][ ( N / S)] k h N ab (4) Figure a. Yearly engue cases istribution from 00 to 07 plotte using ata reporte by Epiemiology Unit, Sri Lanka [7]. so that the progression of infection is escribe approximately, by the expression, S S0 Aexp( t) (5) Equilibrium becomes unstable an an epiemic situation evelops when the number of mosquitoes per person (n/n) excees the threshol value. Otherwise the equilibrium remainss table without growing to an outbreak. The moel oes not fix the value of S 0, implying that the enemic equilibrium can be at any level. An alternative way of unerstaning this issue is as follows: The general solution of equations () an () can be written as t t S Ae Be (6) When n/n <(kh/ab) (N/S) is satisfie, S 0 as t, implying that the above conition leas to an natural elimination of the isease. However, if n/n = (kh/ ab) (N/S), (6) reuces to S A Be (7) ( kh) t Here the system approaches an equilibrium S = A which is not essentially zero. An analysis of extensive ata on engue cases reporte by the Epiemiology Unit of the Ministry of Health, inicates that the pattern of evelopment of an outbreak follows the tren of the equation (5) to a significant level of confience [8]. The istribution of engue cases in Sri Lanka from 00 to 07 are plotte in Figurea. It is seen that the annual incience of the isease shows two peaks, occurring aroun June-July an October-December, correlating with South-Western an North-Eastern monsoon perios. The October-December peak of the previous year wanes uring the February-March ry season an signs of a more pronounce June-July peak begins aroun the 5 th week (March/April) of the year. Figure b. Fitting of miyear engue cases istribution 00 to 07 accoring to the equation [5]. 60 Ceylon Meical Journal

4 Figure b. Shows the regression analysis fitting of the ata for each year from aroun 5 th week (March/April) an up to the June/July peak to the equation (4). The fitting parameters S, A, an R-square confience levels 0, = are given in Table.. Table. Parameters extracte from curve fitting process for each year Year Weeks S O A [months] [months - ] R-square % It is seen that ata fits the expression (5) more significantly when the outbreak has been large (in 05 the June/July engue peak was almost absent an here the R-square value is 67%). The parameter varies from months - (equivalently varying from months), clearly larger epiemics correspons to larger values of (smaller values of ). The year 07 epiemic ha the highest value of or equivalently the smallest value of the time constant. Another peculiarity of 07 istribution is the large value of S 0 (= 3355), whereas in all the other years this parameter ha varie in the range The larger value of S 0 in 07 implicates a situation where pervious outbreak (October-December 06) has equilibrate at an unusually high groun level. The parameters of the expression (5) obtaine by regression analysis of engue cases reporte on istrict basis for the 07 mi-year outbreak are presente in Table. Here the correlation of the time evolution of the epiemic with expression (5) is highly significant (except in some istricts, mostly in the Northern an Eastern Province). Presumably, istrict statistics correspon to localities of uniform conitions, notably the characteristic parameters of In the istrict base ata, is seen to vary between 0.5 an month - ( ~ months). Most significantly variable factors in the expression for given by (4) are the number of mosquitoes per person (n/n), the fraction of susceptibles (S/N) an the survival time of infecte mosquitoes (same orer as that of uninfecte ones) ~h -. The above analysis implies that interventions for preventing an epiemic shoul be aime at controlling the mosquito population per person ratio (n/n) to a level below a critical value (n/n) c =(kh/a)(n/s). Accoring to the moel (n/n) is epenent only upon parameters k, h, a, b efining the ynamics of the system as well as the fraction (S/N). A higher value for (n/n) c, implies that the system coul maintain its resistance to an epiemic uring averse weather conitions causing an increase of the mosquito population. It is important to realize, that not only mosquito eraication but measures elevating (n/n) c are also important in controlling engue. An increase of k an/or h an ecrease of a an/or b will raise the magnitue of (n/n) c. As the inverse of h is the mean life time of infecte mosquitoes, ecreasing mosquito survival time will increase (n/n) c. The life time of Aees aegypti is about two weeks to one month epening on environmental conitions. There is no evience that acquiring the virus has a significant influence on longevity of Aees aegypti. The effect of climatic conitions on survival of Aees aegypti has been stuie [-]. Extreme temperatures an torrential rains lower mosquito survival [3-4]. Nature of the microenvironment seems to play an important role in longevity of mosquitoes. It has been observe that longevity of Aees aegypti is significantly higher in isorganize an thickly populate urban areas compare to more organize an planne urban wellings [5]. Presence of moist shay resting an hiing places enhance mosquito breeing an survival. Consequently, accoring to the moel, the critical mosquito population per person (n /N) c sufficient to initiate a engue out-break in congeste urban area is relatively low. It is interesting to estimate the threshol value (n/n) c for reasonable values of the parameters of the theory. The average values of k an h are the orer of 0. an 0.05 ays - respectively. Biting rates of Aees aegypti uner varying conitions have been stuie, we choose a b = 0.0 ays - base on average of a range reporte in literature [6]. If we assume a situation where the population consists of 75% who are susceptible, the equation (4) yiels (n/n) c = 7. Thus persistence of a engue vector population of aroun 0 per person for a perio of the orer of the time constant ( = month) woul turn an enemic situation to an epiemic. Vol. 63, No., June 08 6

5 Table. Parameters extracte from curve fitting process for ifferent istricts (cases reporte 07). The total cases reporte until July is also given. In istricts marke X ata oes not fit into the equation (5). District population ensity is enote N District Total cases Population ensity N 0 A [months] [months - ] R (N) Colombo Gampaha Kalutara Kany Matale Nuwara Eliya Galle Hambantota Matara Jaffna X X X X X Kilinochchi X X X X X Mannar X X X X X Vavuniya X X X X X Mulativu X X X X X Batticaloa X X X X X Ampara Trincomalee Kurunegala Puttalam Anurahapura Polonnaruwa Baulla Moneragala Ratnapura Kegalle Colombo Municipality Other areas of Colombo The rate constant a efining the extent to which humans are infecte when bitten by infecte mosquitos, occur in the enominator of the expression (4), therefore ecrease of a will increase (n/n) c. Use of mosquito nets, meshing the winows, wearing cloths to cover boy extremities, application an spraying of mosquito repellants, coul effectively lower the value of a. Such personal preventive care not only lessen the probability of an iniviual contracting engue, but also greatly helps to increase the critical mosquito population neee to initiate an explosive infection. The engue virus occurs as four antigenically istinct stereotypes. When a person contracts engue of one type an recovers, he or she acquires life-time immunity to that virus an immunity to other three types happens to be temporary an last only for two to three months. Consequently, sometime after an epiemic of one type, the propensity of an outbreak of an infection of another type is greatly increase. In this situation the susceptibles are excessive, which ecrease fraction (N/S) an therefore (n/n) c ecreases, implying that even a relatively small population ensity of mosquitoes coul trigger an epiemic. Generally, when a engue outbreak winles, mosquito control activities an other precautionary measures also become less. This can lea to more severe epiemics in the future. Although people consciously take precautions to avoi contracting engue, there is often ignorance about the nee to prevent mosquitos acquiring engue from humans. Generally, engue patients bes are not covere with mosquito nets uring the ay, nor o they apply mosquito repellents. Reucing the constant b by aopting safeguars to minimize passing of the infection to mosquitos is equally important to maintain (n/n) c at a level sufficiently high to escape the onset of an epiemic. A feature of the yearly istribution of engue cases in Sri Lanka is a progressive increase in the number of reporte cases in the lull perio of the outbreak approximately between 0 th an 0 th weeks (Figurea). Presumably the yearly enemic equilibrium (the lull perio) has continue to increase from 00 to 07. The situation appears to be parallel to the increase in the value of A in equation (7). The important question is what causes this behavior? The following may be a plausible explanation. 6 Ceylon Meical Journal

6 Demographic changes make the environment more conucive to breeing of mosquitoes increasing the number of infectives. More infectives result in more virus replications an therefore a higher probability of creating genetic variations in the virus. This process progressively increases the genetic iversity of the virus (a wie spectrum genetic strains within each stereotype). The iversity of the virus enables it to overcome the immune response in the human host. Discussion The main fining of the moel is emonstration of the existence of a threshol for the number of mosquitos per (n/n) c above which a community coul become susceptible to an outbreak of engue. The threshol epens on ynamic constants of the system an the fraction of susceptibles in the human population. As complete extermination of mosquitos is a very ifficult task, a more practical engue control strategy woul be the aoption of measures to maintain the value of (n/n) c as high as possible. This precautionary measure will ensure that even if there is a suen introuction of an infestation, it will ecay, without eveloping into an epiemic. Mosquito meiate iseases, malaria an filaria have been eraicate in Sri Lanka an elsewhere. It is important to realize that these successes are not entirely the result of effective mosquito control programs. Emptying of human pathogen reservoir by rug treatment of infecte humans has playe a crucial role. Similarly, for vaccination suppresse yellow fever in Africa. In controlling engue, more emphasis shoul be place on measures that reuce the human reservoir of the virus in situations where mosquitoes cannot be aequately eliminate. In the absence of rugs an vaccines, promoting methos of minimizing mosquito-human contact may be an option. In terms of the moel such efforts will lower the values of the parameters a an b raising the threshol number of mosquitoes per person necessary to initiate an epiemic. The moel reinforces the warning that mosquito control efforts shoul not be neglecte as October-December peak of engue incience in Sri Lanka subsies. Control programs uring this perio will curtail rising of the mosquito population up to threshol an beyon. Historically, Sri Lanka has been successful in controlling an also eliminating infectious iseases [3-5]. In the challenging case of engue, attacking the problem from every angle is essential. Here mathematical moels an analysis of ata on incience of engue in relation to climate, emographic factors an eraication programs will play a vital role. The following section summarizes important factors which shoul be consiere by the engue control programme. These factors were ientifie by the moel. The moel emonstrates the effectiveness of the National Dengue Control Programs in reucing the mosquito population. To prevent the onset of an epiemic, the mosquito eraication work nees to be initiate at the first sign of an increasing tren in the cases of engue, which is more effective than escalation of these activities uring an epiemic. The main component of this effort is insecticie spraying an work geare towars elimination of mosquito breeing stagnant water. The moel clearly inicates that the threshol value (n/n) c is inversely proportional to the mean life time of the mosquito (k - ). The longevity of mosquitos is significantly higher in amp shay environments. Thus the engue control program shoul also make a concerte effort to eucate the general public an conuct activities to clear shrubs an shay locations in close vicinity of living areas. Not only stagnation, wetness an shae also greatly contribute to sprea of engue via reuction of the threshol mosquito population neee to initiate an epiemic situation. It is also important to pay more attention to measures which minimise the passing of the infection from humans to mosquitoes. We nee to highlight the importance of engue patients using mosquito nets uring ay time an the provision of such facilities to hospitals, will reuce the parameter b in the expression (n/n) c = (kh/ab )(N/S), increasing the threshol. Urban an suburban congestion will also reuce the threshol (i.e. by increasing N). National housing construction an planning programs shoul keep this in min when implementing evelopmental work. Frequently, schools are foun to be foci of engue, this can be unerstoo as they are places of potentially low threshol (because of large N). Extra precautions nee to be aopte in a situation with low potential threshol. Conflicts of Interest There are no conflicts of interests. References. Ooi EE, Goh, KT, Gubler DJ. Dengue Prevention an 35 Years of Vector Control in Singapore. Emerg Infect Dis 006; : Falope O, Hanson K, Azizan, A. 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