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Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2016, Article ID 9352725, 29 pages http://dx.doi.org/10.1155/2016/9352725 Research Article A Mathematical Model with Quarantine States for the Dynamics of Ebola Virus Disease in Human Populations 1 2 Gideon A. Ngwa and Miranda I. Teboh-Ewungkem Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA Correspondence should be addressed to Miranda I. Teboh-Ewungkem; mit703@lehigh.edu Received 12 January 2016; Revised 30 May 2016; Accepted 8 June 2016 Academic Editor: Chen Yanover Copyright © 2016 G. A. Ngwa and M. I. Teboh-Ewungkem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A deterministic ordinary differential equation model for the dynamics and spread of Ebola Virus Disease is derived and studied. The model contains quarantine and nonquarantine states and can be used to evaluate transmission both in treatment centres and in the community. Possible sources of exposure to infection, including cadavers of Ebola Virus victims, are included in the model derivation and analysis. Our model’s results show that there exists a threshold parameter,𝑅 , with the property that when its value is above unity, an endemic equilibrium exists whose value and size are determined by the size of this threshold parameter, and when its value is less than unity, the infection does not spread into the community. eTh equilibrium state, when it exists, is locally and asymptotically stable with oscillatory returns to the equilibrium point. The basic reproduction number, 𝑅 ,isshown to be strongly dependent on the initial response of the emergency services to suspected cases of Ebola infection. When intervention measures such as quarantining are instituted fully at the beginning, the value of the reproduction number reduces and any further infections can only occur at the treatment centres. Eeff ctive control measures, to reduce 𝑅 to values below unity, are discussed. 1. Introduction and Background The Ebola Virus Disease (EVD), formally known as Ebola haemorrhagic fever and caused by the Ebola Virus, is very eTh worldhas been rivetedbythe 2014 outbreak of theEbola lethal with case fatalities ranging from 25% to 90%, with a Virus Disease (EVD) that aeff cted parts of West Africa with mean of about 50% [2]. The 2014 EVD outbreak, though not Guinea, Liberia, and Sierra Leone being the most hard hit the rfi st but one of many other EVD outbreaks that have areas. Isolated cases of the disease did spread by land to Sene- occurred in Africa since the first recorded outbreak of 1976, is gal and Mali (localized transmission) and by air to Nigeria. theworst in termsofthe numbersofEbola casesand related Some Ebola infected humans were transported to the US deaths and the most complex [2]. About 9 months after the (except the one case that traveled to Texas and later on died) identification of a mysterious killer disease killing villagers in and other European countries for treatment. An isolated case a small Guinean village as Ebola, the 2014 West African Ebola occurred in Spain, another in Italy (a returning volunteer outbreak,asofDecember24, 2014,had up to 19497Ebola health care worker), and a few cases in the US and the UK cases resulting in 7588 fatalities [1, 3, 4], a case fatality rate of [1–3]. Though dubbed the West African Ebola outbreak, the about38.9%.ByDecember2015, thenumberofEbola Virus movement of patients and humans between countries, if not cases (including suspected, probable, and confirmed) stood at handled properly, could have led to a global Ebola pandemic. 28640 resulting in 11315 fatalities, a case fatality rate of 39.5% There was also a separate Ebola outbreak aeff cting a remote [3, 5]. region in the Democratic Republic of Congo (formerly Zaire), and it was only by November 21, 2014 that the outbreak was Ebola Virus, the agent that causes EVD, is hypothesised to be introduced into the human population through contact reported to have ended [2]. 2 Computational and Mathematical Methods in Medicine with the blood, secretions, u fl ids from organs, and other symptoms,and also theeeff ctiveprotectionbythe patient’s body parts of dead or living animals infected with the virus immune response [7]. Some of the disease management (e.g., fruit bats, primates, and porcupines) [2, 6]. Human-to- strategies include hydrating patients by administering intra- human transmission can then occur through direct contact venous u fl ids and balancing electrolytes and maintaining the (via broken skin or mucous membranes such as eyes, nose, patient’s blood pressure and oxygen levels. Other schemes or mouth) with Ebola Virus infected blood, secretions, and used include blood transfusion (using an Ebola survivor’s fluids secreted through organs or other body parts, in, for blood) and the use of experimental drugs on such patients example, saliva, vomit, urine, faeces, semen, sweat, and breast (e.g., ZMAPP whose safety and efficacy have not yet milk. Transmission can also be as a result of indirect contact been tested on humans). eTh re are some other promising with surfaces and materials, in, for example, bedding, cloth- drugs/vaccines under trials [2]. Studies show that once a ing, andfloorareas,orobjects such as syringes,contaminated patient recovers from EVD they remain protected against the with the aforementioned u fl ids [2, 6]. diseaseand areimmunetoitatleast foraprojectedperiod When a healthy human (considered here to be suscepti- because they develop antibodies that last for at least 10 years ble) who has no Ebola Virus in them is exposed to the virus [7]. Once recovered, lifetime immunity is unknown or (directly or indirectly), the human may become infected, if whether a recovered individual can be infected with another transmission is successful. The risk of being infected with the Ebola strain is unknown. However, aeft r recovery, a person Ebola Virus is (i) very low or not recognizable where there is can potentially remain infectious as long as their blood and casual contact with a feverish, ambulant, self-caring patient, body u fl ids, including semen and breast milk, contain the for example, sharing the same public place, (ii) low where virus. In particular, men can potentially transmit the virus there is close face-to-face contact with a feverish and ambu- through their seminal u fl id, within the rfi st 7 to 12 weeks lant patient, for example, physical examination, (iii) high aer ft recovery from EVD [2]. Table 1 shows the estimated wherethere is closeface-to-facecontact withoutappropriate time frames and projected progression of the infection in an personal protective equipment (including eye protection) average EVD patient. with a patient who is coughing or vomiting, has nose- Given that there is no approved drug or vaccine out bleeds, or hasdiarrhea, and(iv)veryhighwhere thereis yet, local control of the Ebola Virus transmission requires percutaneous, needle stick, or mucosal exposure to virus- a combined and coordinated control eor ff t at the individual contaminated blood, body u fl ids, tissues, or laboratory speci- level, the community level, and the institutional/health/gov- mens in severely ill or known positive patients. eTh symptoms ernment level. Institutions and governments need to educate of EVD may appear during the incubation period of the the public and raise awareness about risk factors, proper hand disease, estimated to be anywhere from 2 to 21 days [2, 7– washing, proper handling of Ebola patients, quick reporting 9], with an estimated 8- to 10-day average incubation period, of suspected Ebola cases, safe burial practices, use of public although studies estimate it at 9–11 days for the 2014 EVD transportation, and so forth. eTh se education eoff rts need outbreak in West Africa [10]. Studies have shown that, during to be communicated with community/chief leaders who are theasymptomaticpartofthe EbolaVirus Disease, ahuman trusted by members of the communities they serve. From a infected with the virus is not infectious and cannot transmit global perspective, a good surveillance and contact tracing thevirus.However,withthe onsetofsymptoms, thehuman program followed by isolation and monitoring of probable can transmit the virus and is hence infectious [2, 7]. eTh onset and suspected cases, with immediate commencement of of symptoms commences the course of illness of the disease disease management for patients exhibiting symptoms of whichcan lead to death6–16dayslater [8,9]orimprovement EVD, is important if we must, in the future, elude a global of health with evidence of recovery 6–11 days later [8]. epidemic and control of EVD transmission locally and In the rst fi few days of EVD illness (estimated at days 1–3 globally [2]. It was by eeff ctive surveillance, contact tracing, [11]), a symptomatic patient may exhibit symptoms common and isolation and monitoring of probable and suspected cases to those like the malaria disease or the u fl (high fever, followed by immediate supportive care for individuals and headache, muscle and joint pains, sore throat, and general families exhibiting symptoms that the EVD was brought weakness). Without eeff ctive disease management, between under control in Nigeria [17], Senegal, USA, and Spain [1]. days 4 and 5 to 7, the patient progresses to gastrointestinal Efficient control and management of any future EVD symptoms such as nausea, watery diarrhea, vomiting, and outbreaks can be achieved if new, more economical, and real- abdominal pain [10, 11]. Some or many of the other symp- izable methods are used to target and manage the dynamics toms, such as low blood pressure, headaches, chest pain, of spread as well as the population sizes of those communities shortness of breath, anemia, exhibition of a rash, abdominal that may be exposed to any future Ebola Virus Disease pain, confusion, bleeding, and conjunctivitis, may develop outbreak. More realistic mathematical models can play a role [10, 11] in some patients. In the later phase of the course of in this regard, since analyses of such models can produce clear the illness, days 7–10, patients may present with confusion insight to vulnerable spots on the Ebola transmission chain and may exhibit signs of internal and/or visible bleeding, where control eor ff ts can be concentrated. Good models progressing towards coma, shock, and death [10, 11]. could also help in the identification of disease parameters that Recovery from EVD can be achieved, as evidenced by the can possibly influence the size of the reproduction number less than 50% fatality rate for the 2014 EVD outbreak in West of EVD. Existing mathematical models for Ebola [14, 16, 18– Africa. With no known cure, recovery is possible through 21] have been very instrumental in providing mathematical effective disease management, the treatment of Ebola-related insight into the dynamics of Ebola Virus transmission. Many Computational and Mathematical Methods in Medicine 3 Table 1: A possible progression path of symptoms from exposure to the Ebola Virus to treatment or death. Table shows a suggested transition and time frame in humans, of the virus, from exposure to incubation to symptoms development and recovery or death. This table is adapted based on the image in the Huffington post, via [11]. Superscript a: for the 2014 epidemic, the average incubation period is reported to be between 9 and 11 days [10]. Superscript b: other studies reported a mean of 4–10 days [8, 9]. Incubation Course of illness Recovery or death period Exposure Range: 6to16daysfromthe endofthe incubation Range: 2 to 21 period Recovery: by the Death: by the end days from end of days 6–11 of days 6–16 Probable Early symptomatic Late symptomatic exposure Days 1–3 Days 4–7 Days 7–10 Patients progress to gastrointestinal symptoms: for An individual example, nausea, Patients may comes in watery diarrhea, Average of 8–11 present with contact with an vomiting, and days before confusion and may Ebola infected Patients exhibit abdominal pain. symptoms are exhibit signs of Some patients may recover, while individual (dead malaria-like or Other symptoms evident. internal and/or others will die. or alive) or have flu-like symptoms: may include low Another visible bleeding, Recovery typically requires early been in the for example, fever blood pressure, estimate reports potentially intervention. vicinity of and weakness. anemia, headaches, an average of progressing someone who chest pain, 4–10 days. towards coma, has been shortness of breath, shock, and death. exposed. exhibition of a rash, confusion, bleeding, and conjunctivitis. of these models have also been helpful in that they have the basic reproduction number of Ebola that depends on the provided methods to derive estimates for the reproduction disease parameters. number for Ebola based on data from the previous outbreaks. The rest of the paper is divided up as follows. In Section 2, However, few of the models have taken into account the fact we outline the derivation of the model showing the state that institution of quarantine states or treatment centres will variables and parameters used and how they relate together aeff ct the course of the epidemic in the population [16]. It is in a conceptual framework. In Section 3, we present a our understanding that the way the disease will spread will be mathematical analysis of the derived model to ascertain that determined by the initial and continual response of the health theresults arephysicallyrealizable. We then reparameterise services in the event of the discovery of an Ebola disease the model and investigate the existence and linear stability of case. eTh objective of this paper is to derive a comprehensive steady state solution, calculate the basic reproduction num- mathematical model for the dynamics of Ebola transmission ber, and present some special cases. In Section 4, we present taking into consideration what is currently known of the adiscussiononthe parameters of themodel.InSection 5, we disease. The primary objective is to derive a formula for the carry out some numerical simulations based on the selected reproduction number for Ebola Virus Disease transmission feasible parameters for the system and then round up the in view of providing a more complete and measurable index paper with a discussion and conclusion in Section 6. for the spread of Ebola and to investigate the level of impact of surveillance, contact tracing, isolation, and monitoring of 2. The Mathematical Model suspected cases, in curbing disease transmission. The model is formulated in a way that it is extendable, with appropriate 2.1. Description of Model Variables. We divide the human modifications, to other disease outbreaks with similar char- population into 11 states representing disease status and acteristics to Ebola, requiring such contact tracing strategies. quarantine state. At any time𝑡 there are the following. Ourmodel dieff rs from othermathematicalmodelsthathave (1) Susceptible Individuals. Denoted by 𝑆 ,thisclass also been used to study the Ebola disease [14, 15, 18, 20–22] in that includes false probable cases, that is, all those individuals who it captures the quarantined Ebola Virus Disease patients and wouldhavedisplayed earlyEbola-likesymptomsbut who provides possibilities for those who escaped quarantine at the eventually return a negative test for Ebola Virus infection. onset of the disease to enter quarantine at later stages. To the best of our knowledge, this is the rfi st integrated ordinary (2) Suspected Ebola Cases. eTh class of suspected EVD differential equation model for this kind of communicable patients comprises those who have come in contact with, or diseaseofhumans. Ourfinalresultwould be aformula for been in the vicinity of, anybody who is known to have been 4 Computational and Mathematical Methods in Medicine sick or died of Ebola. Individuals in this class may or contact [7]. eTh refore, the cycle of infection really stops only may not show symptoms. Two types of suspected cases are when a cadaver is properly buried or cremated. us Th mem- included: the quarantined suspected cases, denoted by 𝑆 , bers from class, 𝑅 , representing dead bodies or cadavers 𝑄 𝐷 and the nonquarantined suspected case, denoted by𝑆 .Thus of EVD victims are considered removed from the infection a suspected case is either quarantined or not. chain, and consequently from the system, only when they have been properly disposed of. eTh class, 𝑅 ,ofindividuals (3) Probable Cases. eTh class of probable cases comprises all who beat the odds and recover from their infection are those persons who at some point were considered suspected considered removed because recovery is accompanied with cases and who now present with fever and at least three other the acquisition of immunity so that this class of individuals early Ebola-like symptoms. Two types of probable cases are are then protected against further infection [7] and they no included: the quarantined probable cases, denoted by 𝑃 , longer join the class of susceptible individuals. eTh third and the nonquarantined probable cases,𝑃 .Thus aprobable type of removal is obtained by considering individuals who case is either quarantined or not. Since the early Ebola-like die naturally or due to other causes other than EVD. es Th e symptoms of high fever, headache, muscle and joint pains, individuals are counted as𝑅 . sore throat, and general weakness can also be a result of other eTh statevariables aresummarizedinNotations. infectious diseases such as malaria or u fl , we cannot be certain at this stage whether or not the persons concerned have 2.2. eTh Mathematical Model. A compartmental framework Ebola infection. However, since the class of probable persons is used to model the possible spread of EVD within a is derived from suspected cases, and to remove the uncer- population. eTh model accounts for contact tracing and quar- tainties, we will assume that probable cases may eventually antining, in which individuals who have come in contact or turn outtobeEVD patients andifthatweretobethe case, have been associated with Ebola infected or Ebola-deceased since they are already exhibiting some symptoms, they can humans are sought and quarantined. They are monitored for be assumedtobemildlyinfectious. twenty-one days during which they may exhibit signs and (4) Conrfi med Early Symptomatic Cases .Theclass of con- symptoms of the Ebola Virus or are cleared and declared free. rfi med early asymptomatic cases comprises all those persons We assume that most of the quarantining occurs at designated who at some point were considered probable cases and a makeshift, temporal, or permanent health facilities. However, conrfi matory laboratory test has been conducted to conrfi m it has been documented that others do not get quarantined, that there is indeed an infection with Ebola Virus. This class is becauseoffearofdying withoutalovedone near them or fear called confirmed early symptomatic because all that they have that if quarantined they may instead get infected at the centre, as symptoms are the early Ebola-like symptoms of high fever, as well as traditional practices and belief systems [14, 16, 22]. headache, muscle and joint pains, sore throat, and general u Th s, there may be many within communities who remain weakness. Two types of conrfi med early symptomatic cases nonquarantined, and we consider these groups in our model. are included: the quarantined confirmed early symptomatic In all the living classes discussed, we will assume that natural cases 𝐶 and the nonquarantined confirmed early symp- death, or death due to other causes, occurs at constant rate𝜇 tomatic cases 𝐶 . us Th a confirmed early symptomatic case where1/𝜇 is approximately the life span of the human. is either quarantined or not. eTh class of confirmed early symptomatic individuals may not be very infectious. 2.2.1. eTh Susceptible Individuals. The number of susceptible (5) Confirmed Late Symptomatic Cases .Theclassofconrfi med individuals in the population decreases when this population late symptomatic cases comprises all those persons who at is exposed by having come in contact with or being associated some point were considered confirmed early symptomatic with any of the possibly infectious cases, namely, infected cases and in addition the persons who now present with most probable case, conrfi med case, or the cadaver of a conrfi med or all of the later Ebola-like symptoms of vomiting, diarrhea, case. eTh density increases when some false suspected indi- stomach pain, skin rash, red eyes, hiccups, internal bleeding, viduals (a proportion of1−𝜃 of nonquarantined and1−𝜃 2 6 and external bleeding. Two types of confirmed late symp- of quarantined) and probable cases (a proportion of1−𝜃 tomatic cases are included: the quarantined confirmed late of nonquarantined individuals and 1−𝜃 of quarantined symptomatic cases𝐼 and the nonquarantined confirmed late 𝑄 individuals) are eliminated from the suspected and probable symptomatic cases 𝐼 .Thus aconrfi medlatesymptomatic 𝑁 case list. We also assume a constant recruitment rateΠ as well case is either quarantined or not. eTh class of confirmed as natural death, or death due to other causes. er Th efore the late symptomatic individuals may be very infectious and any equation governing the rate of change with time within the bodily secretions from this class of persons can be infectious class of susceptible individuals may be written as to other humans. =Π−𝜆𝑆+(1−𝜃 )𝛽𝑃 +(1−𝜃)𝛼𝑆 (6) Removed Individuals. Three types of removals are consid- 3 𝑁 𝑁 2 𝑁 𝑁 (1) ered,but only twoare relatedtoEVD.Theremovalsrelated to the EVD are confirmed individuals removed from the system +(1−𝜃)𝛼𝑆 +(1−𝜃)𝛽𝑃 −𝜇,𝑆 6 𝑄 𝑄 7 𝑄 𝑄 through disease induced death, denoted by𝑅 ,orconrfi med cases that recover from the infection denoted by 𝑅 .Now,it where𝜆 is the force of infection and the rest of the parameters is known that unburied bodies or not yet cremated cadavers are positive and are defined in Notations. We identify two of EVDvictims caninfectother susceptiblehumansupon types of total populations at any time 𝑡 :(i) thetotal living 𝑑𝑡 𝑑𝑆 Computational and Mathematical Methods in Medicine 5 population,𝐻 , and (ii) the total living population including In the context of this model we make the assumption the cadavers of Ebola Virus victims that can take part in the that once quarantined, the individuals stay quarantined until spread of EVD,𝐻 .Thus at each time 𝑡 we have clearance and are released, or they die of the infection. Notice that𝜃 =𝜃 +𝜃 ,sothat1−𝜃 +𝜃 +𝜃 =1. 2 2𝑎 2𝑏 2 2𝑎 2𝑏 𝐻 ( 𝑡 )=(𝑆+𝑆 +𝑆 +𝑃 +𝑃 +𝐶 +𝐶 +𝐼 𝐿 𝑁 𝑄 𝑁 𝑄 𝑁 𝑄 𝑁 (2) +𝐼 +𝑅 ) ( 𝑡 ) , 2.2.3. eTh Probable Cases. The fractions 𝜃 and 𝜃 of sus- 2 6 𝑄 𝑅 pected cases that become probable cases increase the number 𝐻 ( 𝑡 )=(𝑆+𝑆 +𝑆 +𝑃 +𝑃 +𝐶 +𝐶 +𝐼 +𝐼 𝑁 𝑄 𝑁 𝑄 𝑁 𝑄 𝑁 𝑄 of individuals in the probable case class. eTh population of (3) probable cases is reduced (at rates 𝛽 and 𝛽 )whensome 𝑁 𝑄 +𝑅 +𝑅 ) ( 𝑡 ) . 𝑅 𝐷 of these are conrm fi ed to have the Ebola Virus through laboratory tests at rates 𝛼 and 𝛼 .For some,proportions 𝑁 𝑄 Since the cadavers of EVD victims that have not been 1−𝜃 and1−𝜃 , the laboratory tests are negative and the properly disposed of are very infectious, the force of infection 3 7 probable individuals revert to the susceptible class. From the must then also take this fact into consideration and be proportion 𝜃 of nonquarantined probable cases whose tests weighted with 𝐻 instead of 𝐻 . eTh force of infection takes are positive for the Ebola Virus (i.e., conrfi med for EVD), a the following form: fraction, 𝜃 , become quarantined while the remainder, 𝜃 , 3𝑏 3𝑎 remain nonquarantined. So𝜃 =𝜃 +𝜃 .Thus theequation 3 3𝑎 3𝑏 𝜆= (𝜃𝜌 𝑃 +𝜃 𝜌 𝑃 +𝜏 𝐶 +𝜉 𝐼 +𝜏 𝐶 3 𝑁 𝑁 7 𝑄 𝑄 𝑁 𝑁 𝑁 𝑁 𝑄 𝑄 𝐻 governing the rate of change within the classes of probable (4) cases takes the following form: +𝜉 𝐼 +𝑎 𝑅 ), 𝑄 𝑄 𝐷 𝐷 where 𝐻>0 is defined above and the parameters 𝜌 , 𝑁 𝑁 =𝜃 𝛼 𝑆 −(1−𝜃)𝛽𝑃 −𝜃 𝛽 𝑃 2𝑎 𝑁 𝑁 3 𝑁 𝑁 3𝑎 𝑁 𝑁 𝜌 , 𝜏 , 𝜏 , 𝜉 , 𝜉 ,and 𝑎 are positive constants as den fi ed 𝑄 𝑁 𝑄 𝑁 𝑄 𝐷 in Notations. eTh re are no contributions to the force of −𝜃 𝛽 𝑃 −𝜇𝑃 , 3𝑏 𝑁 𝑁 𝑁 infection from the 𝑅 class because it is assumed that (6) once a person recovers from EVD infection, the recovered individual acquires immunity to subsequent infection with =𝜃 𝛼 𝑆 +𝜃 𝛼 𝑆 −(1−𝜃)𝛽𝑃 −𝜃 𝛽 𝑃 2𝑏 𝑁 𝑁 6 𝑄 𝑄 7 𝑄 𝑄 7 𝑄 𝑄 the same strain of the virus. Although studies have suggested that recovered men can potentially transmit the Ebola Virus −𝜇𝑃 . through seminal u fl ids within the first 7–12 weeks of recovery [2], and mothers through breast milk, we assume, here, that, 2.2.4. eTh Conrfi med Early Symptomatic Cases. The fractions with education, survivors who recover would have enough 𝜃 and𝜃 oftheprobablecasesbecomeconrfi medearlysymp- 3 7 information to practice safe sexual and/or feeding habits to tomatic cases thus increasing the number of confirmed cases protect their loved ones until completely clear. u Th s recovered with early symptoms. eTh population of early symptomatic individuals are considered not to contribute to the force of individuals is reduced when some recover at rates 𝑟 for infection. the nonquarantined cases and 𝑟 for the quarantined cases. Others may see their condition worsening and progress and 2.2.2. eTh Suspected Individuals. A fraction 1−𝜃 of the become late symptomatic individuals, in which case they exposed susceptible individuals get quarantined while the enterthe full blownlatesymptomatic stages of thedisease. remaining fraction are not. Also, a fraction𝜃 (resp.,𝜃 )ofthe 2 6 We assume that this progression occurs at rates 𝛾 or 𝛾 , 𝑁 𝑄 nonquarantined (resp., quarantined) suspected individuals respectively,whicharethereciprocalofthemeantimeittakes become probable cases at rate 𝛼 while the remainder 1− for the immune system to either be completely overwhelmed 𝜃 (resp., 1−𝜃 )donot developintoprobablecases and 2 6 by the virus or be kept in check via supportive mechanism. return to the susceptible pool. For the quarantined indi- A fraction 1−𝜃 of the conrfi med nonquarantined early viduals, we assume that they are being monitored, while symptomatic cases will be quarantined as they become late the suspected nonquarantined individuals are not. However, symptomatic cases, while the remaining fraction 𝜃 escape as they progress to probable cases (at rates 𝛼 and 𝛼 ), 𝑁 𝑄 quarantine due to lack of hospital space or fear and belief a fraction 𝜃 of thesehumanswillseekthe health care 2𝑏 customs [16, 22] but become conrfi med late symptomatic services as symptoms commence and become quarantined cases in the community. u Th s the equation governing the while the remainder 𝜃 remain nonquarantined. us Th the 2𝑎 rate of change within the two classes of confirmed early equation governing the rate of change within the two classes symptomatic cases takes the following form: of suspected persons then takes the following form: 𝑑𝐶 𝑁 𝑁 =𝜃 𝜆𝑆−(1−𝜃 )𝛼𝑆 −𝜃 𝛼 𝑆 −𝜃 𝛼 𝑆 =𝜃 𝛽 𝑃 −𝑟 𝐶 −𝜃 𝛾 𝐶 1 2 𝑁 𝑁 2𝑎 𝑁 𝑁 2𝑏 𝑁 𝑁 3𝑎 𝑁 𝑁 𝑁 4 𝑁 𝑁 −𝜇𝑆 , (5) −(1−𝜃)𝛾𝐶 −𝜇𝐶 , (7) 𝑁 4 𝑁 𝑁 𝑁 𝑑𝐶 =(1−)𝜆 𝜃 𝑆−(1−)𝛼 𝜃 𝑆 −𝜃 𝛼 𝑆 −𝜇𝑆 . =𝜃 𝛽 𝑃 +𝜃 𝛽 𝑃 −𝑟 𝐶 −𝛾 𝐶 −𝜇𝐶 . 1 6 𝑄 𝑄 6 𝑄 𝑄 𝑄 3𝑏 𝑁 𝑁 7 𝑄 𝑄 𝑄 𝑄 𝑄 𝑄 𝑑𝑡 𝑑𝑡 𝐸𝑄 𝑑𝑆 𝑑𝑡 𝑑𝑡 𝐸𝑁 𝑑𝑆 𝐸𝑄 𝐸𝑁 𝑑𝑡 𝑑𝑃 𝑑𝑡 𝑑𝑃 6 Computational and Mathematical Methods in Medicine 2.2.5. The Confirmed Late Symptomatic Cases. The frac- disposed at rate 𝑏 . es Th e collection classes satisfy the equa- tions 𝜃 and 𝜃 of conrfi med early symptomatic cases who tions 4 8 progress to the late symptomatic stage increase the number of confirmed late symptomatic cases. eTh population of late symptomatic individuals is reduced when some of these =𝜇𝐻 , individuals are removed. Removal could be as a result of (10) recovery at rates proportional to 𝑟 and 𝑟 or as a result of death because the EVD patient’s conditions worsen and =𝑏𝑅 . the Ebola Virus kills them. eTh death rates are assumed proportional to 𝛿 and 𝛿 . Additionally, as a control eo ff rt 𝑁 𝑄 or a desperate means towards survival, some of the nonquar- Putting all the equations together we have antined late symptomatic cases are removed and quarantined at rate 𝜎 . In our model, we assume that Ebola-related death only occurs at the late symptomatic stage. Additionally, we assume that the conrfi med late symptomatic individuals ̃ ̃ ̃ =Π−𝜆𝑆+ 𝜃 𝛽 𝑃 +𝜃 𝛼 𝑆 +𝜃 𝛼 𝑆 3 𝑁 𝑁 2 𝑁 𝑁 6 𝑄 𝑄 who are eventually put into quarantine at this late period (11) (removingthemfromthe community) maynot have long to +𝜃 𝛽 𝑃 −𝜇,𝑆 7 𝑄 𝑄 live butmay have aslightlyhigherchanceatrecoverythan when in the community and nonquarantined. Since recovery confers immunity against the particular strain of the Ebola =𝜃 𝜆𝑆−(𝛼 +𝜇)𝑆 , (12) 1 𝑁 𝑁 Virus, individuals who recover become refractory to further infection and hence are removed from the population of (13) = 𝜃 𝜆𝑆−(𝛼 +𝜇)𝑆 , susceptibleindividuals.Thus theequationgoverning the 1 𝑄 𝑄 rate of change within the two classes of confirmed late symptomatic cases takes the following form: =𝜃 𝛼 𝑆 −(𝛽+𝜇)𝑃 , (14) 2𝑎 𝑁 𝑁 𝑁 𝑁 =𝜃 𝛾 𝐶 −𝛿 𝐼 −𝜎 𝐼 −𝑟 𝐼 −𝜇𝐼 , 4 𝑁 𝑁 𝑁 𝑁 𝑁 𝑁 𝑁 𝑁 (15) =𝜃 𝛼 𝑆 +𝜃 𝛼 𝑆 −(𝛽+𝜇)𝑃 , 2𝑏 𝑁 𝑁 6 𝑄 𝑄 𝑄 𝑄 𝑄 (8) 𝑑𝐶 =(1−)𝛾 𝜃 𝐶 +𝜎 𝐼 +𝛾 𝐶 −𝛿 𝐼 −𝑟 𝐼 4 𝑁 𝑁 𝑁 𝑁 𝑄 𝑄 𝑄 𝑄 𝑄 (16) =𝜃 𝛽 𝑃 −(𝑟 +𝛾 +𝜇)𝐶 , 3𝑎 𝑁 𝑁 𝑁 𝑁 −𝜇𝐼 . 𝑄 𝑑𝐶 (17) =𝜃 𝛽 𝑃 +𝜃 𝛽 𝑃 −(𝑟 +𝛾 +𝜇)𝐶 , 3𝑏 𝑁 𝑁 7 𝑄 𝑄 𝑄 𝑄 2.2.6. eTh Cadavers and the Recovered Persons. The dead bodies of EVD victims are still very infectious and can (18) =𝜃 𝛾 𝐶 −(𝑟 +𝛿 +𝜇)𝐼 , 4 𝑁 𝑁 𝑁 𝑁 still infect susceptible individuals upon eeff ctive contact [2]. Diseaseinduced deaths from theclass of conrfi med late = 𝜃 𝛾 𝐶 +𝜎 𝐼 +𝛾 𝐶 symptomaticindividuals occuratrates 𝛿 and 𝛿 and the 4 𝑁 𝑁 𝑁 𝑁 𝑄 𝑄 𝑁 𝑄 (19) cadavers aredisposedofvia burial or cremationatrate 𝑏 . −(𝑟 +𝛿 +𝜇)𝐼 , eTh recovered class contains all individuals who recover from 𝑄 𝑄 EVD. Since recovery is assumed to confer immunity against the2014strain(theZaire Virus) [7]ofthe EbolaVirus, (20) =𝑟 𝐶 +𝑟 𝐼 +𝑟 𝐶 +𝑟 𝐼 −𝜇𝑅 , 𝑁 𝑁 𝑄 𝑄 𝑅 once an individual recovers, they become removed from the population of susceptible individuals. u Th s the equation (21) =𝛿 𝐼 +𝛿 𝐼 −𝑏𝑅 , governing the rate of change within the two classes of 𝑁 𝑁 𝑄 𝑄 𝐷 recovered persons and cadavers takes the following form: =𝜇𝐻 (22) =𝑟 𝐶 +𝑟 𝐼 +𝑟 𝐶 +𝑟 𝐼 −𝜇𝐼 , 𝑁 𝑁 𝑄 𝑄 𝑁 (9) =𝑏𝑅 , (23) =𝛿 𝐼 +𝛿 𝐼 −𝑏𝑅 . 𝑁 𝑁 𝑄 𝑄 𝐷 eTh population of humans who die either naturally or due where𝜃 =1−𝜃 and all other parameters and state variables ∗ ∗ to other causes is represented by the variable 𝑅 and keeps are as in Notations. track of all natural deaths, occurring at rate 𝜇 ,fromall the Suitable initial conditions are needed to completely spec- living population classes. This is a collection class. Another ify the problem under consideration. We can, for example, collectionclassistheclassofdisposedEbola-relatedcadavers, assume that we have a completely susceptible population, and 𝑑𝑡 𝑑𝑅 𝑑𝑡 𝑑𝐷 𝑑𝑡 𝐿𝑄 𝐸𝑄 𝐿𝑁 𝐸𝑁 𝑑𝑡 𝑑𝑅 𝑑𝑅 𝑑𝑡 𝑑𝑅 𝑑𝑡 𝐿𝑄 𝐸𝑄 𝐿𝑁 𝐸𝑁 𝑑𝑅 𝐿𝑄 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝐿𝑁 𝑑𝐼 𝑑𝑡 𝐸𝑄 𝑑𝑡 𝑑𝑡 𝐸𝑁 𝐿𝑄 𝑑𝐼 𝑑𝑡 𝑑𝑡 𝐿𝑁 𝑑𝑃 𝑑𝐼 𝑑𝑡 𝑑𝑃 𝑑𝑡 𝑑𝑆 𝑑𝑡 𝑑𝑆 𝑑𝑡 𝑑𝑆 𝑑𝑡 𝑑𝐷 𝐿𝑄 𝐿𝑁 𝑑𝑡 𝑑𝑅 Computational and Mathematical Methods in Medicine 7 anumberofinfectiouspersons areintroducedintothe aeff ct mostly health care providers and use that branch of population at some point. We can, for example, have that thedynamicstostudy theeeff ct of thetransmissionofthe infections to health care providers who are here considered 𝑆 ( 0)=𝑆 , 0 part of the total population. In what follows we do not explicitly single out the infectivity of those in quarantine but 𝐼 ( 0)=𝐼 , 𝑁 0 study general dynamics as derived by the current modelling exercise. 𝑆 ( 0)=𝑆 ( 0)=𝑃 ( 0)=0, (24) 𝑁 𝑄 𝑁 𝑃 ( 0)=𝐶 ( 0)=𝐶 ( 0)=𝐼 ( 0)=𝑅 ( 0)=𝑅 ( 0) 𝑄 𝑁 𝑄 𝑄 𝐷 𝑅 3. Mathematical Analysis =𝑅 ( 0)=𝐷 ( 0)=0. 𝑁 𝐷 3.1. Well-Posedness, Positivity, and Boundedness of Solution. In this subsection we discuss important properties of the Class 𝐷 is used to keep count of all the dead that are model such as well-posedness, positivity, and boundedness of properly disposed of, class 𝑅 is used to keep count of all thesolutions.Westart by denfi ing what we mean by arealistic the deaths due to EVD, and class𝑅 is used to keep count of solution. the deaths due to causes other than EVD infection. The rate of change equation for the two groups of total populations Den fi ition 1 (realistic solution). A solution of system (27) or is obtained by using (2) and (3) and adding up the relevant equivalently system comprising (11)–(21) is called realistic if equations from (11) to (23) to obtain it is nonnegative and bounded. It is evident that a solution satisfying Den fi ition 1 is (25) =Π−𝐻𝜇 −𝛿 𝐼 −𝛿 𝐼 , 𝐿 𝑁 𝑁 𝑄 𝑄 physically realizable in the sense that its values can be measured through data collection. For notational simplicity, =Π−𝐻𝜇− (𝑏−𝜇 ) 𝑅 , (26) 𝐷 we use vector notation as follows: let x =(,𝑆𝑆 ,𝑆 , 𝑁 𝑄 𝑇 11 𝑃 ,𝑃 ,𝐶 ,𝐶 ,𝐼 ,𝐼 ,𝑅 ,𝑅 )be a column vector in R 𝑁 𝑄 𝑁 𝑄 𝑁 𝑄 𝑅 𝐷 where 𝐻 is the total living population and 𝐻 is the aug- containing the 11 state variables, so that, in this nota- mented total population adjusted to account for nondisposed tion, 𝑥 =𝑆,𝑥 =𝑆 ,...,𝑥 =𝑅 .Let f( x)= 1 2 𝑁 11 𝐷 cadavers that are known to be very infectious. On the other 𝑇 (𝑓( x),( 𝑓x),...,𝑓( x)) be the vector valued function 1 2 11 hand, if we keep count of all classes by adding up (11)–(23), the defined in R so that in this notation𝑓 ( x) is the right-hand total human population (living and dead) will be constant if side of the differential equation for rfi st variable 𝑆 ,𝑓 ( x) is the Π=0. In what follows, we will use the classes 𝑅 , 𝑅 ,and 𝐷 𝑁 right-hand side of the equation for the second variable 𝑥 = 𝐷 , comprising classes of already dead persons, only as place 𝑆 ,...,and𝑓 ( x) is the right side of the differential equation 𝑁 11 holders, andstudy theproblem containing thelivinghumans for the 11th variable𝑥 =𝑅 , and so is precisely system (11)– 11 𝐷 and their possible interactions with cadavers of EVD victims (21) in that order with prototype initial conditions (24). We as oeft n is thecaseinsomeculturesinAfrica, andsowe then write the system in the form cannot have a constant total population. Note that (26) can also be written as𝑑𝐻/𝑑𝑡 = Π− −𝑏𝑅 . dx 𝐿 𝐷 = f x , x 0 = x , (27) () () 2.2.7. Infectivity of Persons Infected with EVD. Ebola is a where x :[0,∞)→ R is a column vector of state variables highly infectious disease and person to person transmission 11 11 and f : R → R is the vector containing the right- is possible whenever a susceptible person comes in contact hand sides of each of the state variables as derived from with bodily u fl ids from an individual infected with the Ebola corresponding equations in (11)–(21). We can then have the Virus. We therefore define effective contact here generally following result. to mean contact with these u fl ids. The level of infectivity of an infected person usually increases with duration of Lemma 2. The function f in (27) is Lipschitz continuous in x. theinfection andseverityofsymptomsand thecadavers of EVD victims are the most infectious [23]. us Th we will Proof. Sinceall theterms in theright-handsideare linear assume in this paperthatprobablepersons whoindeed are polynomials or rational functions of nonvanishing polyno- infected with the Ebola Virus are the least infectious while mial functions, and since the state variables, 𝑆 , 𝑆 , 𝑃 , 𝐶 , ∗ ∗ ∗ confirmed late symptomatic cases are very infectious and the 𝐼 ,and𝑅 , are continuously differentiable functions of 𝑡 ,the ∗ ∗ level of infectivity will culminate with that of the cadaver components of the vector valued function f of (27) are all of an EVD victim. While under quarantine, it is assumed continuously dieff rentiable. Further, let L( x, y;𝜃)=x{+𝜃( y− that contact between the persons in quarantine and the x):0≤𝜃≤1.Th}en L( x, y;𝜃)is a line segment that joins susceptible individuals is minimal. u Th s though the potential points x to the point y as 𝜃 ranges on the interval [0,1].We infectivity of the corresponding class of persons in quarantine apply the mean value theorem to see that increases with disease progression, their eeff ctive transmis- f ( y)− f ( x) = f ( z; y − x), sion to members of the public is small compared to that from ∞ ∞ (28) the nonquarantined class. It is therefore reasonable to assume z ∈ L( x, y;𝜃),ameanvalue point, that any transmission from persons under quarantine will 𝑑𝑡 𝜇𝐻 𝑑𝑡 𝑑𝐻 𝑑𝑡 𝑑𝐻 8 Computational and Mathematical Methods in Medicine where f ( z; y− x) is the directional derivative of the function and so 𝑆 (𝑡)>0 contradicting the assumption that 𝑆 (𝑡)< 1 1 f at themeanvalue point z in the direction of the vector y−x. 0.Sonosuch𝑡 exist. eTh same argument can be made for all But, the state variables. It is now a simple matter to verify, using techniques as explained in [24], that whenever we start system (27), with nonnegative initial data in R ,the solution will + f ( z; y − x) = ∑(∇𝑓 z ⋅( y − x)) e () 𝑘 𝑘 ∞ remain nonnegative for all𝑡>0 and that if x = 0,the 0 𝑘=1 solution will remain x = 0 ∀𝑡 > 0 ,and theregion R is (29) indeed positively invariant. ≤ ∑∇𝑓 ( z) y − x , ∞ 𝑘=1 The last two theorems have established the fact that, from a mathematical and physical standpoint, the differential equa- where e is the 𝑘 th coordinate unit vector in R .Itisnow 𝑘 + tion (27) is well-posed. We next show that the nonnegative a straightforward computation to verify that since R is a unique solutions postulated by eo Th rem 3 are indeed realistic convex set, and taking into consideration the nature of the in the sense of Den fi ition 1. functions 𝑓 , 𝑖 = 1,...,11 , all the partial derivatives are bounded and so there exist𝑀>0 such that Theorem 5 (boundedness). The nonnegative solutions char- acterized by eTh orems 3 and 4 are bounded. 11 ∑∇𝑓 ( z) ≤𝑀 ∀ z ∈ L( x, y;𝜃 )∈ R , (30) + Proof. It suffices to prove that the total living population size 𝑘=1 is bounded for all𝑡>0 . We show that the solutions lie in the bounded region and so there exist𝑀>0 such that f ( y)− f ( x) ≤𝑀 y − x (31) ∞ ∞ Ω = { 𝐻 ( 𝑡 ):0≤𝐻 ( 𝑡 )≤ } ⊂ R . (33) 𝐻 𝐿 𝐿 and hence f is Lipschitz continuous. From the den fi ition of 𝐻 given in (2), if 𝐻 is bounded, 𝐿 𝐿 Theorem 3 (uniqueness of solutions). The differential equa- the rest of the state variables that add up to 𝐻 will also be tion (27) has a unique solution. bounded. From (25) we have Proof. By Lemma 2, the right-hand side of (27) is Lips- =Π−𝐻𝜇 −𝛿 𝐼 −𝛿 𝐼 ≤Π−𝐻𝜇 ⇒ chitzian; hence a unique solution exists by existence and 𝐿 𝑁 𝑁 𝑄 𝑄 𝐿 uniqueness theorem of Picard. See, for example, [24]. (34) Π Π −𝜇𝑡 𝐻 ( 𝑡 )≤ +(𝐻 ( 0)− )𝑒 . 𝐿 𝐿 Theorem 4 (positivity). The region R wherein solutions 𝜇 𝜇 defined by (11)–(21) are defined is positively invariant under the ofl wdenfi edbythatsystem. u Th s, from (34), we see that, whatever the size of 𝐻 (0) ,𝐻 (𝑡) 𝐿 𝐿 is bounded above by a quantity that converges toΠ/𝜇 as𝑡→ Proof. We show that each trajectory of the system starting in ∞.Inparticular, if (0),th<eΠn𝐻 (𝑡) is bounded above 𝐿 𝐿 11 11 R will remain in R . Assume for a contradiction that there byΠ/𝜇 , and for all initial conditions + + exists a point 𝑡 ∈[0,∞) such that 𝑆(𝑡)=0 , 𝑆 (𝑡)<0 1 1 1 (where the prime denotes differentiation with respect to time) Π Π −𝜇𝑡 𝐻 𝑡 ≤ lim sup( +(𝐻 0 − )𝑒 ). () () (35) 𝐿 𝐿 but for0<𝑡<𝑡 , 𝑆(𝑡) >,an0d 𝑆 (𝑡) ,>𝑆 0(𝑡) ,> 0 𝑡→∞ 1 𝑁 𝑄 𝜇 𝜇 𝑃 (𝑡) ,>𝑃 0(𝑡) ,>𝐶 0(𝑡) ,>𝐶 0(𝑡) ,>𝐼 0(𝑡) ,> 0 𝑁 𝑄 𝑁 𝑄 𝑁 𝐼 (𝑡), 𝑅> 0(𝑡),a>nd0 𝑅 (𝑡).S>o,0at thepoint𝑡=𝑡 , Thus 𝐻 (𝑡) is nonnegative and bounded. 𝑄 𝑅 𝐷 1 𝑆(𝑡) is decreasing from the value zero in which case it will go negative. If such an 𝑆 will satisfy the given dieff rential Remark 6. Starting from the premise that 𝐻 (𝑡) ≥for0 all𝑡>0 , eo Th rem 5 establishes boundedness for the total equation, then we have living population and thus by extension verifies the positive invariance of the positive octant in R as postulated by =Π−𝜆𝑆(𝑡 )+(1)𝛽−𝜃 𝑃 (𝑡) 1 3 𝑁 𝑁 1 Theorem 4, since each of the variables functions 𝑆 ,𝑆 ,𝑃 ,𝐶 , 𝑡=𝑡 1 ∗ ∗ ∗ 𝐼 ,and𝑅 ,where∗∈{𝑁,,𝑄 𝑅} ,isasubset of𝐻 . ∗ ∗ 𝐿 +(1−𝜃)𝛼𝑆 (𝑡)+(1)𝛼−𝜃 𝑆 (𝑡) 2 𝑁 𝑁 1 6 𝑄 𝑄 1 3.2. Reparameterisation and Nondimensionalisation. The +(1−𝜃)𝛽𝑃 (𝑡)−𝜇𝑆()𝑡 7 𝑄 𝑄 1 1 (32) only physical dimension in our system is that of time. But =Π+(1−)𝛽 𝜃 𝑃 (𝑡) we have state variables which depend on the density of 3 𝑁 𝑁 1 humans and parameters which depend on the interactions +(1−𝜃)𝛼𝑆 (𝑡)+(1)𝛼−𝜃 𝑆 (𝑡) between the different classes of humans. A state variable or 2 𝑁 𝑁 1 6 𝑄 𝑄 1 parameter that measures the number of individuals of certain +(1−𝜃)𝛽𝑃 (𝑡)>0 7 𝑄 𝑄 1 type has dimension-like quantity associated with it [25]. To 𝑑𝑡 𝑑𝑆 𝜇𝐻 𝑑𝑡 𝑑𝐻 Computational and Mathematical Methods in Medicine 9 𝜃 𝛽 𝑃 remove the dimension-like character on the parameters and 0 3𝑎 𝑁 𝑁 𝐶 = , variables, we make the following change of variables: 𝑟 +𝛾 +𝜇 𝑠= , 𝜃 𝛽 𝑃 0 3𝑏 𝑁 𝑆 𝐶 = , 𝑟 +𝛾 +𝜇 𝑠 = , 𝜃 𝛾 𝐶 0 4 𝑁 𝑁 𝐼 = , 𝑟 +𝛿 +𝜇 𝑠 = , 𝑞 0 𝑆 (1−)𝛾 𝜃 𝐶 0 4 𝑁 𝑁 𝐼 = , 𝑟 +𝛿 +𝜇 𝑝 = , 0 0 0 0 𝑅 =𝑟 𝐶 𝑇 , 𝑅 𝑁 𝑃 0 𝛿 𝐼 0 𝑁 N 𝑝 = , 𝑅 = , 𝑃 𝐷 0 0 0 𝑁 𝑅 =𝜇𝑇 𝐻 , 𝑁 𝐿 𝑐 = , 0 0 0 𝐷 =𝑏𝑇 𝑅 , 𝐷 𝐷 𝑐 = , 𝑞 Π 0 0 0 0 𝐻 =𝐻 = =𝑆 , ℎ= , 1 𝑇 = . (36) 𝜇 (37) 𝑖 = , 𝐼 We then define the dimensionless parameter groupings 𝑖 = , 𝜃 𝜌 𝑃 3 𝑁 𝑁 𝜌 = , 𝑅 𝑛 𝑟 = , 𝜃 𝜌 𝑃 7 𝑄 𝐷 𝜌 = , 𝑟 = , 0 𝜏 𝐶 𝑅 𝑁 𝑁 𝜏 = , 𝑟 = , 𝑛 𝑛 0 𝜏 𝐶 𝐷 𝑄 𝑄 𝑑 = , 𝜏 = , 0 𝑞 𝑡 0 𝜉 𝐼 𝑡 = , 𝑁 0 𝜉 = , 𝑇 𝑛 ℎ = , 𝑙 𝜉 𝐼 𝜉 = , where 𝑎 𝑅 𝐷 𝐷 𝑆 = , 𝑎 = , 0 0 0 𝑆 =𝑆 =𝑆 , 𝛼 =(𝛼+𝜇)𝑇 , 𝑁 𝑄 𝑛 𝑁 𝜃 𝛼 𝑆 𝛼 =(𝛼+𝜇)𝑇 , 0 2𝑎 𝑁 𝑁 𝑞 𝑄 𝑃 = , 𝛽 +𝜇 𝛽 =(𝛽+𝜇)𝑇 , 𝑛 𝑁 𝜃 𝛼 𝑆 0 2𝑏 𝑁 𝑁 𝑃 = , 𝛽 +𝜇 𝛽 =(𝛽+𝜇)𝑇 , 𝑞 𝑄 𝜇𝐻 𝜇𝐻 𝜇𝐻 𝜇𝐻 𝜇𝐻 𝜇𝐻 𝜇𝐻 𝐸𝑁 𝐿𝑄 𝐿𝑁 𝐸𝑄 𝐸𝑁 10 Computational and Mathematical Methods in Medicine 𝜇 =𝑏𝑇 , The force of infection 𝜆 then takes the form (1−)𝛽 𝜃 𝑃 3 𝑁 𝑝 𝑐 𝑝 𝑐 𝑏 = , 𝑞 𝑞 𝑛 𝑛 𝜆=𝜌 ()+𝜌( )+𝜏()+𝜏() 𝑛 𝑞 𝑛 𝑞 ℎ ℎ ℎ ℎ (39) (1−)𝛼 𝜃 𝑆 2 𝑁 𝑁 𝑏 = , 𝑖 𝑖 𝑟 2 𝑞 0 𝑛 𝑑 +𝜉 ()+𝜉()+𝑎(). 𝑛 𝑞 𝑑 ℎ ℎ ℎ (1−)𝛼 𝜃 𝑆 6 𝑄 𝑄 𝑏 = , This leads to the equivalent system of equations (1−)𝛽 𝜃 𝑃 7 𝑄 𝑏 = , =1−𝜆𝑠+𝑏 𝑝 +𝑏 𝑠 +𝑏 𝑠 +𝑏 𝑝 −𝑠, (40) 1 𝑛 2 𝑛 3 𝑞 4 𝑞 𝛿 𝐼 𝑁 𝑁 𝑏 = , 5 =𝜃 𝜆𝑠−𝛼 𝑠 , (41) 1 𝑛 𝑛 𝛿 𝐼 𝑄 𝑄 (42) 𝑏 = , =(1−)𝜆 𝜃 𝑠−𝛼 𝑠 , 0 1 𝑞 𝑞 (𝑏−) 𝜇 𝑅 (43) 𝑏 = , =𝛽 (𝑠−𝑝 ), 𝑛 𝑛 𝑛 𝜃 𝛼 𝑆 6 𝑄 (44) =𝛽 (𝑠+𝑎 𝑠 −𝑝 ), 𝑎 = , 𝑞 𝑛 1 𝑞 𝑞 𝜃 𝛼 𝑆 2𝑏 𝑁 𝑁 0 𝑛 𝜃 𝛽 𝑃 =𝛾 (𝑝−𝑐 ), (45) 7 𝑄 𝑄 𝑛 𝑛 𝑛 𝑎 = , 𝜃 𝛽 𝑃 3𝑏 𝑁 (46) =𝛾 (𝑝+𝑎 𝑝 −𝑐 ), 𝜎 𝐼 𝑞 𝑛 2 𝑞 𝑞 𝑁 𝑁 𝑎 = , (𝑟 +𝛿 +𝜇)𝐼 𝑑𝑖 (47) =𝛿 (𝑐−𝑖 ), 𝑛 𝑛 𝑛 𝛾 𝐶 𝑄 𝑄 𝑎 = , (𝑟 +𝛿 +𝜇)𝐼 𝑑𝑖 (48) =𝛿 (𝑐+𝑎 𝑖 +𝑎 𝑐 −𝑖 ), 𝑞 𝑛 3 𝑛 4 𝑞 𝑞 𝑟 𝐼 𝑎 = , 𝑟 𝐶 𝑑𝑟 (49) =𝑐 +𝑎 𝑖 +𝑎 𝑐 +𝑎 𝑖 −𝑟 , 𝑛 5 𝑛 6 𝑞 7 𝑞 𝑟 𝑟 𝐶 𝑎 = , 𝑟 𝐶 𝑑𝑟 =𝜇 (𝑖+𝑎 𝑖 −𝑟 ), (50) 𝑑 𝑛 8 𝑞 𝑑 𝑟 𝐼 𝑎 = , 0 𝑑𝑟 𝑟 𝐶 𝑁 =ℎ, (51) 𝛿 𝐼 𝑄 𝑄 𝑎 = , 8 𝐷 =𝑑 , (52) 𝛿 𝐼 𝑁 𝐷 𝛾 =(𝑟 +𝛾 +𝜇)𝑇 , 𝑛 𝑁 and the total populations satisfy the scaled equation 𝛾 =(𝑟 +𝛾 +𝜇)𝑇 , 𝑞 𝑄 𝑑ℎ 𝛿 =(𝑟 +𝛿 +𝜇)𝑇 , 𝑞 𝑄 𝑙 (53) =1−ℎ −𝑏 𝑖 −𝑏 𝑖 , 𝑙 5 𝑛 6 𝑞 𝛿 =(𝑟 +𝛿 +𝜇)𝑇 . 𝑛 𝑁 𝑑ℎ (54) =1−ℎ−𝑏 𝑟 , (38) 8 𝑑 𝑑𝑡 𝐿𝑁 𝑑𝑡 𝐿𝑄 𝐸𝑄 𝐸𝑁 𝑑𝑡 𝑑𝑑 𝑑𝑡 𝐸𝑁 𝐿𝑄 𝑑𝑡 𝐸𝑁 𝐸𝑄 𝑑𝑡 𝐸𝑁 𝐿𝑁 𝑑𝑡 𝐿𝑄 𝑑𝑡 𝐿𝑄 𝑑𝑡 𝑑𝑐 𝑑𝑡 𝑑𝑐 𝑑𝑡 𝑑𝑝 𝑑𝑡 𝜇𝐻 𝑑𝑝 𝑑𝑡 𝜇𝐻 𝑑𝑠 𝑑𝑡 𝜇𝐻 𝑑𝑠 𝑑𝑡 𝜇𝑆 𝑑𝑠 𝜇𝑆 𝜇𝑆 𝜇𝑆 Computational and Mathematical Methods in Medicine 11 𝛼 +𝜇 where𝑏 >0 if it is assumed that the rate of disposal of Ebola 𝛼 = , Virus Disease victims, 𝑏 , is larger than the natural human death rate,𝜇 .Thescaledordimensionlessparametersarethen 𝛽 +𝜇 as follows: 𝑁 𝛽 = , 𝛽 +𝜇 𝛽 = , 𝜃 𝜃 𝜌 𝛼 𝑞 3 2𝑎 𝑁 𝑁 𝜌 = , (𝛽+𝜇)𝜇 𝑟 +𝛾 +𝜇 𝛾 = , 𝜃 𝜃 𝜌 𝛼 7 2𝑏 𝑄 𝑁 𝜌 = , (𝛽+𝜇)𝜇 𝑟 +𝛾 +𝜇 𝛾 = , 𝜃 𝜃 𝜏 𝛼 𝛽 3𝑎 2𝑎 𝑁 𝑁 𝑁 𝜏 = , (𝛽+𝜇)(+𝛾𝑟 +𝜇)𝜇 𝑁 𝑁 𝑟 +𝛿 +𝜇 𝛿 = , 𝜃 𝜃 𝜏 𝛼 𝛽 3𝑏 2𝑎 𝑄 𝑁 𝑁 𝜏 = , 𝜃 𝛼 (𝛽+𝜇)(+𝛾𝑟 +𝜇)𝜇 𝑁 𝑄 6 𝑄 𝑎 = , 𝜃 𝛼 2𝑏 𝑁 𝜃 𝜃 𝜃 𝜉 𝛾 𝛽 𝛼 3𝑎 2𝑎 4 𝑁 𝑁 𝑁 𝑁 𝜉 = , 𝜃 𝜃 𝛽 (𝛽+𝜇) (𝛽+𝜇)(+𝛾𝑟 +𝜇)(+𝛿𝑟 +𝜇)𝜇 7 2𝑏 𝑄 𝑁 𝑁 𝑁 𝑁 𝑎 = , 𝜃 𝜃 (𝛽+𝜇)𝛽 2𝑎 3𝑏 𝑄 𝑁 𝜃 𝜃 (1−)𝜉 𝜃 𝛾 𝛽 𝛼 3𝑎 2𝑎 4 𝑄 𝑁 𝑁 𝑁 𝜉 = , 𝑞 𝜃 𝜎 4 𝑁 (𝛽+𝜇)(+𝛾𝑟 +𝜇)(+𝛿𝑟 +𝜇)𝜇 𝑁 𝑁 𝑄 𝑎 = , (1−)( 𝜃 +𝛿𝑟 +𝜇) 4 𝑁 𝜃 𝜃 𝜃 𝛿 𝛾 𝛽 𝛼 𝑎 2𝑎 3𝑎 4 𝑁 𝑁 𝑁 𝑁 𝐷 𝑎 = , 𝜃 𝛾 (𝑟 +𝛾 +𝜇) 3𝑏 𝑄 𝑁 (𝛽+𝜇)(+𝛾𝑟 +𝜇)(+𝛿𝑟 +𝜇)𝜇𝑏 𝑁 𝑁 𝑁 𝑎 = , (1−)𝜃 𝜃 𝛾 (𝑟 +𝛾 +𝜇) 4 3𝑎 𝑁 𝑄 (1−)𝜃 𝜃 𝛽 𝛼 3 2𝑎 𝑁 𝑁 𝑏 = , 𝑟 𝜃 𝛾 4 𝑁 (𝛽+𝜇)𝜇 𝑎 = , 𝑟 (𝑟 +𝛿 +𝜇) (1−)𝛼 𝜃 2 𝑁 𝑏 = , 𝜇 = , (1−)𝛼 𝜃 6 𝑄 𝑏 = , 𝜃 𝑟 (𝑟 +𝛾 +𝜇) 3𝑏 𝑁 𝑎 = , 𝜃 𝑟 (𝑟 +𝛾 +𝜇) 3𝑎 𝑄 (1−)𝜃 𝜃 𝛽 𝛼 7 2𝑏 𝑄 𝑁 𝑏 = , 𝑟 (1−)𝛾 𝜃 (𝛽+𝜇)𝜇 4 𝑁 𝑎 = , 𝑟 (𝑟 +𝛿 +𝜇) 𝜃 𝜃 𝜃 𝛿 𝛾 𝛽 𝛼 4 2𝑎 3𝑎 𝑁 𝑁 𝑁 𝑁 𝑏 = , (1−)𝛿 𝜃 (𝑟 +𝛿 +𝜇) (𝑟 +𝛿 +𝜇)(+𝛾𝑟 +𝜇)(+𝜇𝛽 )𝜇 4 𝑄 𝑁 𝑁 𝑁 𝑁 𝑎 = . 𝜃 𝛿 (𝑟 +𝛿 +𝜇) 4 𝑁 𝑄 (1−)𝜃 𝜃 𝜃 𝛿 𝛾 𝛽 𝛼 4 2𝑎 3𝑎 𝑄 𝑁 𝑁 𝑁 (55) 𝑏 = , (𝑟 +𝛿 +𝜇)(+𝛾𝑟 +𝜇)(+𝜇𝛽 )𝜇 𝑄 𝑁 𝑁 3.3. eTh Steady State Solutions and Linear Stability. The steady (𝑏−) 𝜇 𝜃 𝜃 𝜃 𝛼 𝛽 𝛾 𝛿 4 2𝑎 3𝑎 𝑁 𝑁 𝑁 𝑁 state of the system is obtained by setting the right-hand side of 𝑏 = , (𝛽 +𝜇)(+𝛾𝑟 +𝜇)(+𝛿𝑟 +𝜇) thescaledsystemtozeroand solvingfor thescalarequations. 𝑁 𝑁 𝑁 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Let x =(,𝑠𝑠 ,𝑠 ,𝑝 ,𝑝 ,𝑐 ,𝑐 ,𝑖 ,𝑖 ,𝑟 ,𝑟 ,ℎ ) be a steady 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑟 𝑑 𝑟 +𝛿 +𝜇 state solution of the system. en, Th (43), (45), and (47) indicate 𝛿 = , that 𝛼 +𝜇 ∗ ∗ ∗ ∗ 𝛼 = , 𝑠 =𝑝 =𝑐 =𝑖 𝑛 (56) 𝑛 𝑛 𝑛 𝑛 𝐿𝑁 𝐿𝑁 𝐸𝑁 𝑏𝜇 𝐸𝑁 𝐿𝑄 𝐿𝑄 𝐸𝑁 𝐿𝑁 𝐿𝑁 𝐿𝑄 𝐸𝑁 𝐿𝑄 𝐸𝑄 𝐸𝑁 𝐸𝑁 𝐸𝑄 𝐿𝑁 𝐸𝑁 𝐿𝑁 𝐸𝑄 𝐿𝑁 𝐸𝑁 𝐸𝑁 𝐿𝑁 𝐿𝑄 𝐸𝑁 𝐿𝑁 𝐸𝑁 𝐸𝑄 𝐿𝑄 𝐸𝑁 𝐸𝑄 𝐸𝑁 12 Computational and Mathematical Methods in Medicine andwecanuseanyoftheseasaparametertoderivethevalues (42). It is easy to verify from reparameterisation (38) that the of the other steady state variables. We use the variables𝑝 and parameter groupings𝐵 and𝐵 are both nonnegative. In fact, 3 4 𝑠 as parameters to obtain the expressions 𝐵 =1 𝛽 𝜃 (𝜃−1)𝛽 𝜃 (𝜃−1) 𝑄 2𝑏 7 𝑁 𝑁 2𝑎 3 ∗ ∗ ∗ ∗ ∗ + ( + +𝜃 ) 𝑝 (𝑝,𝑠 )=𝑝+𝑎 𝑠 , 𝑞 𝑛 𝑞 𝑛 𝑞 𝜇 𝛽 +𝜇 𝛽 +𝜇 𝑁 𝑄 (59) ∗ ∗ ∗ ∗ ∗ 𝑐 (𝑝,𝑠 )=𝐴𝑝 +𝐴 𝑠 , >0 since 𝜃 =𝜃 +𝜃 , 1 2 𝑞 𝑛 𝑞 𝑛 𝑞 2 2𝑎 2𝑏 ∗ ∗ ∗ ∗ ∗ 𝛼 𝜃 (𝛽𝜃 +𝜇)+𝜇(𝛽+𝜇) 𝑖 (𝑝,𝑠 )=𝐴𝑝 +𝐴 𝑠 , 𝑄 6 𝑄 7 𝑄 3 4 𝑞 𝑛 𝑞 𝑛 𝑞 𝐵 = >0, 𝜇(𝛽 +𝜇) ∗ ∗ ∗ ∗ ∗ 𝑟 (𝑝,𝑠 )=𝐴𝑝 +𝐴 𝑠 , 5 6 𝑟 𝑛 𝑞 𝑛 𝑞 showing that𝐵 >0 and𝐵 >0. (57) 3 4 ∗ ∗ ∗ ∗ ∗ ∗ ∗ To obtain a value for𝑝 and𝑠 ,wesubstituteallcomputed 𝑟 (𝑝,𝑠 )=𝐴𝑝 +𝐴 𝑠 , 𝑛 𝑞 7 8 𝑑 𝑛 𝑞 𝑛 𝑞 steady state values, (56) and (57), into (41) and (42). The ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ expression for 𝜆 𝑠 in terms of 𝑝 and 𝑠 is obtained from 𝑛 𝑞 ℎ (𝑝,𝑠 )=1−𝑝𝐵 −𝐵 𝑠 , 𝑛 𝑞 1 𝑛 2 𝑞 (39). Performing the aforementioned procedures leads to the two equations ∗ ∗ ∗ ∗ ∗ 𝑠 (𝑝,𝑠 )=1−𝑝𝐵 −𝐵 𝑠 , 𝑛 𝑞 3 𝑛 4 𝑞 ∗ ∗ ∗ ∗ 𝜃 (𝐵𝑝 +𝐵 𝑠 )(1𝑝 −𝐵−𝐵 𝑠 ) 1 5 6 3 4 𝑛 𝑞 𝑛 𝑞 ∗ ∗ ∗ ∗ ∗ ℎ (𝑝,𝑠 )=1−( +𝑏 𝑏 𝐴 )𝑝−𝑏 𝐴 𝑠 , 𝑙 𝑛 𝑞 5 6 3 𝑛 6 4 𝑞 (60) ∗ ∗ ∗ =𝛼 𝑝 (1−𝑝𝐵 −𝐵 𝑠 ), 𝑛 1 2 𝑛 𝑛 𝑞 ∗ ∗ ∗ ∗ (1−)( 𝜃 𝑝 𝐵+𝐵 𝑠 )(1𝑝 −𝐵−𝐵 𝑠 ) 1 5 𝑛 6 𝑞 3 𝑛 4 𝑞 where (61) ∗ ∗ ∗ =𝛼 𝑠 (1−𝑝𝐵 −𝐵 𝑠 ), 𝑞 𝑞 1 𝑛 2 𝑞 𝐴 =1+𝑎 , where 1 2 𝐴 =𝑎 𝑎 , 𝐵 =𝜌 +𝜌 +𝜏 +𝜏 𝐴 +𝜉 +𝜉 𝐴 +𝑎 𝐴 , 2 1 2 5 𝑛 𝑞 𝑛 𝑞 1 𝑛 𝑞 3 𝑑 7 (62) 𝐵 =𝜌 𝑎 +𝜏 𝐴 +𝜉 𝐴 +𝑎 𝐴 . 𝐴 =1+𝑎 +𝑎 𝐴 , 6 𝑞 1 𝑞 2 𝑞 4 𝑑 8 3 3 4 1 𝐴 =𝑎 𝐴 , Next, we solve (60) and (61) simultaneously, which clearly 4 4 2 differ in some of their coefficients, to obtain the expressions ∗ ∗ 𝐴 =1+𝑎 +𝑎 𝐴 +𝑎 𝐴 , for𝑝 and𝑠 .Quickly observethatthe twoequations maybe 5 5 6 1 7 3 𝑛 𝑞 reduced to one such that 𝐴 =𝑎 𝐴 +𝑎 𝐴 , 6 6 2 7 4 ∗ ∗ ∗ ∗ (58) (1−𝐵 𝑝 −𝐵 𝑠 )( 𝛼 𝑝 (1−𝜃 )−𝛼 𝑠 𝜃 )=0. (63) 1 𝑛 2 𝑞 𝑛 𝑛 1 𝑞 𝑞 1 𝐴 =1+𝑎 𝐴 , 7 8 3 ∗ ∗ Two possibilities arise: either (i)1−𝐵 𝑝 −𝐵 𝑠 =0 or (ii) 1 2 𝑛 𝑞 𝐴 =𝑎 𝐴 , 8 8 4 ∗ ∗ 𝛼 𝑝 (1 −)−𝛼 𝜃 𝑠 𝜃 =0. eTh rfi st condition leads to the 𝑛 𝑛 1 𝑞 𝑞 1 system 𝐵 =𝑏 𝐴 , 1 8 7 ∗ ∗ 1−𝐵 𝑝 −𝐵 𝑠 =0, 𝐵 =𝑏 𝐴 , 2 8 8 3 𝑛 4 𝑞 (64) ∗ ∗ 1−𝐵 𝑝 −𝐵 𝑠 =0. 𝐵 =𝛼 −𝑏 −𝑏 −𝑏 , 1 2 3 𝑛 1 2 4 𝑛 𝑞 𝐵 =𝛼 −𝑏 −𝑏 𝑎 . However, the two equations are equivalent to ℎ =0 and 4 𝑞 3 4 1 𝑠 =0 (see (57)), which are unrealistic, based on our constant population recruitment model. Hence, we only consider the second possibility, which yields the relation Here the solution for the scaled total living (ℎ )and scaled living and Ebola-deceased (ℎ)populations is,respectively, 𝛼 𝜃 𝑞 1 ∗ ∗ obtained by equating the right-hand sides of (53) and (54) to 𝑝 =( )𝑠, (65) 𝑛 𝑞 𝛼 (1−) 𝜃 𝑛 1 zero, while that for𝑠 is obtained by adding up (40), (41), and Computational and Mathematical Methods in Medicine 13 so that substituting (65) into (60) yields Thus, if (1−)𝛼 𝜃 (𝐵−𝐵 )>𝜃𝛼 (𝐵−𝐵 ) ,then𝑧>1 . 1 𝑛 4 2 1 𝑞 1 3 This will hold if 𝐵 >𝐵 .Inthe case where 𝐵 <𝐵 ,wewill 3 1 3 1 𝑠 =0, require that𝐵 −𝐵 be greater than(𝜃𝛼 /(1−)𝛼 𝜃 )(𝐵 −𝐵 ). 4 2 1 𝑞 1 𝑛 1 3 (1−)𝛼 𝜃 (𝑅−1) ∗ 7 1 𝑛 0 We identify 𝑅 as theuniquethreshold parameterofthe or 𝑠 = = (66) system as follows. (𝜃𝛼 𝐵 +(1−𝜃)𝛼𝐵 )( R−1) 1 𝑞 1 1 𝑛 2 𝑅 −1 Lemma 8. The parameter 𝑅 defined in (69) is the unique =𝑥( ), R−1 threshold parameter of the system whenever𝑧>1 . where Proof. If𝑧>1 ,then R =𝑧𝑅 >1 whenever 𝑅 >1 and the 0 0 existence or nonexistence of a realistic solution of the form of (1−)𝐵 𝜃 𝜃 𝐵 1 5 1 6 𝐵 =𝛼 𝛼 (1−)( 𝜃 (+ )−1) (66) is determined solely by the size of𝑅 . 7 𝑞 𝑛 1 𝛼 𝛼 𝑛 𝑞 eTh rest of the steady states are then obtained by using ∗ ∗ =𝛼 𝛼 (1−)( 𝜃 −1𝑅), 𝑞 𝑛 1 0 these values for 𝑝 and 𝑠 givenby(66)in(65)and (57) to 𝑛 𝑞 obtain the following: 𝜃 𝐵 (1−)𝐵 𝜃 1 6 1 5 𝐵 =𝛼 [( + ) 8 𝑞 𝛼 𝛼 𝑅 −1 𝑛 𝑞 ∗ 0 𝑠 =𝑥( ), R−1 ⋅(𝜃𝛼 𝐵 +𝛼 (1−)𝐵 𝜃 ) 𝑞 1 3 𝑛 1 4 𝑅 −1 ∗ ∗ ∗ ∗ 0 (67) 𝑖 =𝑐 =𝑠 =𝑝 =𝑦( ), −(𝐵𝜃 𝛼 +𝛼 (1−)𝐵 𝜃 )] = 𝛼(𝜃𝐵 𝛼 𝑛 𝑛 𝑛 𝑛 1 1 𝑞 𝑛 1 2 𝑞 1 1 𝑞 R−1 𝑅 −1 ∗ 0 𝜃 𝐵 (1−)𝐵 𝜃 𝑝 =(𝑦+𝑎 𝑥)( ), 1 6 1 5 1 +𝛼 (1−)𝐵 𝜃 )((+ ) R−1 𝑛 1 2 𝛼 𝛼 𝑛 𝑞 𝑅 −1 ∗ 0 𝛼 𝜃 𝐵 +𝛼 (1−)𝐵 𝜃 𝑐 =(𝐴𝑦+𝐴 𝑥)( ), 𝑞 1 3 𝑛 1 4 𝑞 1 2 ⋅( )−1)=(𝜃𝛼 𝐵 𝛼 R−1 𝑞 1 1 𝑞 𝜃 𝐵 𝛼 +𝛼 (1−)𝐵 𝜃 1 1 𝑞 𝑛 1 2 𝑅 −1 𝑖 =(𝐴𝑦+𝐴 𝑥)( ), (71) 3 4 +𝛼 (1−)𝐵 𝜃 )( 𝑧−1𝑅 ), 𝑞 𝑛 1 2 0 R−1 with 𝑅 −1 ∗ 0 𝑟 =(𝐴𝑦+𝐴 𝑥)( ), 𝑟 5 6 R−1 (1−)𝛼 𝜃 1 𝑛 𝑥= , 𝑅 −1 (𝜃𝛼 𝐵 +(1−𝜃)𝛼𝐵 ) 1 𝑞 1 1 𝑛 2 𝑟 =(𝐴𝑦+𝐴 𝑥)( ), 𝑑 7 8 R−1 𝜃 𝛼 𝑥 1 𝑞 𝑦= , (68) (1−)𝛼 𝜃 𝑅 −1 ∗ 0 1 𝑛 𝑠 =1−(𝑦+𝐵𝐵 𝑥)( ), 3 4 R−1 (𝜃𝛼 𝐵 +(1−𝜃)𝛼𝐵 ) 1 𝑞 3 1 𝑛 4 𝑧= , 𝑅 −1 𝜃 𝛼 𝐵 +(1−𝜃)𝛼𝐵 ∗ 1 𝑞 1 1 𝑛 2 ℎ =1−(𝑦+𝐵𝐵 𝑥)( ), 1 2 R−1 𝜃 𝐵 (1−)𝐵 𝜃 1 6 1 5 𝑅 = + , 0 where 𝑥 , 𝑦 ,and 𝑧 are as defined in (68). We have proved the 𝛼 𝛼 𝑛 𝑞 following result. (69) 𝑅 (𝜃𝛼 𝐵 +(1−𝜃)𝛼𝐵 ) 0 1 𝑞 3 1 𝑛 4 R = =𝑧𝑅 . Theorem 9 (on the existence of equilibrium solutions). 𝜃 𝛼 𝐵 +(1−𝜃)𝛼𝐵 1 𝑞 1 1 𝑛 2 System (40)–(52) has at least two equilibrium solutions: the disease-free equilibrium x =𝐸 = Remark 7. It canbeshown that𝐵 <𝐵 .Infact, 2 4 (1,0,0,0,0,0,0,0,0,0,0,1an) d an endemic equilibrium ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x =𝐸 =(,𝑠𝑠 ,𝑠 ,𝑝 ,𝑝 ,𝑐 ,𝑐 ,𝑖 ,𝑖 ,𝑟 ,𝑟 ,ℎ ) .The 𝑒𝑒 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑟 𝑑 𝜃 𝛼 𝜃 𝛽 6 𝑄 7 𝑄 endemic equilibrium, 𝐸 , exists and is realistic only when 𝐵 =𝑏 𝑎 𝑎 𝑎 𝑎 = ((1) − 𝑒𝑒 2 8 8 4 1 2 𝜇 (𝛽+𝜇) 𝑏 the threshold parameters 𝑅 and R,given by (69),are of appropriate magnitude. 𝛿 𝛾 𝜃 𝛼 𝑄 𝑄 6 𝑄 ⋅ )< (70) (𝑟 +𝛿 +𝜇)(𝑟 +𝛾 +𝜇) 𝜇 The stability of the steady states is governed by the sign of 𝑄 𝑄 the eigenvalues of the linearizing matrix near the steady state ∗ ∗ 𝜃 𝛽 𝜃 𝛼 (𝜃𝛽 +𝜇) 7 𝑄 6 𝑄 7 𝑄 solutions. If𝐽( x ) is the Jacobian matrix at the steady state x , ⋅ < +1=𝐵 . (𝛽+𝜇) 𝜇 (𝛽+𝜇) then we have 𝑄 𝑄 𝐸𝑄 𝐿𝑄 𝑑𝑓 14 Computational and Mathematical Methods in Medicine −1 𝑏 𝑏 (𝑏−𝜌 )( −𝜌 𝑏 )−𝜏 −𝜏 −𝜉 −𝜉 0−𝑎 0 2 3 1 𝑛 4 𝑞 𝑛 𝑞 𝑛 𝑞 𝑑 0−𝛼 0𝜃 𝜌 𝜃 𝜌 𝜃 𝜏 𝜃 𝜏 𝜃 𝜉 𝜃 𝜉 0𝑎 𝜃 0 𝑛 1 𝑛 1 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 𝑑 1 ( ) ̃ ̃ ̃ ̃ ̃ ̃ ̃ ( ) 00 −𝛼 𝜃 𝜌 𝜃 𝜌 𝜃 𝜏 𝜃 𝜏 𝜃 𝜉 𝜃 𝜉 0𝑎 𝜃 0 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 𝑑 1 ( ) ( ) (0𝛽 0−𝛽 0 0 0 0 0 000 ) 𝑛 𝑛 ( ) ( ) (0𝛽 𝑎 𝛽 0−𝛽 00 0 0 0 0 0 ) 𝑞 1 𝑞 𝑞 ( ) ( ) 00 0 𝛾 0−𝛾 00 0 0 0 0 ( 𝑛 𝑛 ) 𝐽 = ( ), (72) dfe ( ) 00 0 𝛾 𝑎 𝛾 0−𝛾 0 0 000 ( 𝑞 2 𝑞 𝑞 ) ( ) ( ) 00 0 0 0 𝛿 0−𝛿 00 0 0 𝑛 𝑛 ( ) ( ) ( ) 00 0 0 0 𝛿 𝑎 𝛿 𝑎 𝛿 −𝛿 000 𝑞 4 𝑞 3 𝑞 𝑞 ( ) ( ) (00 0 0 0 1 𝑎 𝑎 𝑎 −1 0 0 ) 6 5 7 00 0 0 0 0 0 𝜇 𝜇 𝑎 0−𝜇 0 𝑑 𝑑 8 𝑑 00 0 0 0 0 0 0 0 0 −𝑏 −1 ( ) where𝜃 =1−𝜃 .Thus if 𝜁 is an eigenvalue of the linearized an equation involving a polynomial of degree 12 in 𝜁 ,where 1 1 system at the disease-free state, then 𝜁 is obtained by the 𝑃 (𝜁) is a polynomial of degree9 in𝜁 ,given by solvability condition 3 (73) 𝑃 ( 𝜁 )= 𝐽 −𝜁 I = ( 𝜁+1 )𝑃 ( 𝜁 )=0, dfe 9 −𝛼 −𝜁 0 𝜃 𝜌 𝜃 𝜌 𝜃 𝜏 𝜃 𝜏 𝜃 𝜉 𝜃 𝜉 𝑎 𝜃 𝑛 1 𝑛 1 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 𝑑 1 ̃ ̃ ̃ ̃ ̃ ̃ ̃ 0−𝛼 −𝜁 𝜃 𝜌 𝜃 𝜌 𝜃 𝜏 𝜃 𝜏 𝜃 𝜉 𝜃 𝜉 𝑎 𝜃 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 1 𝑛 1 𝑞 𝑑 1 𝛽 0−𝛽 −𝜁 0 0 0 0 0 0 𝑛 𝑛 𝛽 𝑎 𝛽 0−𝛽 −𝜁 0 0 0 0 0 𝑞 1 𝑞 𝑞 𝑃 ( 𝜁 )= 00 𝛾 0−𝛾 −𝜁 0 0 0 0 . (74) 9 𝑛 𝑛 00 𝛾 𝑎 𝛾 0−𝛾 −𝜁 0 0 0 𝑞 2 𝑞 𝑞 00 0 0 𝛿 0−𝛿 −𝜁 0 0 𝑛 𝑛 00 0 0 𝛿 𝑎 𝛿 𝑎 𝛿 −𝛿 −𝜁 0 𝑞 4 𝑞 3 𝑞 𝑞 00 0 0 0 0 𝜇 𝜇 𝑎 −𝜇 −𝜁 𝑑 𝑑 8 𝑑 Now, all we need to know at this stage is whether there is where solution of (73) for 𝜁 with positive real part which will then 𝑐 =1, indicate the existence of unstable perturbations in the linear 𝑐 =𝛼 +𝛼 +𝛽 +𝛽 +𝛾 +𝛾 +𝛿 +𝛿 +𝜇 , regime. eTh coecffi ients of polynomial (73) can give us vital 8 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑑 information about the stability or instability of the disease- free equilibrium. For example, by Descartes’ rule of signs, . (76) a sign change in the sequence of coefficients indicates the 𝜃 𝐵 (1−)𝐵 𝜃 1 6 1 5 presence of a positive real root which in the linear regime 𝑐 =𝛽 𝛽 𝛾 𝛾 𝛿 𝛿 𝜇 𝛼 𝛼 {1− − } 0 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑑 𝑛 𝑞 𝛼 𝛼 signifies the presence of exponentially growing perturbations. 𝑛 𝑞 We can write polynomial equation (73) in the form =𝛽 𝛽 𝛾 𝛾 𝛿 𝛿 𝜇 𝛼 𝛼 (1−), 𝑅 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑑 𝑛 𝑞 0 and we can see that𝑐 changessignfrompositivetonegative 9 when 𝑅 increases from values of 𝑅 <1 through 𝑅 =1 to 0 0 0 3 𝑖 𝑃 ( 𝜁 )= ( 𝜁+1 )∑𝑐 𝜁 , (75) values of𝑅 >1 indicating a change in stability of the disease- 𝑖 0 𝑘=0 free equilibrium as𝑅 increases from unity. 0 Computational and Mathematical Methods in Medicine 15 −1+𝜆𝑠−𝑏 𝑝 −𝑏 𝑠 −𝑏 𝑠 −𝑏 𝑝 +𝑠 3.4. eTh Basic Reproduction Number. A threshold parameter 1 𝑛 2 𝑛 3 𝑞 4 𝑞 that is of essential importance to infectious disease trans- 𝛼 𝑠 𝑛 𝑛 mission is the basic reproduction number denoted by 𝑅 . 𝑅 0 0 ( ) ( ) measures the average number of secondary clinical cases of ( ) 𝛼 𝑠 ( 𝑞 𝑞 ) infection generated in an absolutely susceptible population ( ) ( ) by a single infectious individual throughout the period ( ) 𝛽 (−𝑠+𝑝 ) 𝑛 𝑛 𝑛 ( ) within which the individual is infectious [26–29]. Generally, ( ) ( ) the disease eventually disappears from the community if ( ) 𝛽 (−𝑠−𝑎 𝑠 +𝑝 ) 𝑞 𝑛 1 𝑞 𝑞 ( ) 𝑅 <1 (and in some situations there is the occurrence ( ) ( ) of backward bifurcation) and may possibly establish itself ( 𝛾 (−𝑝+𝑐 ) ) 𝑛 𝑛 𝑛 ( ) within the community if 𝑅 >1.Thecriticalcase 𝑅 =1 0 0 V = , ( ) ( ) represents the situation in which the disease reproduces itself 𝛾 (−𝑝−𝑎 𝑝 +𝑐 ) ( ) 𝑞 𝑛 2 𝑞 𝑞 ( ) thereby leaving the community with a similar number of ( ) infection cases at any time. The definition of 𝑅 specicfi ally ( ) 0 𝛿 (−𝑐+𝑖 ) ( 𝑛 𝑛 𝑛 ) ( ) requires that initially everybody but the infectious individual ( ) in the population be susceptible. us, Th this definition breaks ( ) 𝛿 (−𝑐−𝑎 𝑖 −𝑎 𝑐 +𝑖 ) 𝑞 𝑛 3 𝑛 4 𝑞 𝑞 ( ) down within a population in which some of the individ- ( ) ( ) uals are already infected or immune to the disease under ( ) −𝑐 −𝑎 𝑖 −𝑎 𝑐 −𝑎 𝑖 +𝑟 𝑛 5 𝑛 6 𝑞 7 𝑞 𝑟 ( ) consideration. In such a case, the notion of reproduction ( ) number R becomes useful. Unlike𝑅 which is xfi ed, R may −𝜇 𝑖 −𝜇 𝑎 𝑖 +𝜇 𝑟 0 𝑑 𝑛 𝑑 8 𝑞 𝑑 𝑑 vary considerably with disease progression. However, R is −1+ℎ+𝑏 𝑟 bounded from above by 𝑅 anditiscomputedatdieff rent ( 8 𝑑 ) points depending on the number of infected or immune cases (77) in the population. where the force of infection 𝜆 is given by (39). To obtain the One way of calculating 𝑅 is to determine a threshold −1 next-generation operator, 𝐹𝑉 ,wemustcalculate (𝐹)= condition for which endemic steady state solutions to the ̃ ̃ system under study exist (as we did to derive (69)) or 𝜕 F /𝜕𝑥 and (𝑉)=𝜕 V /𝜕𝑥 evaluated at the disease-free 𝑖 𝑗 𝑖 𝑗 for which the disease-free steady state is unstable. Another equilibrium position, where𝑠=1=ℎ , 𝑠 =𝑠 =𝑝 = 𝑛 𝑞 𝑛 method is the next-generation approach where 𝑅 is the 𝑝 =𝑐 =𝑐 =𝑖 =𝑖 =𝑟 =𝑟 =0.Thebasic 𝑞 𝑛 𝑞 𝑛 𝑞 𝑟 𝑑 spectral radius of the next-generation matrix [26]. Using the reproduction number is then the spectral radius of the next- −1 −1 next-generation approach, we identify all state variables for generation matrix𝐹𝑉 .Thusif (𝐹𝑉 ) is the spectral radius −1 the infection process, 𝑝 , 𝑝 , 𝑐 , 𝑐 , 𝑖 , 𝑖 , 𝑟 , 𝑟 ,and ℎ.The 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑟 𝑑 of the matrix𝐹𝑉 ,then transitions from 𝑠 , 𝑠 to 𝑝 , 𝑝 are not considered new 𝑛 𝑞 𝑛 𝑞 −1 infections but rather a progression of the infected individuals 𝑅 =(𝐹𝑉 )= [𝛼 𝜃 {(1 0 𝑞 1 𝛼 𝛼 𝑛 𝑞 through the different stages of disease compartments. Hence, we identify terms representing new infections from the above +(1+𝑎+(1+𝑎)𝑎)𝑎 3 2 4 𝑑 equations and rewrite the system as the difference of two ̃ ̃ ̃ vectors F and V,where F consists of all new infections and +𝜉 (𝑎𝑎 +𝑎 +𝑎 +1)+𝑎𝜏 +𝜉 +𝜌 +𝜌 𝑞 2 4 3 4 2 𝑞 𝑛 𝑛 𝑞 V consists of the remaining terms or transitions between (78) +𝜏 +𝜏 )}−𝛼(𝜃−1)( 𝜌 𝑎 states. at Th is, we set ẋ = F − V,where x is the vector 𝑛 𝑞 𝑛 1 1 𝑞 of state variables corresponding to new infections: x = +𝑎 [𝑎 𝑎 (𝑎𝑎 +𝜉 )+𝑎𝜏 ])] (𝑠,,𝑠𝑠 ,𝑝 ,𝑝 ,𝑐 ,𝑐 ,𝑖 ,𝑖 ,𝑟 ,𝑟 ,ℎ) . This gives rise to 1 2 4 8 𝑑 𝑞 2 𝑞 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑛 𝑞 𝑟 𝑑 (1−)𝐵 𝜃 𝜃 𝐵 1 5 1 6 = + , 𝛼 𝛼 𝑛 𝑞 𝜃 𝜆𝑠 as computed before. The expression for 𝑅 has two parts. The first part ( ) 0 (1−)𝜆 𝜃 𝑠 ( 1 ) measures the number of new EVD cases generated by ( ) ( ) an infected nonquarantined human. It is the product of ( ) ( ) 𝜃 (the proportion of susceptible individuals who become ( ) ( ) 0 suspected but remain nonquarantined), 𝐵 (which indicates ̃ ( ) 5 F = , ( ) the contacts from this proportion of individuals with infected ( ) ( ) individuals at various stages of the disease), and1/𝛼 (which ( 0 ) ( ) is the average duration a human remains as a suspected ( ) ( ) nonquarantined individual). In the same way, the second part ( ) can be interpreted likewise. The stability of the endemic steady state is obtained by calculating the eigenvalues of the linearized matrix evaluated ( ) 𝑖𝑗 𝑖𝑗 16 Computational and Mathematical Methods in Medicine at the endemic state. eTh computations soon become very where the variables𝑠 andthe augmentedpopulationℎ satisfy complicated because of the size of the system and we proceed the differential equations (81) and (85), respectively. with a simplification of the system. System (81)–(85)has thesamesteadystatessolutions as the original system if we combine it with (80). However, on its own, it represents a pseudo-steady state approximation 3.5. Pseudo-Steady State Approximation. Ebola Virus Disease [30] of theoriginalsystem. Clearlythe reducedsystemhas is a very deadly infection that normally kills most of its vic- two realistic steady states: 𝐸 and 𝐸 ,sothatif 𝐸 ∗ = tims within about 21 days of exposure to the infection. u Th s dfe 𝑒𝑒 x ∗ ∗ ∗ ∗ ∗ (𝑠,𝑠 ,𝑠 ,𝑟 ,ℎ ) is a steady state solution, then when compared with the life span of the human, the elapsed 𝑛 𝑞 𝑟 time representing the progression of the infection from rfi st exposure to death is short when compared to the total time required as the life span of the human. u Th s we set 𝜇≈ 𝐸 = ( 1,0,0,0,1) , dfe 1/life span of human, so that the rates 𝛼 , 𝛽 ,and so forth ∗ ∗ (87) ∗ ∗ ∗ ∗ ∗ will all be such that 1/rate ≈ resident time in given state, 𝐸 =(,𝑠𝑠 ,𝑠 ,𝑟 ,ℎ ), 𝑒𝑒 𝑛 𝑞 𝑟 some of which will be short compared with the life span of the human. It is therefore reasonable to assume that 𝜇 𝜇+ rate ≈ small ⇒ ≈ large, (79) where, following the same method as was done in the full 𝜇+ rate 𝜇 system, so that the scaling above renders some of the state variables essentially at equilibrium. aTh t is, the quantities, 1/𝛽 , 1/𝛽 , 𝑛 𝑞 1/𝛾 , 1/𝛾 , 1/𝛿 , 1/𝛿 ,and 𝑏/𝜇 , may be regarded as small 𝑛 𝑞 𝑛 𝑞 ∗ 7 parameters so that, in the corresponding equations (43)–(48) 𝑠 = , and(50), thestate variablesmodelledbythese equationsare 8 essentially in equilibrium and we can evoke the Michaelis- 𝛼 𝜃 𝑞 1 ∗ ∗ Menten pseudo-steady state hypothesis [30]. To proceed, we 𝑠 =( )𝑠, 𝑛 𝑞 𝛼 (1−) 𝜃 𝑛 1 make the pseudoequilibrium approximation (88) ∗ ∗ ∗ 𝑝 =𝑐 =𝑖 =𝑠 , 𝑟 =𝐴 𝑠 +𝐴 𝑠 , 𝑛 𝑛 𝑛 𝑛 5 6 𝑟 𝑛 𝑞 ∗ ∗ ∗ 𝑝 =𝑠 +𝑎 𝑠 , 𝑞 𝑛 1 𝑞 𝑠 =1−𝐵 𝑠 −𝐵 𝑠 , 3 𝑛 4 𝑞 ∗ ∗ ∗ 𝑐 =𝐴 𝑠 +𝐴 𝑠 , (80) 𝑞 1 𝑛 2 𝑞 ℎ =1−𝐵 𝑠 −𝐵 𝑠 . 1 2 𝑛 𝑞 𝑖 =𝐴 𝑠 +𝐴 𝑠 , 𝑞 3 𝑛 4 𝑞 𝑟 =𝐴 𝑠 +𝐴 𝑠 All coefficients are as defined in (58). When steady states (88) 𝑑 7 𝑛 8 𝑞 are rendered in parameters of the reduced system, taking into to have the reduced system consideration the fact that, from (68),𝑧=𝐵 𝑦+𝐵 𝑥 and 3 4 𝐵 𝑦+𝐵 𝑥=1 ,weget (81) 1 2 =1−𝜆𝑠+ ( 𝛼 −𝐵 ) 𝑠 +( 𝛼 −𝐵 )𝑠 −𝑠, 𝑛 3 𝑛 𝑞 4 𝑞 (82) =𝜃 𝜆𝑠−𝛼 𝑠 , 1 𝑛 𝑛 𝑧−1 𝑠 =( ), R−1 (83) = (1−𝜃 ) 𝜆𝑠−𝛼 𝑠 , 𝑅 −1 1 𝑞 𝑞 ∗ 0 𝑠 =𝑦( ), R−1 𝑑𝑟 (84) =𝐴 𝑠 +𝐴 𝑠 −𝑟 , 𝑅 −1 5 𝑛 6 𝑞 𝑟 ∗ 𝑠 =𝑥( ), (89) R−1 𝑑ℎ (85) =1−ℎ−𝐵 𝑠 −𝐵 𝑠 , 𝑅 −1 1 𝑛 2 𝑞 ∗ 0 𝑟 =(𝐴𝑦+𝐴 𝑥)( ), 5 6 R−1 and the total population and the force of infection also reduce accordingly. In particular, we have ( 𝑧−1 ) 𝑅 ∗ 0 ℎ =( ). 𝑝 𝑐 R−1 𝑝 𝑐 𝑞 𝑞 𝑛 𝑛 𝜆𝑠 = (𝜌 ()+𝜌( )+𝜏()+𝜏() 𝑛 𝑞 𝑛 𝑞 ℎ ℎ ℎ ℎ 𝑖 𝑟 𝑠 𝑛 𝑑 𝑛 The stability of the steady states is determined by the (86) +𝜉 ()+𝜉()+𝑎())𝑠 = (𝐵 () 𝑛 𝑞 𝑑 5 ℎ ℎ ℎ ℎ eigenvalues of the linearized matrix of the reduced system ∗ ∗ ∗ ∗ ∗ ∗ ∗ evaluated at the steady state x =(,𝑠𝑠 ,𝑠 ,𝑟 ,ℎ ) .If𝐽( x ) is 𝑠 𝑛 𝑞 𝑟 +𝐵 ())𝑠, the Jacobian of the system near the steady state x ,then 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑠 𝑑𝑡 𝑑𝑠 𝑑𝑡 𝑑𝑠 Computational and Mathematical Methods in Medicine 17 ∗ ∗ ∗ ∗ −𝐶 ( x )−1 −𝐶( x )+𝛼−𝐵 −𝐶 ( x )+𝛼−𝐵 0−𝐶 ( x ) 3 4 𝑛 3 5 𝑞 4 6 ∗ ∗ ∗ ∗ 𝜃 𝐶 ( x)𝜃𝐶 ( x )−𝛼 𝜃 𝐶 ( x)0 𝐶𝜃 ( x ) 1 3 1 4 𝑛 1 5 1 6 ( ) ∗ ∗ ∗ ∗ ̃ ̃ ̃ ̃ 𝐽( x )=( 𝜃 𝐶 ( x ) 𝜃 𝐶 ( x ) 𝜃 𝐶 ( x )−𝛼 0 𝜃 𝐶 ( x ) ), (90) 1 3 1 4 1 5 𝑞 1 6 0𝐴 𝐴 −1 0 5 6 0−𝐵 −𝐵 0−1 ( ) 1 2 ∗ ∗ ∗ ̃ ̃ ̃ +𝛼 (𝛼+𝜃 𝐶 ( x )−𝐶( x ) 𝜃 +1)−𝐶( x ) where𝜃 =1−𝜃 and 1 1 𝑛 𝑞 1 3 5 1 5 ⋅𝜃 , ∗ ∗ ∗ ∗ 𝑄 ( x )=𝛼𝜃 (𝐵𝐶 ( x )+𝐵𝐶 ( x )−𝐶( x )) 0 𝑞 1 1 6 3 3 4 𝑠 𝛼 𝑦(𝑅 −1) ∗ 𝑛 0 𝐶 ( x )=𝐵 +𝐵 = , ∗ ∗ ∗ 3 5 6 ̃ ∗ ∗ +𝛼 ( 𝜃 (𝐵𝐶 ( x )+𝐵𝐶 ( x )−𝐶( x ))). +𝛼 ℎ ℎ 𝜃 (𝑧−1 ) 𝑛 1 2 6 4 3 5 𝑞 ∗ (93) 𝑠 𝐵 ∗ 5 𝐶 ( x )=𝐵 = , 4 5 ℎ 𝑅 (91) Now, the signs of the zeros of (92) will depend on the signs of 𝑠 𝐵 ∗ 6 the coecffi ients 𝑄 , 𝑖 ∈ {0,1,2} . We now examine these. 𝐶 ( x )=𝐵 = , 𝑖 5 6 ∗ ∗ ∗ ∗ ∗ ℎ 𝑅 At the disease-free state where 𝑠 =1=ℎ , 𝑠 =𝑠 = 𝑛 𝑞 ∗ ∗ ∗ ∗ 𝑟 =0,orequivalently 𝑅 =1,wehave x = x = ∗ ∗ 0 dfe 𝑠 𝑠 𝑠 𝑠 𝛼 𝑦(𝑅 −1) ∗ 𝑛 𝑛 0 (1,0,0,0,1 so) that 𝐶 ( x )=0 , 𝐶 ( x )=𝐵, 𝐶 ( x )= 𝐶 ( x )=−(𝐵 +𝐵 )=− . 3 dfe 4 dfe 5 5 dfe 6 5 6 2 2 ∗ ∗ 𝜃 ( 𝑧−1 ) 𝑅 ℎ ℎ 1 0 𝐵 ,𝐶 ( x )=0 , and (92) becomes 6 6 dfe 3 2 𝑃 (𝜁, x )=𝜁+1 (𝜁+𝑄 𝜁+𝑅 )=0, (94) ( ) 5 dfe dfe dfe The asterisk is used to indicate that the quantities so cal- culated are evaluated at the steady state. We can perform a where stability analysis on the reduced system by noting that if 𝜁 is an eigenvalue of (90), then𝜁 satisefi s the polynomial equation 𝑄 =𝛼 +𝛼 −𝐵 𝜃 −𝐵 (1−) 𝜃 dfe 𝑛 𝑞 5 1 6 1 𝛼 −𝛼 𝑛 𝑞 =𝛼 (1−)+𝛼 𝑅 +(1−𝜃)𝐵( ), 𝑛 0 𝑞 1 6 (95) 𝑃 (𝜁, x ) 𝑅 =𝛼 (𝛼−𝐵 𝜃 )+𝛼𝐵 (𝜃−1) dfe 𝑞 𝑛 5 1 𝑛 6 1 2 3 ∗ 2 ∗ ∗ = (𝜁+1 )(𝜁+𝑄 ( x )𝜁+𝑄 ( x )𝜁+𝑄 ( x )) (92) 2 1 0 =𝛼 𝛼 (1−). 𝑅 𝑞 𝑛 0 =0, eTh roots of (94) are −1, −1, −1,and (−𝑄 ± dfe √𝑄 −4𝛼 𝛼 (1−))/2 𝑅 , showing that there is one 𝑞 𝑛 0 dfe positive real solution as 𝑅 increases beyond unity and the where disease-free equilibrium loses stability at 𝑅 =1.For the local stability when 𝑅 ≤1, the additional requirement 𝑄 >0 is necessary. dfe At the endemic steady state, and in the original scaled ∗ ∗ ∗ ∗ 𝑄 ( x )=𝛼+𝛼 +𝐶 ( x )−𝐶( x )𝜃−𝐶 ( x ) 𝜃 2 𝑛 𝑞 3 4 1 5 1 parameter groupings of the system, x = x = 𝑒𝑒 ∗ ∗ ∗ ∗ ∗ (𝑠,𝑠 ,𝑠 ,𝑟 ,ℎ ) , the coefficients of (92) simplify accordingly 𝑛 𝑞 𝑟 +1, and we have 𝑄 ( x )=𝛼 1 𝑞 ∗ 2 3 2 𝑃 (𝜁, x )=( 𝜁+1 )(𝜁+𝑃 𝜁 +𝑄 𝜁+𝑅 )=0,(96) ∗ ∗ ∗ 5 x x x 𝑒𝑒 𝑒𝑒 𝑒𝑒 −𝜃 (−𝐵𝐶 ( x )+𝛼𝐶 ( x )+𝐶( x )) 1 1 6 𝑞 4 4 ∗ ∗ ̃ ̃ +𝐵 𝐶 ( x ) 𝜃 +𝐶 ( x )( 𝜃 +𝐵𝛼 𝜃 +𝐵 𝜃 ) where 2 6 1 3 𝑞 1 3 1 4 1 18 Computational and Mathematical Methods in Medicine 𝐵 𝜃 𝜃 ( 𝑧−1 ) (𝛼−𝛼 )+𝛼𝑅 (𝛼(𝑅−1)𝑦+(+1𝛼 )( 𝜃 𝑧−1 ) ) 6 1 1 𝑛 𝑞 𝑞 0 𝑛 0 𝑞 1 𝑃 = , 𝑒𝑒 𝛼 𝜃 𝑅 ( 𝑧−1 ) 𝑞 1 0 𝛼 𝜃 𝑄 +𝛼 𝑅 𝑄 𝑛 1 11 0 12 (97) 𝑄 = , 𝑒𝑒 𝛼 𝛼 𝑅 𝑧−1 𝜃 ( ) 𝑛 𝑞 0 1 (𝑅−1) ( R−1) 𝛼 𝛼 0 𝑛 𝑞 𝑅 = , 𝑒𝑒 ( 𝑧−1 ) 𝑅 where and (96) becomes 𝑄 =(𝐵( 𝑧−1 ) 𝜃 (𝛼−𝛼 )+𝛼(𝛼(𝐵𝑥+𝑅 𝑧) 11 6 1 𝑞 𝑛 𝑞 𝑞 2 0 +𝛼 (𝑅−1)( 𝑥+𝛼𝐵 𝑅 𝑥−1))), 𝑛 0 2 𝑞 0 ∗ 2 2 𝑃 (𝜁, x )=(𝜁) +(𝛼𝜁+1 )(𝜁 5 𝑛 (98) 𝑄 =(𝛼(−𝜃(𝐵𝑥+(𝑅 −1)𝑥(+𝐵𝛼 )+1) 12 𝑛 1 2 0 𝑞 3 (𝑅−1)𝑥𝛼 0 𝑞 +(1+ )𝜁 (101) +𝐵 𝑥+1)+𝛼 𝐵 𝜃 (𝑅−1)𝑥). 2 𝑞 3 1 0 𝑧−1 Now the necessary and sufficient conditions that will guaran- R−1 (𝑅−1)𝛼 ( ) 0 𝑞 tee the stability of the nontrivial steady state x will be the +( )), 𝑒𝑒 𝑅 ( 𝑧−1 ) Routh-Hurwitz criteria which, in the present parameteriza- tions, are 𝑃 >0, 𝑒𝑒 showing that all solutions of the equation 𝑃 (𝜁, x )=0 are negative or have negative real parts whenever they are 𝑄 >0, 𝑒𝑒 (99) complex, indicating that the nontrivial steady state is stable 𝑅 >0, to small perturbations whenever 𝑅 >1.Inthiscasewe 𝑒𝑒 0 canregardanincreasein 𝑅 as an increase in the parameter 𝑃 𝑄 −𝑅 >0. x x x grouping𝐵 . 𝑒𝑒 𝑒𝑒 𝑒𝑒 All initial suspected cases escape quarantine: that is,𝜃 = With this characterization, we can then explore special cases 1.Inthiscasewesee that𝑠 =0 and initially we will be on the of intervention. left branch of our flow chart in Figure 1. The strength of the present model is that, based on its derivation, it is possible 3.6. Some Special Cases. All initial suspected cases are quar- for some individuals to eventually enter quarantine as the antined: that is,𝜃 =0.Inthiscasewesee that𝑠 =0 and we 1 𝑛 systems wake up from sleep and control measures kick into have only the right branch of our flow chart in Figure 1. We place. Mathematically, we then have have here a problem involving infections only at the treatment centres. Mathematically, we then have 6 𝐵 𝑅 = , 𝑅 = , 𝑞 𝛼 𝑦=0, 𝑥=0, 𝑥= , 𝑦= , (100) (102) 𝑧= , 𝑧= , 𝛼 𝑥(𝑅 −1) 𝛼 𝑦(𝑅 −1) 𝑞 0 𝑛 0 𝐶 = , 𝐶 = , 3 3 𝑧−1 𝑧−1 𝐶 𝐶 3 3 𝐶 =− 𝐶 =− 6 6 𝑅 𝑅 0 0 (1− )𝜆S r I LQ Q 2b N N 3b N (1− ) C 4 N Computational and Mathematical Methods in Medicine 19 Susceptible, false suspected, and probable cases (1− 𝜃 )𝛽 P 7 Q Susceptibles (S) (1− 𝜃 )𝛽 P 3 N N (1− 𝜃 )𝛼 S (1− 𝜃 )𝛼 S 2 N N 6 Q Q Suspected nonquarantined Suspected cases quarantined cases S S N Q 𝜃 𝛼 S 𝜃 𝛼 S 2a N N 6 Q Q Probable Probable nonquarantined quarantined cases cases 𝜃 𝛽 P 𝜃 𝛽 P 7 Q 3a N N r C r C Confirmed nonquarantined EN N Removal by EQ Q Confirmed quarantined early symptomatic cases recovery (R ) early symptomatic cases 𝜃 𝛾 C 𝛾 C 4 N N Q Q 𝜎 C N N Confirmed nonquarantined Confirmed quarantined 𝛿 I 𝛿 I N N Q Q Removal by death late symptomatic cases late symptomatic cases due to EVD (R ) D (I ) bR Figure 1: Conceptual framework showing the relationships between the different compartments that make up the different population of individuals and actors in the case of an EVD outbreak. Susceptible individuals include false suspected and probable cases. True suspects and probable cases are confirmed by a laboratory test and the confirmed cases can later develop symptoms and die of the infection or recover to become immune to the infection. Humans can also die naturally or due to other causes. Nonquarantined cases can become quarantined through intervention strategies. Others run the course of the illness from infection to death without being quarantined. Flow from compartment to compartment is as explained in the text. and (96) becomes quarantined or nonquarantined. Mathematically, we have =𝛼 and (96) becomes that𝛼 𝑛 𝑞 ∗ 2 2 𝑃 (𝜁, x )=(𝜁) +𝛼𝜁+1 (𝜁 ( ) 5 𝑞 ∗ 2 2 𝑃 (𝜁, x )=(𝜁) +(𝛼𝜁+1 )(𝜁 5 𝑛 (𝑅−1)𝑥𝛼 0 𝑞 (104) +(1+ )𝜁 ( 𝑧−1 ) (1−) 𝜃 (𝑅−1)𝛼𝑦 0 𝑛 +(1+ )𝜁 (103) 𝑧−1 ( R−1) (𝑅−1)𝛼 0 𝑞 +( )). R−1 (𝑅−1)𝛼 𝑅 ( 𝑧−1 ) ( ) 0 𝑛 +( )), 𝑅 ( 𝑧−1 ) 0 ∗ In this case as well, all solutions of the equation 𝑃 (𝜁, x )= 0 either are negative or have negative real parts whenever 𝑅 >1 and𝑧>1 , showing that in this case again the steady state is stable to small perturbations. In this particular case, an increase in 𝑅 canberegardedasanincreaseinthe two again showing that all solutions of the equation 𝑃 (𝜁, x )= parameter groupings𝐵 and𝐵 . 0 are negative or have negative real parts whenever they 5 6 are complex, indicating that the nontrivial steady state is stable to small perturbations whenever 𝑅 >1.Inthis 0 4. Parameter Discussion case we can regard an increase in 𝑅 as an increase in the parameter grouping 𝐵 . 𝑅 in this case appears larger than Some parameter values were chosen based on estimates in 5 0 in the previous cases. [15, 20], on the 2014 Ebola outbreak, while others were The rate at which suspected individuals become probable selected from past estimates (see [9, 14, 18]) and are sum- cases is the same: that is, 𝛼 =𝛼 . In this case the flow marized in Table 2. In [20], an estimate based on data 𝑁 𝑄 from being a suspected case to a probable case is the same primarily from March to August 20th yielded the following in all circumstances, irrespective of whether or not one is average transmission rates and 95% confidence intervals: 0.27 𝜆S r I LN N Nonquarantined Quarantine 20 Computational and Mathematical Methods in Medicine (0.27,0.27) per day in Guinea, 0.45 (0.43,0.48) per day in The parameter 𝛾 measures the rate at which early Sierra Leone, and 0.28(0.28,0.29) per day in Liberia. In [15], symptomatic individuals leave that class. This could be as a the number of cases of the 2014 Ebola outbreak data (up result of recovery or due to increase and spread of the virus until early October) was tte fi d to a discrete mathematical within the human. It takes about 2 to 4 days to progress from the early symptomatic stage to the late symptomatic stage, model, yielding estimates for the contact rates (per day) in the so that 𝛾 ,𝛾 ∈ [1/4,1/2],which canbeassumedtobethe community and hospital (considered quarantined) settings as 𝑁 𝑄 reciprocal of the mean time it takes from when the immune 0.128 in Sierra Leone for nonquarantined cases (and 0.080 system is either completely overwhelmed by the virus or kept for quarantined cases, about a 61% reduction) while the rates in check via supportive mechanism. Severe symptoms are in Liberia were 0.160 for nonquarantined cases (and 0.062 followed either by death aeft r about an average of two to for quarantined cases, close to a 37.5% drop). The models in four days beyond entering the late symptomatic stage or by [15, 20] did not separate transmission based on early or late and 𝛿 are recovery [12], and thus we can assume that 𝛿 𝑁 𝑄 symptomatic EVD cases, which was considered in our model. in the range[1/4,1/2]. If on the other hand the EVD patient Basedonthe information, we will assume that eeff ctive recovers, then it will take longer for patient to be completely contacts between susceptible humans and late symptomatic clear of the virus. Notice that, based on the ranges given EVD patients in the communities fall in the range[0.12,0.48] above, the time frames are [6,16] days from the onset of per day, which contains the range cited in [16]. u Th s 𝜉 ∈ symptoms of Ebola to death or recovery. eTh range from the [0.12,0.48]. However, we will consider scenarios in which this onset of symptoms which commences the course of illness parameter varies. Furthermore, if we assume about a 37.5% to deathwas givenas6–16daysin[8],while therange to 61% reduction in effective contact rates in the quarantined for recovery was cited as 6–11 days [8]. For our baseline settings,thenwecan assume that 𝜉 =𝜙 𝜉 ,where 𝑄 𝑄 𝑁 parameters, the mean time from the onset of the illness to 𝜙 ∈ [0.0375,0.062]. However, to understand how effective death or recovery will be in the range of 6–11 days. quarantining Ebola patients is, general values of 𝜙 ∈ [0,1] Here, we will consider that the recovery rate for quar- can be considered. Under these assumptions, a small value of antined early symptomatic EVD patients lies in the range 𝜙 will indicate that quarantining was effective, while values [0.4829,0.5903] per day with a baseline value of 0.5366 per of𝜙 close to 1 will indicate that quarantining patients had no day, as cited in [16]. u Th s 𝑟 ∈ [0.4829,0.5903]. If we assume effect in minimizing contacts and reducing transmissions. that patients quarantined in the hospital have a better chance SincepatientswithEVD at theonset of symptoms areless of surviving than those in the community or at home, without infectious than EVD patients in the later stages of symptoms the necessary expert care that some of the quarantined EVD [2, 10], we assume that the eeff ctive contact rate between patients may get, then we can consider that the recovery rate confirmed nonquarantined early symptomatic individuals for nonquarantined EVD patients, in the community, would and susceptible individuals, denoted by 𝜏 ,isproportional be slightly lower. u Th s we scale 𝑟 by some proportion𝜔∈ to 𝜉 with proportionality constant 𝑞 ∈ [0,1].Likewise, 𝑁 𝑁 [0,1],sothat 𝑟 =𝜔𝑟 . Since late symptomatic patients we assume that the eeff ctive contact rate between conrfi med have a much lower recovery chance, then both 𝑟 and 𝑟 quarantined early symptomatic individuals and susceptible will be lower than 𝑟 and 𝑟 ,respectively. Hencewewill individuals is proportional to 𝜉 with proportionality con- consider that 𝑟 =𝜅𝑟 and 𝑟 =𝜅𝑟 ,where0<𝜅≪1 . stant 𝑞 ∈ [0,1].Thus, 𝜏 =𝑞 𝜉 and 𝜏 =𝑞 𝜉 ,with 𝑞 𝑁 𝑁 𝑁 𝑄 𝑄 𝑄 Sometimes late symptomatic EVD patients are removed from 0<𝑞 ,𝑞 <1. 𝑁 𝑄 thecommunity andquarantined.Hereweassume ameanof The range, for the parameter 𝑎 ,ofthe eeff ctive con- 2dayssothat𝜎 = 0.5 per day. tact rate between cadavers of conrfi med late symptomatic The fractions 𝜃 measure the proportions of individu- individuals and susceptible individuals was chosen to be als moving into various compartments. If we assume that [0.111,0.489]per day, where 0.111 is the rate estimated in [15] members in the quarantined classes are not left unchecked for Sierra Leone and 0.489 is that for Liberia. With control but have medical professionals checking them and giving measures and education in place, these rates can be much them supportive remedies to boost their immune system to lower. fight the Ebola Virus or enable their recovery, then it will be eTh incubation period of EVD is estimated to be between reasonable to assume that𝜃 ,𝜃 ,𝜃 ,and𝜃 are all greater than 6 7 4 5 2 and 21 days [2, 7–9], with a mean of 4–10 days reported in 0.5. If such an assumption is not made, then the parameters [8, 9]. In [10], a mean incubation period of 9–11 days was canbechosentobeequal or closetoeachother. reported for the 2014 EVD. Here, we will consider a range The parameter Π is chosen to be 555 per day as in [16]. from 4 to 11 days, with a mean of about 10 days used as the Furthermore, the parameters 𝜌 and 𝜌 and the eeff ctive 𝑁 𝑄 baseline value. u Th s we will consider that 𝛼 and𝛼 are in the contact rates between probable nonquarantined and, respec- 𝑁 𝑄 range [1/11,1/4].Atthe endofthe incubation period,early tively, quarantined individuals and susceptible individuals symptoms may emerge 1–3 days later [10], with a mean of 2 will be varied to see their eects ff on the model dynamics. days. us Th the mean rate at which nonquarantined (𝛽) and However, the values chosen will be such that the value of 𝑅 𝑁 0 quarantined(𝛽) suspected cases become probable cases lies computed is within realistic reported ranges. in [1/3,1] [12].About 1or2to 4dayslater aeft r theearly The parameter 𝜇 , the natural death rate for humans, symptoms,moreseveresymptomsmay developsothatthe is chosen based on estimates from [13]. The parameter 𝑏 rates at which nonquarantined (𝛾) and quarantined (𝛾) measures the time it takes from death to burial of EVD 𝑁 𝑄 probable cases become conrfi med cases lie in [1/4,1/2]. patients.Amean valueof2days wascited in [14] forthe 1995 𝐸𝑄 𝐿𝑄 𝐸𝑁 𝐿𝑁 𝐸𝑄 𝐸𝑁 𝐿𝑄 𝐿𝑁 𝐸𝑄 𝐸𝑁 𝐸𝑄 𝐸𝑄 Computational and Mathematical Methods in Medicine 21 Table 2: Parameters, baseline values, and ranges of baseline values with references. Parameters Baseline values Range of values Reference Π 3, 555 Varies 𝜌 Varies Varies 𝜌 Varies Varies 𝜉 0.27 [0.12,0.48] Estimated 𝜏 𝑞 𝜉 ,𝑞 = 0.75 𝑞 ∈ [0,1] Variable 𝑁 𝑁 𝑁 𝑁 𝑁 𝜉 𝜙 𝜉 ,𝜙 = 0.5 𝜙 ∈ [0,1] Estimated 𝑄 𝑄 𝑁 𝑄 𝑄 𝜏 𝑞 𝜉 ,𝑞 = 0.75 𝑞 ∈ [0,1] Variable 𝑄 𝑄 𝑄 𝑄 𝑄 𝜃 𝜃 ∈ [0,1],𝑖 = 1,2,...,7 𝜃 ∈ [0,1],𝑖 = 1,2,...,7 Variable 𝑖 𝑖 𝑖 𝛼 1/10 [1/11,1/4] [8–10] 𝛼 1/10 [1/11,1/4] [8–10] 𝛽 0.5 [1/3,1] [10] 𝛽 1/2 [1/3,1] [10] 𝛾 1/3 [1/4,1/2] [10, 12] 𝛾 1/3 [1/4,1/2] [10, 12] 𝛿 1/3 [1/4,1/2] [10, 12] 𝛿 1/3 [1/4,1/2] [10, 12] −1 𝜇 1/(60×365) [1/(40×365),1/(70da×y365)] [13] −1 𝑏 1/2.5 [1/4.50,1/2]day [14, 15] −1 𝑎 0.3000 [0.111,0.489]day [15] 𝑟 0.5366 [0.4829,0.5903] [16] 𝑟 𝜔𝑟 ,𝜔 = 0.88 𝜔 ∈ [0,1] Estimate 𝑟 𝜅𝑟 ,𝜅 = 0.02 𝜅 ≪ 1 Estimate 𝑟 𝜅𝑟 ,𝜅 = 0.02 𝜅 ≪ 1 Estimate 𝜎 0.5 [1/3,1] Estimate Estimates discussed in Section 4. and 2000 Ebola outbreak epidemics in the Democratic parameters have been assigned; it may be written as 𝑅 = Republic of Congo and Uganda, respectively. For the 2014 𝑟 +𝑟 𝜌 +𝑟 𝜌 ,where𝑟 ,𝑖=0,1,2 , are positive constants that 0 1 𝑄 2 𝑁 𝑖 West AfricanOutbreak, theestimates were 2.01 days in can be shown to be dependent on the other parameters. u Th s Liberia and 4.50 days in Sierra Leone [15]. 𝑅 will increase linearly with increase in any of the parameters 𝜌 and 𝜌 for xfi ed given values of the other parameters. 𝑄 𝑁 Though we have theoretically found, for example, that 𝑠 5. Numerical Simulation of the Scaled becomes infinite when R is near one equivalent to 𝑅 being Reduced Model near1/𝑧 ,thiscasedoesnot arisebecause we have assumedin the analysis that𝑧>1 .Thus thecase 0≤𝑧<1 is linked with The parameter values given in Table 2 were used to carry the trivial steady state. out some numerical simulations for the reduced model, (81)– The situation shown in Figure 3 has important conse- (85), when the constant recruitment term is 555 persons per quences for control strategies. While 𝑠 varies sharply for day. The varying parameters were chosen so that 𝑅 would a narrow band of reproduction numbers, its values do not be within ranges of reported values, which are typically less change much for larger values of 𝑅 . Referring to Figure 3, than 2.5 (see, e.g., [16, 22]). In Figure 2, we show a time an application of a control measure that will reduce 𝑅 say series solution for a representative choice of values for the from a high value of 10 down to 5, a 50% reduction, will not parameters𝜌 and𝜌 . Figures 2(a)–2(e) show the long term 𝑁 𝑄 appreciably aeff ct the rest of the disease transmission. u Th s solutions to the reduced model exhibiting convergence to the thesystemisbestcontrolledwhen𝑅 is small which can occur stable nontrivial equilibrium in the case where 𝑅 >1,the in the early stages of the infection or late in the infection when case with sustained infection in the community. Figure 2(f) some eeff ctive control measures have already been instituted then shows an example of convergence to the trivial steady such as eeff ctive quarantining or prompt removal of EVD state when𝑅 <1, the case where the disease is eradicated. In deceased individuals. Notice that as𝑅 further increases, the that example we notice that as𝑅 <1,𝑠 →0 as𝑛→∞ and 0 𝑛 number of susceptible individuals continues to drop. We note, this in turn implies that𝑠 →0 as𝑛→∞ and eventually the however, that typical values of𝑅 computed for the 2014 Ebola system relaxes to the trivial state(𝑠,,𝑠𝑠 ,𝑟 ,ℎ)=(1,0,0,0,1) 𝑛 𝑞 𝑟 outbreak arelessthan2.5. for large time. Next, we investigate the eeff ct of 𝜉 on 𝑅 and the We note here that the computed value of𝑅 can be shown 0 𝑁 0 to be linear in the variables 𝜌 and 𝜌 , when eventually all model dynamics. In Figure 2(f), we showed an example of 𝑄 𝑁 𝐸𝑄 𝐿𝑄 𝐸𝑁 𝐿𝑁 𝐸𝑄 𝐸𝑁 𝐸𝑄 22 Computational and Mathematical Methods in Medicine 0.0006 0.0020 0.0005 0.0015 0.0004 0.0003 0.0010 0.0002 0.0005 0.0001 0.0000 0.0000 5 101520 510 15 20 t t (a) (b) 1.00 0.0010 0.95 0.0008 0.90 0.0006 0.85 0.0004 0.80 0.0002 0.75 0.0000 510 15 20 510 15 20 t t (c) (d) 1.00 0.0010 0.95 0.0008 0.90 0.0006 0.85 0.0004 0.80 0.0002 0.75 0.0000 510 15 20 0.01 0.02 0.03 0.04 t t (e) (f) Figure 2: (a)–(e) Time series showing convergence of the solutions to the steady states for the nondimensional reduced model when the constant recruitment term is 555 persons per day. In this example, 𝜃 = 0.85 and 𝜌 = 1.15 and 𝜌 = 0.85, and all the other parameters are 1 𝑁 𝑄 as in Table 2, giving values of 𝑅 = 1.102. In this case, the nonzero steady state is stable and the solution converges to the steady state value as given by (89) as𝑡→∞ . (f) Time series showing the long term behaviour of the variable 𝑠 in the reduced model for 𝜌 =𝜌 = 0.8 and 𝑛 𝑁 𝑄 all other values of the parameters are as given in Table 2. In this case𝑅 = 0.88 andsothe only steady stateisthe trivialsteadystate whichis stable. convergence to the trivial steady state for 𝜌 = 0.8 = 𝜌 Figures 4(a) and 4(b) that the disease begins to propagate 𝑁 𝑄 for the nondimensional reduced model when the constant and stabilize within the community. eTh re is a major peak recruitment term is 555 persons per day. The model dynamics whichstartstodecay as EVDdeathsbegin to rise.An yielded an 𝑅 value of 0.88 < 1, when all other parameters estimated size of the epidemic can be computed as the were as giveninTable 2. From this scenario,weincreased difference between 𝑠 andℎ when the disease dynamics settles only 𝜉 from its baseline Table 2 value of 0.27 to 0.453. to its equilibrium state. As more and more persons become This yields an increase in 𝑅 to 1.00038 and we see from infected, 𝑅 increases and the estimated size of the epidemic 0 0 n Computational and Mathematical Methods in Medicine 23 0.6 1.0 0.5 0.8 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0.0 0.0 2468 10 2468 10 R R 0 0 (a) (b) ∗ ∗ ∗ Figure 3: Graph showing the behaviour of the steady states 𝑠 and𝑠 as a function of𝑅 .(a) Thisgraph showsthe form of thesteadystate 𝑠 𝑛 0 ∗ ∗ as a function of𝑅 .(b) Thisgraph showsthe form of thesteadystate 𝑠 as a function of𝑅 . eTh steady state solution 𝑠 varies greatly only in 0 0 𝑛 𝑛 a narrow band of reproduction numbers but saturates for large values of𝑅 . On the other hand,𝑠 continues to drop to zero as𝑅 increases. 0 0 −6 ×10 1.0000 2.5 0.9995 2.0 1.5 0.9990 1.0 0.9985 0.5 0.9980 10 20 30 40 10 20 30 40 t t (a) (b) 0.98 0.0001 0.97 0.00008 0.96 0.00006 0.00004 0.95 0.00002 0.94 q 10 20 30 40 10 20 30 40 t t (c) (d) Figure 4: (a)–(d) Time series plot showing the propagation and stabilization of EVD to a stable nontrivial steady state for the nondimensional reduced model when the constant recruitment term is 555 persons per day and with 𝜌 =𝜌 = 0.8 and 𝜃 = 0.85 as used in Figure 2(f). 𝑁 𝑄 1 Except for 𝜉 that is increased from its baseline value of 0.27, all other parameters are as in Table 2. In graphs (a) and (b), 𝜉 is increased 𝑁 𝑁 to 0.453. This yields 𝑅 = 1.00038, slightly bigger than 1. eTh graphs show that there is a major peak which starts to decay as EVD deaths begin to rise. The size of the epidemic can be estimated as the difference in the areas between the 𝑠 and ℎ curves as the disease settles to its steady state. Graphs (c) and (d) show the model dynamics when𝜉 is further increased to 0.48, which yields𝑅 = 1.01765.Ingraphs(c) and 𝑁 0 (d), the oscillations are more pronounced and the size of the epidemic is larger due to the increased eeff ctive contacts with late symptomatic individuals. s and h s and h s and s n q n∞ s and s n q 24 Computational and Mathematical Methods in Medicine 0.9 0.0007 0.0006 0.8 0.0005 0.0004 0.7 0.0003 0.0002 0.6 0.0001 s 10 20 30 40 10 20 30 40 t t (a) (b) Figure 5: (a)-(b) Time series plot showing the propagation and stabilization of EVD to a stable nontrivial steady state for the nondimensional reduced model when the constant recruitment term is 555 persons per day and with 𝜌 = 1.15,𝜌 = 0.85,and𝜃 = 0.85 as in Figures 2(a)– 𝑁 𝑄 1 2(e). Except for 𝜉 that is increased from 0.27 to 0.36, all the other parameters are as given in Table 2, and the corresponding 𝑅 value is 𝑁 0 𝑅 = 1.159. Notice that, in this case, the disease has a higher frequency of oscillations and the difference between the areas under ℎ and 𝑠 is considerably larger indicating that the size of the disease burden is considerably larger in this case. also increases as is expected. In particular, increasing 𝜉 the eradication of the disease. us Th control which includes further to 0.48 increases 𝑅 to 1.01765, and, from Figures quarantining has to be comprehensive and sustained until 4(c) and 4(d), the difference between 𝑠 and ℎ is visibly eradication is achieved. larger compared to the difference from Figures 4(a) and 4(b). The 2014 Ebola outbreak did not show sustained disease Moreover, the oscillatory dynamics becoming more pro- states. eTh disease dynamics exhibited epidemic fade-outs. nounced indicated a higher back and forth movement activity Here, we show that such fade-outs are possible with our between the𝑠 and𝑠 and𝑠 classes. model. To investigate the epidemic-like fade-outs, we rfi st 𝑛 ℎ note that due to the scaling adopted in our model, our time In Figures 2(a)–2(e), we showed an example of the long scales are large. However, any epidemic-like EVD behaviour term dynamics of the solutions of the reduced model for would be expected to occur over a shorter timescale. u Th s, 𝜌 = 1.15 and 𝜌 = 0.85.For this case,weobtained 𝑁 𝑄 for the results illustrated here, we plot the model dynamics 𝑅 = 1.102 > 1 and the model dynamics show how the in terms of the original variable by simulating the equations reduced model converges to a stable nontrivial equilibrium, that make up the system, (11)–(23). To illustrate that the large when all other parameters are as stated as in Table 2. From time scales do not aeff ct the long term dynamics of the model this point, if we increase only𝜉 from its default value of 0.27 results, we first present a graph of the original system in the to 0.36, we see that 𝑅 increases to approximately 1.159 and case where the parameters are maintained as those used in the model exhibits irregular random oscillations with higher = 0.85 and Figure 6 with Π = 555 persons per day, 𝜃 frequency but eventually stabilizes (Figures 5(a) and 5(b)). 𝜌 =1.15 and 𝜌 = 0.85, and with all the other parameters 𝑁 𝑄 The size of the epidemic is considerably larger in this case. as given in Table 2. The 𝑅 value was1.102 and so a sustained This highlights the importance of reducing contacts between disease with no other effort is possible over a long time frame EVDpatientsand susceptiblehumansincontrolling thesize of more than 10,000 days. of thedisease burden andloweringthe impact of thedisease. When the number of individuals recruited daily reduces, then we can show that, for the case whereΠ=3 persons 5.1. Fade-Outs and Epidemics in Ebola Models. Our model per day, 𝜃 = 0.85 and 𝜌 =2 and 𝜌 =1,and allthe 1 𝑁 𝑄 results as highlighted in Figures 2(a)–2(e), 4, and 5 indicate other parameters remain as in Table 2; then an epidemic-like that it is possible to have a long term endemic situation behaviour is obtained (see Figure 7). for Ebola transmission, if conditions are right. In particular, eTh dynamics of Figures 6 and 7 indicate that quaran- in our model, for the case where we have a relatively large tining alone is not sufficient to eradicate the Ebola epidemic constant recruitment term of 555 persons per day and with especially when there is a relative high number of daily available resources to sustain the quarantine eor ff ts then as recruitment. In fact, quarantining can instead serve as a buffer long as there are people in the community (nonquarantined) zone allowing the possibility of sustained disease dynamics with the possibility to come in contact with infectious when there are a reasonable number of people recruited each day. However, when the daily recruitment is controlled, EVDfluids,thenthe diseasecan be sustainedaslongas 𝑅 >1 (Figures 2(a)–2(e), 4, and 5). Increasing control reduced to a value of 3 per day, then the number of new daily by reducing contacts early enough between suspected and infections is reduced to a low value as depicted in Figures 7(a)–7(c). However, the estimated cumulative number of probable individuals with susceptible individuals can bring down the size of 𝑅 to a value <1 which eventually leads to infections increases daily (see Figure 7(d)). s and h s and s n q Computational and Mathematical Methods in Medicine 25 7000 250 3000 N 1000 Q 0 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 t t (a) (b) 400 500 N 200 0 0 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 t t (c) (d) Figure 6: (a)–(c) Time series showing convergence of the solutions to the steady states for the full model in dimensional form when the constant recruitment term is 555 persons per day. In this example, 𝜃 = 0.85 and 𝜌 = 1.15 and 𝜌 = 0.85, and all the other parameters are 1 𝑁 𝑄 as in Table 2, giving a value of 𝑅 = 1.102.Thegraph showsthe shortscale dynamicsaswellasthe long term behaviourshowing stabilityof the nonzero steady state. 6. Discussion and Conclusion other health workers whose dedication was inspirational and helpful in curbing the 2014 Ebola outbreak in Africa. In this paper we set out to derive a comprehensive model As we now look forward with optimism for a better and for the dynamics of Ebola Virus Disease transmission in a Ebola-free tomorrow, there is work going on in the scientific complex environment where quarantining is not effective, community to develop vaccines [2]. Mathematical modelling meaning that some suspected cases escape quarantine while of the dynamics and transmission of Ebola provides unique others do not. When the West African countries of Liberia, avenues for exploration of possible management scenarios in Guinea,and Sierra LeonecamefacetofacewithEbola the event of an EVD outbreak, since, during an outbreak, Virus Disease infection in 2014, it took the international management of the cases is crucial for containment of the community some time to react to the crises. As a result, spread of the infection within the community. In this paper most of the initial cases of EBV infection escaped monitoring we have presented a comprehensive ordinary differential and entered the community. African belief systems and other equation model that handles management issues of EVD traditional practices further compounded the situation and infection. Our model takes care of quarantine and nonquar- before long large number of cases of EBV infections were antine cases and therefore can be used to predict progression in the community. Even when the international community of disease dynamics in the population. Our analysis has shownthatthe initialresponsetoall suspectedcases of EVD reacted and started putting in place treatment centres, it still tooksometimeforpeopletobesensitizedonthedangersthey infection is crucial. This is captured through the parameter are facing. The consequence was that infections continued 𝜃 which measures the initial fraction of suspected cases in families, during funerals, and even in hospitals. People that are put into quarantine. We have shown that the basic checkedintohospitals andwould nottellthe truthabout reproductionnumber canbeindexed by thisparameterinthe their case histories and as a result some medical practitioners sense that when all cases are initially quarantined the spread gotexposed to theinfection.Acase in pointisthatofDr. of theinfection canonlytakeplace at thetreatment centres, Stella Ameyo Adadevoh, an Ebola victim and everyday hero but in cases where all suspected cases escape quarantine, the reproduction number can be large. Our model has been able [31],who preventedthe spread of EbolainNigeria andpaid with her life. We still pay tribute and honour to her and the to quantify the densities of infected and recovered individuals S and S N Q I and I N Q C and C N Q 26 Computational and Mathematical Methods in Medicine 2000 100 Q 20 100 200 300 400 500 100 200 300 400 500 t t (a) (b) I 25000 I N 0 0 2000 4000 6000 8000 10000 12000 14000 100 200 300 400 500 t t (c) (d) Figure 7: (a)-(b) Time series showing epidemic-like behaviours of the solutions to the steady states for the full model in dimensional form when the constant recruitment term is 3 persons per day over a short time scale. In this example,𝜃 = 0.85 and𝜌 =2 and𝜌 =1, and all the 1 𝑁 𝑄 other parameters are as in Table 2. eTh value of 𝑅 = 1.63455. eTh graph shows the short scale dynamics exhibiting epidemic-like behaviour that fades out. within the population based on baseline parameters identi- that it is possible to control EVD infection in the community fied during the 2014 Ebola Virus Disease outbreak in Africa. provided we reduce and maintain the reproduction number The basic reproduction number in our model depends to below unity. Such control measures are possible if there is on the initial exposure rates including exposure to cadavers effective contact tracing and identification of EVD patients of EVD victims. eTh provision of scope for further quar- and eeff ctive quarantining, since a reduction of the propor- antining during the progression of the infection means that tion of cases that escape quarantine reduces the value of𝑅 . these exposure rates are weighted accordingly, depending Additionally, our model results indicate that when there on whetherornot thesystemwokeupfromslumber and are a high constant number of recruitment into an EVD picked up those persons who initially escaped quarantine. For community, quarantining alone may not be sucffi ient to example, the parameter𝜏 which measures the eecti ff ve con- eradicate the disease. It may serve as a buffer enhancing tact rate between confirmed quarantined early symptomatic a sustained epidemic. However, reducing the number of individuals and susceptible individuals is eventually scaled persons recruited per day can bring the diseases to very low by the proportions 𝜃 and 𝜃 which, respectively, are those values. 3𝑎 2𝑎 proportions of suspected and probable cases that eventually To demonstrate the feasibility of our results, we per- progress to become EVD patients and have escaped quar- formed a pseudoequilibrium approximation to the system antine. us Th our framework can progressively be used at derived based on the assumption that the duration of man- each stage to manage the progression of infections in the ifestation of EVD infection in the community, per individual, community. is short when compared with the natural life span of an Our results show that eventually the system settles down average human. eTh reduced model was used to show that to a nonzero xfi ed point when there is constant recruitment all steady state solutions are stable to small perturbations into the population of 555 persons per day and for 𝑅 >1. and that there can be oscillatory returns to the equilibrium The values of the steady states are completely determined in solution. es Th e results were confirmed with numerical simu- terms of the parameters in this case. Our analysis also shows lations. Given the size of the system, we have not been able to S and S I and I N Q N Q Cumulative I and I N Q C and C N Q Computational and Mathematical Methods in Medicine 27 perform a detailed nonlinear analysis on the model. However 𝑅 (𝑡) : Population density at time𝑡 of all humans whowereonceinfectedwithEVD infection the discussion on the nature of the parameters for the model is and who have recovered from the infection; basedonthe statistics gathered from the2014EVD outbreak this classofpersons arethenimmunetoany in Africa andwebelieve that ourmodel canbeusefulinchar- further infection and are removed from the acterizing and studying a class of epidemics of Ebola-type. susceptible pool We have not yet carried out a complete sensitivity analysis on 𝑅 (𝑡) : Population density at time𝑡 of all humans all the parameters to determine the most crucial parameters whowereonceinfectedwithEVD andwho in our model. This is under consideration. Furthermore, the have died because of the EVD infection; this effect of stochasticity seems relevant to study. This and other class of persons though dead are still aspects of the model are under consideration. infectious 𝑅 (𝑡) : Population density at time𝑡 of all humans whodiednaturally or duetoother causes; Notations this is just a collection class 𝐷 (𝑡) : Population density at time𝑡 of all State Variables and eTh ir Descriptions Ebola-related dead humans removed from the infection cycle because they received proper burial or were cremated. 𝐻 (𝑡) : Total population density of living humans at any time𝑡 𝐻(𝑡) : eTh total population density at time 𝑡 of Parameters, eTh ir Descriptions, and eTh ir Corresponding living humans together with Ebola-related Quasidimension cadavers, that is, those humans who have died of EVD and have not yet been disposed Π: Net constant migration rate of humans, −1 at time𝑡 𝐻 𝑇 𝑆(𝑡) : Population density at time𝑡 of all susceptible 𝜌 : Eeff ctive contact rate between probable humans in the population nonquarantined individuals and susceptible 𝑆 (𝑡) ,𝑆 (𝑡) : Population densities at time𝑡 of all humans individuals; a fraction𝜃 of these contacts 𝑁 𝑄 that are known to have been in contact with are potentially infectious to the susceptible −1 or have a history of association with any humans,𝑇 person known to have once had or died of 𝜌 : Eeff ctive contact rate between probable EVD; these are suspected Ebola Virus patient quarantined individuals and susceptible cases that are either not quarantined𝑆 or individuals; a fraction𝜃 of these contacts are potentially infectious to the susceptible quarantined𝑆 and who are not yet showing −1 any Ebola-like symptoms humans,𝑇 𝑃 (𝑡) ,𝑃 (𝑡) : Population densities at time𝑡 of all persons 𝜏 : Eeff ctive contact rate between conrfi med 𝑁 𝑄 𝑁 suspected of having EVD infection and who nonquarantined early symptomatic −1 present with fever and at least three other individuals and susceptible individuals,𝑇 Ebola-like symptoms; these probable cases 𝜉 : Eeff ctive contact rate between conrfi med are either not quarantined𝑃 or quarantined nonquarantined late symptomatic −1 individuals and susceptible individuals,𝑇 𝐶 (𝑡) ,𝐶 (𝑡) : Population densities at time𝑡 of all probable 𝜏 : Eeff ctive contact rate between conrfi med 𝑁 𝑄 Ebola Virus infected humans who aeft r a lab quarantined early symptomatic individuals −1 test have been confirmed to indeed have and susceptible individuals,𝑇 EVD infection and who still present only 𝜉 : Eeff ctive contact rate between conrfi med early Ebola-like symptoms of fever, aches, quarantined late symptomatic individuals −1 tiredness, and so forth; they are called and susceptible individuals,𝑇 confirmed early symptomatic in the sense 𝑎 : Eeff ctive contact rate between cadavers of explained in the text; these confirmed Ebola confirmed late symptomatic individuals and −1 Virus carriers are either not quarantined𝐶 susceptible individuals,𝑇 or quarantined𝐶 𝜆 : A real function depending on the active 𝐼 (𝑡) ,𝐼 (𝑡) : Population densities at time𝑡 of all 𝑁 𝑄 members of the population representing the −1 conrfi med EVD patients who now present force of infection,𝑇 with full later stage Ebola-like symptoms; 𝜃 :Proportions;0≤𝜃 ≤1, 𝑖=1,2,...,7,1 𝑖 𝑖 they arecalledconrfi medlatesymptomatic 𝛼 : Rate at which nonquarantined suspected in the sense explained in the text; these cases become nonquarantined probable −1 confirmed EVD patients with full blown cases,𝑇 symptoms are either not quarantined𝐼 or 𝛼 : Rate at which quarantined suspected cases −1 quarantined𝐼 become quarantined probable cases,𝑇 𝑄 28 Computational and Mathematical Methods in Medicine 𝛽 : Rate at which nonquarantined probable [6] Center for Disease Control and Prevention, Ebola Virus Disease: cases become nonquarantined conrfi med Transmission,CDC,2014. −1 early symptomatic cases,𝑇 [7] Center for Disease Control and Prevention, Ebola Virus Dis- 𝛽 : Rate at which quarantined probable cases ease: Signs and Symptoms, CDC, 2014, http://www.cdc.gov/vhf/ become quarantined confirmed early ebola/symptoms/. −1 symptomatic cases,𝑇 [8] H. Feldmann and T. W. Geisbert, “Ebola haemorrhagic fever,” 𝛾 : Rate at which nonquarantined confirmed The Lancet ,vol.377,no. 9768,pp. 849–862, 2011. early symptomatic cases become [9] M. Goeijenbier, J. J. A. van Kampen, C. B. E. M. Reusken, nonquarantined confirmed late symptomatic M. P. G. Koopmans, and E. C. M. van Gorp, “Ebola virus −1 disease: a review on epidemiology, symptoms, treatment and cases,𝑇 pathogenesis,” The Netherlands Journal of Medicine ,vol.72, no. 𝛾 : Rate at which quarantined conrfi med early 9, pp.442–448,2014. symptomatic cases become quarantined −1 [10] Center for Disease Control and Prevention, Clinical Presenta- confirmed late symptomatic cases, 𝑇 tion and Clinical Course,CDC,2014. 𝛿 : Rate at which nonquarantined confirmed late −1 [11] A. L. Chan, What Actually Happens when a Person is Infected symptomatic cases die due to the EVD,𝑇 with the Ebola Virus,TheHuffington Post,2014. 𝛿 : Rate at which quarantined conrfi med late −1 symptomatic cases die due to the EVD,𝑇 [12] WHO ER Team, “Ebola virus disease in West Africa—the first −1 9 months of the epidemic and forward projections,” The New 𝜇 : Constant natural death rate for humans,𝑇 England Journal of Medicine,vol.371,no. 16,pp. 1481–1495, 2014. 𝑏 : Rate at which cadavers are removed and −1 [13] Central Intelligence Agency, Country Comparison: Life buried,𝑇 Expectancy at Birth,TheWorld Fact Book,2014. 𝑟 : Rateofrecoveryofnonquarantined −1 [14] J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron, and A. confirmed early symptomatic cases, 𝑇 Flahault, “Understanding the dynamics of Ebola epidemics,” 𝑟 : Rateofrecoveryofnonquarantined −1 Epidemiology and Infection,vol.135,no. 4, pp.610–621,2007. confirmed late symptomatic cases, 𝑇 [15] C. M. Rivers, E. T. Lofgren, M. 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Computational and Mathematical Methods in Medicine – Pubmed Central
Published: Aug 7, 2016
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