A Stochastic Spatial Model of the Spread of Dengue Hemorrhagic Fever

Volume-7, Issue-5, September-October 2017 International Journal of Engineering and Management Research Page Number: 98-104 A Stochastic Spatial Model of the Spread of Dengue Hemorrhagic Fever A. R. Nuha 1, P. Sianturi 2, H. Sumarno 3 Department of Mathematics, Bogor Agricultural University, INDONESIA ABSTRACT The dengue virus is spread the Aedes aegypti mosquito that can breed well in the area of human settlements. The dengue virus has the ability to transmit vertically, from mosquitoes to their eggs. This research applies stochastic model study the spread of dengue virus. The purpose of this research were: to modify the Otero and Solari models, to determine the transition probability of the modified model, calculate the probability of disease outbreaks and to carry out computer simulate of the modified model. This simmulation was used to know the effect of temperature and availability the breeding sites of mosquitos on the spread of dengue virus. The results of this research had been constructed the modification of Otero and Solari models by adding four new subpopulations i.e. infected eggs, infected larvae, infected pupae and infected first adult mosquito, in which the transition probability of this model was analyzed by Markov chain approach with discrete time. The numerical simulation results had showed that the variations in temperature has a great influence on the spread of dengue fever, but the number of breeding sites had a little influence on the spread of dengue fever cases. Keywords Breeding sites, Dengue virus, Spatial, Stochastic model, Temperature I. INTRODUCTION Dengue hemorrhagic fever is a type of infectious disease caused by pathogenic and infectious virus. Dengue virus is spreaded by Aedes aegypti mosquitoes, that can breed well in human settlements and they often lives indoors. The mosquito infected dengue virus can transmited in human body through its biting. Transmission of the virus in the human body can cause dengue hemorrhagic fever. Symptoms dengue fever disease such as headache, joint and bone pain, redness of the skin rash, back pain and abdominal pain accompanied by nausea and vomiting. A major breakthrough in the treatment of dengue was discovered in 1944 when Dr. Albert Sabin successful identified dengue virus. In the next research showed that disease is caused by four viruses called DEN-1, DEN-2 DEN-3 and DEN-4 [1]. The Aedes aegypti mosquitoes can be found almost in all parts of Indonesia. Dengue virus can grow and multiply without causing death within mosquitoes body because it does not form cytopathic effect [5]. So it needs special attention to handle spread of dengue virus, such as with formulation mathematical models that can help provide an overview events or effect caused in the spread of dengue virus. Several researches have been conducted to model the transmission process of dengue virus, among effect of temperature variation in an environment of development Aedes aegypti [3]. Next proceed is analyze effect of availability breeding sites in an environment of development mosquitoes [4]. Otero and Solari [3,4] formulated models that analyze spread of dengue virus with consider effect of temperature variation and spread of Aedes aegypti mosquito egg [2]. Dengue virus has ability to transmitted Vertically i.e. dengue virus transmission from mosquito to mosquito eggs [5]. Therefore, author interest to do some research that modify Otero and Solari model in 2009 [2], with added assumption occurrence transmission of dengue virus from mosquitoes to eggs. In this research, a stochastic model is used to describe and analyze some factors causing the spread of dengue virus. The stochastic model is suitable to apply for this model because it contains adequate coverage to analyze events considered as random variables, which is represented by number of subpopulation in the system. The purpose of this research were: to modify the Otero and Solari models, to analyze transition probability for the subpopulations of mosquito and human of the modified model, calculate the probability of outbreaks and to simulate process of transition in modified model. 98 Copyright 2017. Vandana Publications. All Rights Reserved. II. METHODOLOGY This research was conducted from February 2017 until July 2017. This research is done by modify the model developed by Otero and Solari model about spread of dengue virus in 2009. After modifing the model, the next step is formulating the transition probability, which is used in the numerical simulations aimed to analyse the dynamics of human population in each compartment. Finally, the

probability of an outbreak is determined using a branched process approach. III. MODEL OF DENGUE HEMORRHAGIC FEVER DISEASE The research model used in this study is a model formulated by Otero and Solari in 2009 [2], adding four new subpopulations. The formulation model consists of human population and mosquito population, where population is categorized into several subpopulations based on the disease status given susceptible humans (Hs (i,j) ), exposed humans (He (i,j) ), infectious humans (Hi (i,j) ) and removed humans (Hr (i,j) ). The mosquito population was divided into 14 subpopulations i.e. healthy eggs (E (i,j) ), infected eggs (e (i,j) ), healthy larvae (L (i,j) ), infected larvae (l (i,j) ), healthy pupae (P (i,j) ), infected pupae (p (i,j) ), female adults not having laid eggs according to their disease status: susceptible (A1s (i,j) ) and infectious (A1i (i,j) ), susceptible flyers (Fs (i,j) ), exposed flyers (Fe (i,j) ), infectious flyers (Fi (i,j) ) and female adults having laid eggs acording to their status: susceptible (A2s (i,j) ), exposed (A2e (i,j) ) and infectious (A2i (i,j) ). This research assumes that the location of mosquitos indocated by indicies (i, j), where only mosquito flyers can move or from one location to another. So the spread of mosquitoes on this model is represented by the subpopulation of flyers. If X denotes subpopulation at stage X then X (i, j) declares subpopulation of stage X on index (i, j). Schematic of the dengue virus spread is sowed in Figure 1 and Figure 2. Figure 2: Schematic model of the spread of dengue virus in humans. IV. RESULTS Transitional Probability Let Δt denote very small interval so that at interval is assumed that only one event occurs. Suppose also change random variables and in interval represented with,, and. With Assume and constant over time, the transition probabilities of dengue virus spread consist in several stages: aquatic without virus infected, aquatic with virus infected, Gonotropic cycle 1, Gonotropic cycle 2 and dengue spread in humans. The change of random variables at each stage is expressed in a transition opportunity presented in Table 1 to Table 5. The development of aquatic stage mosquitoes without infected with virus starts from eggs, larvae, pupae, to the formation of adult mosquitoes 1, where at this stage it is assumed subpopulations are infected with dengue virus yet. The transition opportunities at this stage are outlined in Table 1. TABLE 1 TRANSITION LEVEL AT AQUATIC STAGE WITHOUT VIRUS INFECTION Event Description 1. Egg death 2. Egg hatching Transition Process Probability 3. Larva death 4. Pupation 5. Pupa death 6. Suspectible adult 1 emergence 7. Suspectible adult 1 death ( ) Figure 1: The schematic model of modifided model of dengue virus spread Aedes aegypti vector. transmission process, migration, egg deposit. The aquatic stage infected virus has a process similar with previous stage, just subpopulations at this stage have been infected dengue virus. The transition opportunities at this stage are outlined in Table 2. TABLE 2 TRANSITION LEVELS AT THE AQUATIC STAGE ARE INFECTED WITH THE VIRUS Event Description 8. Egg death 9. Egg hatching 10. Larval death 11. Pupatiom Transition Process Probability 99 Copyright 2017. Vandana Publications. All Rights Reserved.

12. Pupal death 13. Infection adult 1 emergence 14. Infection adult 1 dead ( ) If mosquitoes sucked blood from humans infected it will happen transmission of dengue virus from humans. After sucked the blood, female mosquitoes will find a rest place then lay eggs. The process performed by female mosquitoes, ranging from sucking blood until mosquitoes lay eggs is called Gonotropik cycle. The probability of the transition at this stage of the cycle Gonotropik 1 described in Table 3. Gonotropik cycle 2 stage, the female mosquitoes lay eggs sucked human blood again. This process is called Gonotropic cycle 2. The transition opportunities at this stage are described in Table 4. In this research it was assumed spread mosquitoes happen that affected by lack of availability breeding sites. The spread of mosquito is described by index 34 s.d 36. Dengue virus may spread because of the interaction between viruses, humans and mosquitoes. Female mosquitoes that bite humans can transmit the virus, both from human to mosquito or vice versa. The transition opportunities at this stage are outlined in Table 5. In real terms, the mosquito is able to issue a ± 63 eggs in one cycle Gonotropik. As a result, specialized in the transition process subpopulations eggs ( dan ) are assumed to occur 63 incident addition of eggs in the interval Every event that occurs within the was approximated using the discrete time Markov chain approach. TABLE 3 TRANSITION LEVELS AT THE GONOTROPHIC CYCLE STAGE Event Description Transition Process Probability 15. Susceptible adult 1 bites healthy human * + 16. Oviposition of susceptible flyers 17. Susceptible adult 2 dead 18. Susceptible adult 2 bites infection human 19. Oviposition of exposed flyers 20. Exposed adult 2 dead 21. Infectious adult 1 bites sick human * + TABLE 4 TRANSITION LEVELS AT THE GONOTROPHIC CYCLE STAGE 2 Event Description Transition Process Probability 22. Susceptible adult 2 bites healthy human * + 23. Susceptible flyers dead 24. Susceptible adult 2 bites invection human 25. Exposed flyers dead 26. Exposed adult 2 bites human 27. Extrinsic incubation of exposed flyers 28. Infectious flyers dead 29. Extrinsic incubation of exposed adult 2 30. Infectious adult 2 dead 31. Infectious adult 2 bites sick human * + 32. Oviposition of infectious flyers without transmitting the virus to the egg 33. Oviposition of infectious flyers with transmitting the virus to the egg 100 Copyright 2017. Vandana Publications. All Rights Reserved.

www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 TABLE 5 TRANSITION RATES FOR THE SPREAD OF DENGUE VIRUS IN HUMANS Event Description Transition Process Probability 37. Birth of susceptible humans 38. Death of susceptible humans 39. Humans are exposed virus from infectious adult 2 40. Humans are exposed virus from infectious adult 1 41. Death of exposed humans 42. Intrinsic incubation of exposed humans 43. Death of infectious humans 44. Removal of invectious humans 45. Death of removed humans Probability of Outbreaks Outbreaks will occur when the number of infected people increases. In 2012 Allen formulated an outbreak opportunity by using a branched process approach. Through this approach [7] can be formulated opportunities for outbreaks in accordance with this research model. Let, and, where dan describe large population sizes. Define =, for i j and. For, probability generating functions (pgf) of is, pgf of is, pgf of is, pgf of is The fixed point pgf of,, and is the solution of under conditions,. By using substitution and elimination methods, it is obtained: The disease-free probability is determined by {. So the chance of the outbreak is, with,, and. The basic reproduction number can be seen in (5). 101 Copyright 2017. Vandana Publications. All Rights Reserved. Where,,,,,,,. (5) Model Parameters Modification Models of Otero and Solari contain several parameters that related with factors is caused by transmission of dengue virus and the effects to be incurred in transmission event. The model parameters used are taken from the Otero and Solari model parameters of 2009 [2]. The parameter values used in the model are summarized in Table 6 and 7.

TABLE 6 COEFFICIENT, SYMBOLS AND VALUES OF THE PARAMETERS CHARACTERISTIC OF THE SPREAD OF DENGUE VIRUS Coefficient symmbol Value Intrinsic incubation period 5 days Extrinsic incubation period 10 days Viremik period in humans 3 days Probability of transmitting virus from 0.75 human to mosquito Probability of virus transmission from 0.75 mosquito to human Probability of virus transmission from 0.75 mosquitoes to their eggs Human mortality rate 0.009 hari Egg mortality rate 0.011 hari Environmental carrying capacity 1.5 Factor formed adult 0.83 Adult mortality rate 0.091 availability of breeding sites. The simulation for this model uses software RStudio 1.0.136 2009-2016. Condition or initial value is given in subpopulations that 10000, 1000. Numerical simulations carried out to provide a picture or illustration about dynamics spread of dengue fever in field conditions in accordance with initial value both population and parameters. Human Population Dynamics for Breeding Site Conditions ( Observation in this condition is done by giving an initial value of, means that there are 30 points where it can be used as a mosquito to breed and values obtained 0.873. The spread of dengue hemorrhagic disease in this condition is illustrated in Figure 3. The value of mortality rate of larvae and mortality rate of pupa are formulated as follows: ) (1) (2) where T is the temperature. The rate of development of each stage of mosquito subpopulation as follows: { If BS 150 (3) If BS 150 (). (4) where represents the rate of oviposition, is the enthalpy activation of the reaction that is catalyzed by enzyme, is the change in enthalpy associated with high temperature inactivation of the enzyme, is mean development rate at temperature, is the universal gas constant and is temperature at which the enzyme is active and low temperature inactive [6]. The parameter values in equation (3) are summarized in Table 7. TABLE 7 DEVELOPMENT OF MOSQUITO SUBPOPULATION STAGE, MEASURED PER DAY, TEMPERATURE MEASURED IN DEGREES KELVIN. Development rate Hatching Eggs 0.24 10798 100000 14184 Formation of 0.2008 26018 55990 304.6 Pupa Formation of 0.3884 14931-472379 148 Adult I Gonotrophic 0.216 15725 1756481 447.2 cycle II Gonotrophic cycle 0.372 15725 1756481 447.2 V. NUMERICAL SIMULATION This stage is done a simulation of all mosquito and human subpopulations follows the transition process described earlier. Simulations were performed in model by analyzing the temperature change conditions and Figure 3: Dynamics of human population (a) exposed and (b) infectious when, for different temperature level. In this condition, the number of people exposed and infected individuals has increased in the last 10 days, after which it decreases toward the disease free condition. Figure 3 shows that when there are 10000 larvae infected with dengue virus in an environment will result in an outbreak within a period of 0 to 10 days. The development of outbreaks tends to be greater in environments where the average temperature is quite high (303 0 K). Human Population Dynamics for Breeding Site Conditions ( Observation is given initial value obtained value 2.037 which means hatching rate of eggs in this condition is greater than previous condition. The spread of dengue virus in this condition is illustrated in Figure 4. When compared to Figure 3 with the same initial value, it appears that two images almost have the same illustrations. The highpoint number of people 102 Copyright 2017. Vandana Publications. All Rights Reserved.

exposed and infected humans in both images are almost the same, this indicates that the increase in the number of breeding site on the environment does not result in increasing the number of people infected with dengue fever on the environment. The number of people exposed and number of humans infected have the same behavior as previous condition that there is an increase in the beginning of time and then decreases over 20 days. The behavior of model in Figure 3 s.d Figure 5 shows that increase in temperature can lead to an increase in dengue virus inveksi large enough, while the increase in the number of breeding site impact is not so great on the spread of dengue virus. Rate of Development of Mosquito Subpopulation The development of mosquito subpopulation observed were the rate of hatching of the egg, rate of pupa formation, rate of Gonotropic cycle 1, rate of Gonotropic cycle 1 and rate of adult mosquito formation. For the development of mosquito subpopulations are presented in Table 8. TABLE 8 DEVELOPMENT RATE OF MOSQUITO SUBPOPULATIONS Parameter 0.0602 0.1231 0.24 0.3297 0.0082 0.0436 0.184 0.2716 0.0583 0.1546 0.384 0.5919 0.0298 0.0830 0.216 0.3404 0.0513 0.1429 0.372 0.5862 Figure 4: Dynamics of human population (a) exposed and (b) infectious when, for different temperature level. Human Population Dynamics for Breeding Site Conditions ( The last observation is given initial value of obtained value. The spread of dengue virus in this condition is illustrated in Figure 5. Table 8 shows that when the temperature increase the rate of hatching of the egg, the rate of pupa formation, the rate of Gonotropic cycle 1, the rate of Gonotropic cycle 1 and the rate of adult mosquito formation also increase. The increasing rate of development in the mosquito subpopulation can suppress the spread of dengue hemorrhagic fever. Mortality rate Mosquito subpopulation At this stage the relationship between the death rate of larvae and pupae mortality rate with changes in temperature will be studied. Changes in the value of the parameter at each condition can impact both large and small on the system, so that the analysis is done to see the impact of mosquito subpopulations death rate in the spread of dengue fever. The mortality rate of mosquito subpopulations can be seen in Table 9. TABLE 9 THE MORTALITY RATE OF MOSQUITO SUBPOPULATIONS Parameter 0.9825 0.034 0.010595 0.010093 0.9825 0.034 0.010595 0.010093 Figure 5 Dynamics of human population (a) exposed and (b) infectious when, for different temperature level. Table 9 shows that when the temperature is increased the mortality rate of the larvae and the pupae mortality rate decreases. As a result the transmission of dengue virus has increased. Probabilities of Outbreaks with BS = 150 The probability calculation of outbreaks in this research is to see how big the possibility of outbreaks occur in four environments with different temperature conditions. The results of these calculations can be seen in Table 10. 103 Copyright 2017. Vandana Publications. All Rights Reserved.

Table 10 Probabilities of outbreaks with conditions BS = 150 2 1 1 2 150 0.47 0 2 1 1 2 150 1.31 0.762918 2 1 1 2 150 3.41 0.998191 2 1 1 2 150 5.37 0.999801 Based on table 10 can be seen the results of the calculation of the probability of an outbreak that the higher the average temperature in an environment then the chances of the outbreak is also greater. In addition, it is also seen when the mean temperature of the probability of an outbreak indicates a zero, this corresponds to a numerical simulation result when the condition the number of infected persons at a certain time decreases towards to the point of diseasefree stability. [6] Schoolfield RM, Sharpe PJH, Magnuson CE. 1981. Non-linear regression of biological temperaturedependent rate models based on absolute reaction-rate theory. Journal of Theoretical Biology. 88:719-731.doi:10.1016/0022-5193(81)90246-0. [7] Allen LJS, Lahondry GE. 2012. Extinction thresholdsin deterministic and stochastic epidemic models. Journal of Biolog Dynamics. 6(2):590-611.doi:10.1080/17513758.2012.665502. VI. CONCLUSION 1. Modification model of Otero and Solari in cases of dengue virus spread resulted in the addition of four new subpopulations: infected eggs, infected larvae, infected pupae and infected adult 1. 2. The transition probability of modification model Otero and Solari follows a stochastic process approximated by the discrete time Markov chain approach. 3. Through numerical simulations it was found that temperature variations have a great influence in cases of dengue hemorrhagic fever, whereas difference in the number of breeding sites has little effect on the case of dengue hemorrhagic fever. 4. The higher average temperature in the living environment, the probability of outbreak will be higher. REFERENCES [1] Chakraborty T. 2008. Dengue Fever and Other Hemorrhagic Viruses. New York (US): Chelsea House. [2] Otero M, Solari HG. 2009. Stochastic ecoepidemiological model of dengue disease transmission by aedes aegypti mosquito. Mathematical Biosciences. 223:32-46.doi:10.1016/j.mbs.2009.10.005. [3] Otero M, Solari HG, Schweigmann N. 2006. A stochastic population dynamics model for aedes aegypti: formulation and aplication to a city with temperature climate. Bulletin of Mathematical Biology. 68:1945-1974.doi:10.1007/s11538-006-9067-y. [4] Otero M, Schweigmann N, Solari HG. 2008. A stochastic spatial dynamical model for aedes aegypti. Bulletin of Mathematical Biology. doi:10.1007/s11538-008-9300-y. [5] Prasetyowati H, Seran MD. 2012. Transimisi Transovarial Virus Dengue pada Telur Nyamuk Aedes aegypti (L.). Aspirator. 4(2):53-58. 104 Copyright 2017. Vandana Publications. All Rights Reserved.