Integration of Queueing Network into Susceptible Exposed Likely Infected Confirmed Dead Removed Model for Ebola Virus Disease Analysis

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1 Amerian Journal of Applied Mathematis and Statistis, 08, Vol 6, No, Available online at Siene and Eduation Publishing DOI:069/ajams-6--3 Integration of Queueing Network into Suseptible Exposed Likely Infeted Confirmed Dead Removed Model for Ebola Virus Disease Analysis Ikeme John Dike,*, Chinyere Ogohukwu Dike Department of Statistis & Operations Researh, Modibbo Adama University of Tehnology, PMB 076, Yola, Adamawa State, Nigeria Federal College of Eduation PMB 04, Yola, Adamawa State, Nigeria *Corresponding author: Abstrat Ebola Virus Disease (EVD) is a ompliated and unparalleled epidemi disease In reent times, the disease has been ravaging human lives and eonomy in West Afrian nations The most affeted ountries are Guinea, Liberia and Sierra Leone Contagious disease like Ebola transmits through networks, made by bodily interations among persons As a result of the transmission mode and deadly nature of the disease, this paper proposes an EVD transmission and ontrol model whih inorporates queueing network that onsiders all the transmission phases in order to understand the real nature of the disease and predit its behaviour The result for network indiates that the population was vulnerable to large sale epidemis before intervention in the three ountries In network model, ritial transmissibility or epidemi threshold T C is the least transmissibility neessary for an outbreak to turn into a large sale epidemi The T C for the three ountries was Before intervention, the transmissibility for Guinea was 0365, Liberia, 0093 and Sierra Leone, 035 After intervention, the transmissibility for Guinea beame 00595, Liberia, and Sierra Leone, 0057 The before and after transmissibility results in omparison with the T C indiate that intervention was effetive in ontaining the EVD epidemi The vulnerability dereased drastially after intervention Keywords: Ebola virus disease, epidemi, queueing network, transmission, ontrol Cite This Artile: Ikeme John Dike, and Chinyere Ogohukwu Dike, Integration of Queueing Network into Suseptible Exposed Likely Infeted Confirmed Dead Removed Model for Ebola Virus Disease Analysis Amerian Journal of Applied Mathematis and Statistis, vol 6, no (08): doi: 069/ajams-6--3 Introdution Ebola Virus Disease has unique transmission phases, namely: Suseptible, Exposed, Likely Infeted, Confirmed Infeted, Dead, Removed-SEI L I C DR [6] Contagious disease like Ebola transmits through networks, made by bodily interations among persons Nevertheless, EVD is a ontagious epidemi ase with high transmission rate one an outbreak ours Hene, open queueing network is appliable in EVD epidemi, sine open queueing networks are systems in whih infeted person are able to enter and depart the system [] This an be justified from the developed SEI L I C DR model whih is an open queueing network Infeted individual move from Suseptible to Exposed, to Likely Infeted, to Confirm Infeted and finally either Die or Reover The transmission of EVD is a system with a random, unsequential and multiple disorder pattern This paper applies the priniples of multiple hannel open Jakson queueing networks to desribe the EVD transmission dynamis The network involves the multiple ontat stations An urban ontat network offers a high degree of realism, whih involves stations like households, shopping entres, religious entres, shools, workplaes, hospitals and so on in a given ommunity Queueing theory is the mathematial study of waiting lines, or queues [4] As a result of the transmission mode and deadly nature of the disease, this paper inoporates network into the SEI L I C DR model that onsiders all the transmission phases in order to understand the real nature of the disease and predit its behaviour Network Model SEI L I C DR Queueing Networks The priniple of queueing networks assumes that the treatment of an individual in an epidemi transmission/ontrol system is not omplete until the individual has passed through all the available stages or states [] For instane, in SEI L I C DR model an individual has to pass through the S E IL IC D/ R stages The transmission of EVD is a system with a random, unsequential, and multiple disorder patterns Therefore, multiple hannel open Jakson queueing networks was applied in this paper

55 Amerian Journal of Applied Mathematis and Statistis Multiple Channel Open Jakson Queueing Networks Interonneted olletion of stations is defined as queueing networks, that is, inventions with

ommuniation systems Jakson []; Gordon and Newell [7] indiate that open and losed networks with one job lass, exponential servie time distributions and First Come First Served (FCFS) sheduling in all

2 55 Amerian Journal of Applied Mathematis and Statistis Multiple Channel Open Jakson Queueing Networks Interonneted olletion of stations is defined as queueing networks, that is, inventions with queues, in whih there is movement of jobs from one station to the subsequent station demanding servie Over the last five deades queueing networks have been known as models of manufaturing omputer and ommuniation systems Jakson []; Gordon and Newell [7] indiate that open and losed networks with one job lass, exponential servie time distributions and First Come First Served (FCFS) sheduling in all stations have produt form solution Multiple job lasses in open, mixed and losed networks is an extension by Kelly [3] Baskett et al [] give produt form solutions for servie requirement distributions that have rational Laplae transforms, denoted by stages Multiple job lasses an be suitable, and the servie neessity distributions an rely on the lass Multitask performane is treated by single hannel, serial proessing models as an issue of task seletion and sheduling The multiple hannel, parallel proessing models in ontrast, treat multitask performane as an issue of parallel allotment and splition of proessing resoures among onurrent jobs [6] 3 Contat Network Contat exists in designated stations A ontat network is a group of persons who an all rapidly get in touh with one another It does not have assurane of any type of assistane or aess to knowledge Contat network models endeavor to depit eah interative ontat that an possibly lead to disease spread in a ommunity [7] In this paper, the urban network was used in the ontat network analysis, sine it offers a high degree of realism 4 Urban Contat Network An urban ontat network involves stations like households, shopping entres, religious entres, shools, workplaes, hospitals, and so on in a given ommunity In the ommunity, every individual whih serves as a node or a vertex in the ontat network denotes an individual host An edge or line between two nodes denotes an interation (that has happened at any point in time for any period) through whih disease spread The number of edges (ie ontats) is the node s degree onneted to the node [7] Node degree is a fundamental quantity in network theory Figure shows individuals ontat network in an urban setting Figure indiates that all individuals emanate from household That is individuals from shools, hospitals, workplaes, shopping entres and religious entres originate from households Quantifiation of degree of possible ontat among individuals in the above plaes ould enhane the knowledge of EVD transmission The assumptions that were employed to quantify the degree of ontat are desribed in the next paragraph Figure Urban Contat Network

3 Amerian Journal of Applied Mathematis and Statistis 56 A redible ontat network was generated applying omputer simulations in this study for an urban loation The data for the ountry of Liberia was used for simulation However, households were hosen at random from Liberia household distribution of size N=000 [], whih generated 5000 persons The ages of members of households are assigned in aordane with Liberia measured age struture, and then alloated to shools in aordane with shool and lass distributions size [4]; to oupations in aordane with unemployment data [5], to shopping entres in aordane with business transations; to hospitals inluding patients as well as aregivers in aordane with hospital employment as well as bed data [4, 0, 8]; and to religious entres in aordane with data derived from [] The above assumptions are also appliable to Guinea and Sierra Leone respetively [8, 9,, 3] Moreover, in eah loation, a random onnetion was reated between persons with probabilities of for households, 035 for shools, 05 for hospitals, 040 for shopping entres, 08 for workplaes and 098 for religious entres Eah shool or hospital or shopping entre or workplae or religious entre is shared into smaller lassrooms or wards or setions or departments or types Pairs of students or patients or people within these sub-units were related with probability that is higher than pairs related with sub-units that are not the same Teahers are alloated to lassrooms and related stohastially to suitable students Caregivers are alloated wards and at this point related to patients appropriately Shopping entres are alloated to different setions Workplaes are alloated to different departments and religion is alloated to different types or groups The probability neighborhood ontats between persons from various households are also low This network is quite omplex but it offers a high degree of realism Dots denote nodes [5] That is, persons and lines between dots denote ontats between persons that possibly, ould lead to spread of disease The rate of ontat of individuals in the designated stations with respet to population is shown in Figure, while the likelihood of transmission of EVD by ontat in the designated plaes is shown in Figure 3 It indiates that the transmission rate dereases in this order: household, religious entre, shopping entre, shool, workplae and hospital The finding supports the idea that all EVD exposed individuals should be gathered in the hospital (Quarantine ) sine the possibility of spread of the disease is lowest in the hospital Also, the finding is in line with the submission of Chowell et al [3] that the use of Quarantine for observation and treatment of EVD affeted individuals is a dependable ontrol measure of the disease outbreak Population (Millions) Guinea Liberia Sierra Leone Household Shool Hospital Shopping Plaes Work Plae Religious Figure Degree of Contat in the Network Stations Probability Guinea Liberia Household Shool Hospital Shopping Plaes Work Plae Religious Figure 3 Probability Degree of Contat in the Network Stations

4 57 Amerian Journal of Applied Mathematis and Statistis 3 Transmissibility of EVD Transmissibility of a disease T is the mean probability that an infetious person will transmit the disease to a suseptible person with whom they have ontat Transmissibility T reaps main faets of transmission of disease as well as the rate at whih ontats take plae between persons, the probability that a ontat will lead to transmission, the duration of the infetious period, and the suseptibility of persons to EVD Critial transmissibility or epidemi threshold T C is the value of transmissibility whih a population is vulnerable to large sale epidemis when the basi reprodutive number R 0 is Aording to Meyers et al [7]; Brauer and Castillo-Chavez [0], the basi reprodution number and epidemi threshold is given by R0 = T TC = () () where is the mean degree and is the mean square degree of the network T C is also alled the least transmissibility ( T ) neessary for an outbreak to turn into a large sale epidemi Cumulative degree distribution against degree of ontat for urban ontat network is shown in Figure 4 The figure desribes the number of ontats (degree) speified on the horizontal axis, whih a randomly hosen individual (vertex) possesses Figure 4 Cumulative Degree Distribution for Urban Contat Network 4 Epidemiologial Analysis of EVD Probability Generating Funtions (PGF), on the other hand is used to foreast the fate of an outbreak quantity that depits probability distributions Aording to Meyers [7] PGF for a degree distribution is defined as G0 ( h) ph = = (3) The probability of attaining a node or vertex of degree, or exess degree ( ), by following a random edge is proportional to and thus, the probability that a vertex at the end of a random edge has exess degree ( ) is a onstant multiple of p with the onstant hosen to make the sum over of the probabilities equal to Then, the probability that a vertex has exess degree ( ) is G ( h) = = = p h p (4) The mean degree equals the derivatives of these terms at h = respetively are and mean exess degree ( ) = p = = (5) p = e = = ( ) p (6) Newman [9] derived the value of the epidemi threshold T C, the foreasted mean outbreak size e and epidemi probability E A PGF an be generated for the size of the outbreak through enlosing PGF for new infetions number originating from the node or vertex that are infeted, and thus, the mean size of an outbreak ( ) ( ) T e = + (7) T e Equ 7 differs one an outbreak turns out to be a large-sale epidemi and it is graded by epidemi threshold T C The probability of a single infetion is onsidered, whih will lead to merely an outbreak (initial value) instead of a full-blown epidemi Therefore, the fullblown epidemi probability E is obtained by subtrating the initial value from one E = p ( + ( l ) T) = (8) The individual at the end of an edge or line does not have the disease, l is the probability at that point and the result is given as l = = ( + ( ) ) p l T = p (9) Full-blown epidemi probability E and probability of a single infetion whih will lead to merely an outbreak l, is solved and these outomes extended to foreast the outome of an outbreak based on the number of

5 Amerian Journal of Applied Mathematis and Statistis 58 individuals that are infeted First person in a ommunity that is infeted is known as patient zero A patient zero with degree who will initiate an epidemi at a probability r, is the same as probability that spread of the disease along at least one of the edges originating from the new node or vertex will lead to an epidemi The probability that the disease does not get transmitted along the edge is T Tl is the probability that even if disease is transmitted to the sueeding node or vertex, it does not progress into a full-blown epidemi, for any one of its edges Hene ( ) r = T + Tl (0) The probability that an outbreak of size N will kindle N an epidemi is i ( r i ), where = i is the degree of person i It is done by subtrating one from the probability that none of the N infeted persons flikers an epidemi The good approximation for the probability of an epidemi is omputed likewise, if the number of reent inidents are known but not their ontat patterns with r s substituted with the probability that eah of the ( ) i a typial infeted person does not fliker an epidemi The number of edges via whih a type of infeted person an initiate an epidemi is given by the exess degree PGF, and the probability that any of those edges will not lead to an inrease in an epidemi is T + Tl () Hene, the probability that no edges will be a hannel to an epidemi is p T Tl = p = ( + ), () and the probability that an outbreak of size N flikers an epidemi is N p ( T + Tl) = p = (3) Lastly a person s risk of infetion for the period of an epidemi as a funtion of an individual degree was derived The probability f that a person with degree will turn out to be infeted for the period of an epidemi is equal to one minus the probability that no individual ontat will transmit the disease to an individual The probability that a ontat does not transmit the disease is the same to the probability that the ontat is infeted, but did not transmit the disease, ( l)( T) (4) in addition to the probability that the ontat was not infeted in the first plae, l Hene, a randomly seleted node or vertex of degree will turn out to be infeted with probability f = r = ( T + Tl) (5) Aordingly, mathematial expresssions above are integrated to generate probability of the network (households, religious entres, shopping entres, shools, workplaes and hospitals) infeted Epifire software was used to generate the mean degree in urban ontat network The values of transmissibility and their probabilities of network infeted for Guinea, Liberia and Sierra Leone are above ritial transmissibility (00644) and probability of network infeted (030) before intervention After intervention, the values redued whih are below ritial transmissibility and probability of network infeted as shown in Table Figure 5 and Figure 6 show the probability of network infeted for dissimilar values of transmissibility before and after intervention respetively The results indiate that the population was vulnerable to large sale epidemis before intervention in the three ountries The vulnerability dereased drastially after intervention Figure 5 Network simulation for transmissibility before intervention

6 59 Amerian Journal of Applied Mathematis and Statistis Figure 6 Network simulation for transmissibility after intervention Table Critial Transmissibility, Transmissibility and Probability of Network of Infeted Before Intervention After Intervention Country Transmissibility Probability of Network Infeted Guinea Liberia Sierra Leone Guinea Liberia Sierra Leone Critial Transmissibility = 00644, Probability = Conlusion The paper demonstrated the effetiveness of integrating queueing network into SEI L I C DR model for EVD The result for network indiates that the population was vulnerable to large sale epidemis before intervention in T for the three ountries was the three ountries The C Before intervention, the transmissibility for Guinea was 0365, Liberia, 0093 and Sierra Leone, 035 After intervention, the transmissibility for Guinea redued to 00595, Liberia, and Sierra Leone, 0057 The before and after transmissibility results in omparison with the T C indiate that intervention was effetive in ontaining the EVD epidemi The vulnerability dereased drastially after intervention Referenes [] Baskett, F, Chandy, KM, Muntz, RR, Palaios, FG: Open, Closed and Mixed Networks of Queues with Different Classes of Customers J of the Asso for Comput Mah (), (975) [] Central Intelligene Ageny CIA world fatbook (06) Retrieved nd January 07 [3] Chowell, G, Hengartner, NW, Castillo-Chavez, C, Fenimore, PW, Hyman, JM: The basi reprodution number of Ebola and the effets of publi health measures: the ases of Congo and Uganda J of Theoret Bio 9(), 9-6 (004) [4] CHRD: for Health Researh and Diagnostis Guinea wwwdwuapg>indexphp>entre (05) Retrieved nd January 07 [5] De Sterk, H, Miller, T, Manteuffel, T, Sanders, G: Top-level aeleration of adaptive Algebrai Multilevel Methods for Steady- State Solution to Markov Chains Advan Comput Math 35, (0) [6] Dike, CO, Zainuddin, ZM, Ikeme John Dike, IJ: Suseptible Infeted Removed Epidemi Model Extension for Effiient Analysis of Ebola Virus Disease Transmission Amerian Sientifi Publishers [7] Gordon, WJ, Newell, GF: Closed queueing systems with exponential servers Operations Researh 5(), (967) [8] Guinea Age Struture: wwwindexmundiom/guinea/age_struturehtml (06) Retrieved nd January 07 [9] Guinea Unemployment Rate: instant&ion=&espv=&ie=utf- 8#q=guinea+unemployment+rate (07) Retrieved nd January 07 [0] Hernandez-Suarez CM, Castillo-Chavez C, Lopez OM, and Hernandez-Cuevas K: An appliation of queueing theory to SIS and SEIS epidemi models Math Biosi and Eng 7(4), (00) [] International Religious Freedom Report: Liberia United States Department of State, November 7, 00 Retrieved Otober 9, 06 [] Jakson, JR: Jobshop-like queueing systems, management siene 0(l), 3-4 (963) [3] Kelly, F P: Networks of queues with ustomers of different types J of Appl Probab, (975) [4] Liberia Age Struture: wwwindexmundiom/liberia/age_struturehtml (06) Retrieved nd January 07

7 Amerian Journal of Applied Mathematis and Statistis 60 [5] Liberia Unemployment Rate: instant&ion=&espv=&ie=utf- 8#q=liberia+unemployment+rate (07) Retrieved nd January 07 [6] Liu, Y: Queueing networks as models of human performane and human-omputer interation Proeedings of the 994 Symposium on Human Interation with Complex Systems Department of Industrial and Operations Engineering, University of Mihigan, USA Tehnial Report [7] Meyers, LA, Pourbohloul, B, Newman, MEJ, Skowronski, DM, Brunham, RC: Network theory and SARS: Preditng outbreak diversity J of Theoret Bio 3(), 7-8 (005) [8] MIDC: Establishment of Medial Imaging and Diagnosti in Freetown wwwrvonl>projeten>establishment (009) Retrieved nd January 07 [9] Newman, MEJ (00) The spread of epidemi disease on networks Phys Review E 66, 068 [0] NPHIL: National Publi Health Institute of Liberia wwwianphiorg>newprofiles>liberia (04) Retrieved nd January 07 [] Ross, SM Introdution to probability models (9 th ed) Aademi Press, Elsevier In: USA (007) [] Sierra Leone Age Struture: wwwindexmundiom/sierra_leone/age_struturehtml (06) Retrieved nd January 07 [3] Sierra Leone Unemployment Rate: instant&ion=&espv=&ie=utf- 8#q=sierra+leone+unemployment+rate (07) Retrieved nd January 07 [4] Sztrik, J: Basi Queueing Theory (0) Retrieved September, 05

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