Research Article | Open Access
Gideon A. Ngwa, Miranda I. Teboh-Ewungkem, "A Mathematical Model with Quarantine States for the Dynamics of Ebola Virus Disease in Human Populations", Computational and Mathematical Methods in Medicine, vol. 2016, Article ID 9352725, 29 pages, 2016. https://doi.org/10.1155/2016/9352725
A Mathematical Model with Quarantine States for the Dynamics of Ebola Virus Disease in Human Populations
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, , 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. The equilibrium state, when it exists, is locally and asymptotically stable with oscillatory returns to the equilibrium point. The basic reproduction number, , is shown 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. Effective control measures, to reduce to values below unity, are discussed.
1. Introduction and Background
The world has been riveted by the 2014 outbreak of the Ebola Virus Disease (EVD) that affected parts of West Africa with Guinea, Liberia, and Sierra Leone being the most hard hit areas. Isolated cases of the disease did spread by land to Senegal and Mali (localized transmission) and by air to Nigeria. Some Ebola infected humans were transported to the US (except the one case that traveled to Texas and later on died) and other European countries for treatment. An isolated case occurred in Spain, another in Italy (a returning volunteer health care worker), and a few cases in the US and the UK [1–3]. Though dubbed the West African Ebola outbreak, the movement of patients and humans between countries, if not handled properly, could have led to a global Ebola pandemic. There was also a separate Ebola outbreak affecting a remote region in the Democratic Republic of Congo (formerly Zaire), and it was only by November 21, 2014 that the outbreak was reported to have ended .
The Ebola Virus Disease (EVD), formally known as Ebola haemorrhagic fever and caused by the Ebola Virus, is very lethal with case fatalities ranging from 25% to 90%, with a mean of about 50% . The 2014 EVD outbreak, though not the first but one of many other EVD outbreaks that have occurred in Africa since the first recorded outbreak of 1976, is the worst in terms of the numbers of Ebola cases and related deaths and the most complex . About 9 months after the identification of a mysterious killer disease killing villagers in a small Guinean village as Ebola, the 2014 West African Ebola outbreak, as of December 24, 2014, had up to 19497 Ebola cases resulting in 7588 fatalities [1, 3, 4], a case fatality rate of about 38.9%. By December 2015, the number of Ebola Virus cases (including suspected, probable, and confirmed) stood at 28640 resulting in 11315 fatalities, a case fatality rate of 39.5% [3, 5].
Ebola Virus, the agent that causes EVD, is hypothesised to be introduced into the human population through contact with the blood, secretions, fluids from organs, and other body parts of dead or living animals infected with the virus (e.g., fruit bats, primates, and porcupines) [2, 6]. Human-to-human transmission can then occur through direct contact (via broken skin or mucous membranes such as eyes, nose, or mouth) with Ebola Virus infected blood, secretions, and fluids secreted through organs or other body parts, in, for example, saliva, vomit, urine, faeces, semen, sweat, and breast milk. Transmission can also be as a result of indirect contact with surfaces and materials, in, for example, bedding, clothing, and floor areas, or objects such as syringes, contaminated with the aforementioned fluids [2, 6].
When a healthy human (considered here to be susceptible) who has no Ebola Virus in them is exposed to the virus (directly or indirectly), the human may become infected, if transmission is successful. The risk of being infected with the Ebola Virus is (i) very low or not recognizable where there is casual contact with a feverish, ambulant, self-caring patient, for example, sharing the same public place, (ii) low where there is close face-to-face contact with a feverish and ambulant patient, for example, physical examination, (iii) high where there is close face-to-face contact without appropriate personal protective equipment (including eye protection) with a patient who is coughing or vomiting, has nosebleeds, or has diarrhea, and (iv) very high where there is percutaneous, needle stick, or mucosal exposure to virus-contaminated blood, body fluids, tissues, or laboratory specimens in severely ill or known positive patients. The symptoms of EVD may appear during the incubation period of the disease, estimated to be anywhere from 2 to 21 days [2, 7–9], with an estimated 8- to 10-day average incubation period, although studies estimate it at 9–11 days for the 2014 EVD outbreak in West Africa . Studies have shown that, during the asymptomatic part of the Ebola Virus Disease, a human infected with the virus is not infectious and cannot transmit the virus. However, with the onset of symptoms, the human can transmit the virus and is hence infectious [2, 7]. The onset of symptoms commences the course of illness of the disease which can lead to death 6–16 days later [8, 9] or improvement of health with evidence of recovery 6–11 days later .
In the first few days of EVD illness (estimated at days 1–3 ), a symptomatic patient may exhibit symptoms common to those like the malaria disease or the flu (high fever, headache, muscle and joint pains, sore throat, and general weakness). Without effective disease management, between days 4 and 5 to 7, the patient progresses to gastrointestinal symptoms such as nausea, watery diarrhea, vomiting, and abdominal pain [10, 11]. Some or many of the other symptoms, such as low blood pressure, headaches, chest pain, shortness of breath, anemia, exhibition of a rash, abdominal pain, confusion, bleeding, and conjunctivitis, may develop [10, 11] in some patients. In the later phase of the course of the illness, days 7–10, patients may present with confusion and may exhibit signs of internal and/or visible bleeding, progressing towards coma, shock, and death [10, 11].
Recovery from EVD can be achieved, as evidenced by the less than 50% fatality rate for the 2014 EVD outbreak in West Africa. With no known cure, recovery is possible through effective disease management, the treatment of Ebola-related symptoms, and also the effective protection by the patient’s immune response . Some of the disease management strategies include hydrating patients by administering intravenous fluids and balancing electrolytes and maintaining the patient’s blood pressure and oxygen levels. Other schemes used include blood transfusion (using an Ebola survivor’s blood) and the use of experimental drugs on such patients (e.g., ZMAPP whose safety and efficacy have not yet been tested on humans). There are some other promising drugs/vaccines under trials . Studies show that once a patient recovers from EVD they remain protected against the disease and are immune to it at least for a projected period because they develop antibodies that last for at least 10 years . Once recovered, lifetime immunity is unknown or whether a recovered individual can be infected with another Ebola strain is unknown. However, after recovery, a person can potentially remain infectious as long as their blood and body fluids, including semen and breast milk, contain the virus. In particular, men can potentially transmit the virus through their seminal fluid, within the first 7 to 12 weeks after recovery from EVD . Table 1 shows the estimated time frames and projected progression of the infection in an average EVD patient.
Given that there is no approved drug or vaccine out yet, local control of the Ebola Virus transmission requires a combined and coordinated control effort at the individual level, the community level, and the institutional/health/government level. Institutions and governments need to educate the public and raise awareness about risk factors, proper hand washing, proper handling of Ebola patients, quick reporting of suspected Ebola cases, safe burial practices, use of public transportation, and so forth. These education efforts need to be communicated with community/chief leaders who are trusted by members of the communities they serve. From a global perspective, a good surveillance and contact tracing program followed by isolation and monitoring of probable and suspected cases, with immediate commencement of disease management for patients exhibiting symptoms of EVD, is important if we must, in the future, elude a global epidemic and control of EVD transmission locally and globally . It was by effective surveillance, contact tracing, and isolation and monitoring of probable and suspected cases followed by immediate supportive care for individuals and families exhibiting symptoms that the EVD was brought under control in Nigeria , Senegal, USA, and Spain .
Efficient control and management of any future EVD outbreaks can be achieved if new, more economical, and realizable methods are used to target and manage the dynamics of spread as well as the population sizes of those communities that may be exposed to any future Ebola Virus Disease outbreak. More realistic mathematical models can play a role in this regard, since analyses of such models can produce clear insight to vulnerable spots on the Ebola transmission chain where control efforts can be concentrated. Good models could also help in the identification of disease parameters that can possibly influence the size of the reproduction number of EVD. Existing mathematical models for Ebola [14, 16, 18–21] have been very instrumental in providing mathematical insight into the dynamics of Ebola Virus transmission. Many of these models have also been helpful in that they have provided methods to derive estimates for the reproduction number for Ebola based on data from the previous outbreaks. However, few of the models have taken into account the fact that institution of quarantine states or treatment centres will affect the course of the epidemic in the population . It is our understanding that the way the disease will spread will be determined by the initial and continual response of the health services in the event of the discovery of an Ebola disease case. The objective of this paper is to derive a comprehensive mathematical model for the dynamics of Ebola transmission taking into consideration what is currently known of the disease. The primary objective is to derive a formula for the reproduction number for Ebola Virus Disease transmission in view of providing a more complete and measurable index for the spread of Ebola and to investigate the level of impact of surveillance, contact tracing, isolation, and monitoring of suspected cases, in curbing disease transmission. The model is formulated in a way that it is extendable, with appropriate modifications, to other disease outbreaks with similar characteristics to Ebola, requiring such contact tracing strategies. Our model differs from other mathematical models that have been used to study the Ebola disease [14, 15, 18, 20–22] in that it captures the quarantined Ebola Virus Disease patients and provides possibilities for those who escaped quarantine at the onset of the disease to enter quarantine at later stages. To the best of our knowledge, this is the first integrated ordinary differential equation model for this kind of communicable disease of humans. Our final result would be a formula for the basic reproduction number of Ebola that depends on the disease parameters.
The rest of the paper is divided up as follows. In Section 2, we outline the derivation of the model showing the state variables and parameters used and how they relate together in a conceptual framework. In Section 3, we present a mathematical analysis of the derived model to ascertain that the results are physically realizable. We then reparameterise the model and investigate the existence and linear stability of steady state solution, calculate the basic reproduction number, and present some special cases. In Section 4, we present a discussion on the parameters of the model. In Section 5, we carry out some numerical simulations based on the selected feasible parameters for the system and then round up the paper with a discussion and conclusion in Section 6.
2. The Mathematical Model
2.1. Description of Model Variables
We divide the human population into 11 states representing disease status and quarantine state. At any time there are the following.
(1) Susceptible Individuals. Denoted by , this class also includes false probable cases, that is, all those individuals who would have displayed early Ebola-like symptoms but who eventually return a negative test for Ebola Virus infection.
(2) Suspected Ebola Cases. The class of suspected EVD patients comprises those who have come in contact with, or been in the vicinity of, anybody who is known to have been sick or died of Ebola. Individuals in this class may or may not show symptoms. Two types of suspected cases are included: the quarantined suspected cases, denoted by , and the nonquarantined suspected case, denoted by . Thus a suspected case is either quarantined or not.
(3) Probable Cases. The class of probable cases comprises all those persons who at some point were considered suspected cases and who now present with fever and at least three other early Ebola-like symptoms. Two types of probable cases are included: the quarantined probable cases, denoted by , and the nonquarantined probable cases, . Thus a probable case is either quarantined or not. Since the early Ebola-like symptoms of high fever, headache, muscle and joint pains, sore throat, and general weakness can also be a result of other infectious diseases such as malaria or flu, we cannot be certain at this stage whether or not the persons concerned have Ebola infection. However, since the class of probable persons is derived from suspected cases, and to remove the uncertainties, we will assume that probable cases may eventually turn out to be EVD patients and if that were to be the case, since they are already exhibiting some symptoms, they can be assumed to be mildly infectious.
(4) Confirmed Early Symptomatic Cases. The class of confirmed early asymptomatic cases comprises all those persons who at some point were considered probable cases and a confirmatory laboratory test has been conducted to confirm that there is indeed an infection with Ebola Virus. This class is called confirmed early symptomatic because all that they have as symptoms are the early Ebola-like symptoms of high fever, headache, muscle and joint pains, sore throat, and general weakness. Two types of confirmed early symptomatic cases are included: the quarantined confirmed early symptomatic cases and the nonquarantined confirmed early symptomatic cases . Thus a confirmed early symptomatic case is either quarantined or not. The class of confirmed early symptomatic individuals may not be very infectious.
(5) Confirmed Late Symptomatic Cases. The class of confirmed late symptomatic cases comprises all those persons who at some point were considered confirmed early symptomatic cases and in addition the persons who now present with most or all of the later Ebola-like symptoms of vomiting, diarrhea, stomach pain, skin rash, red eyes, hiccups, internal bleeding, and external bleeding. Two types of confirmed late symptomatic cases are included: the quarantined confirmed late symptomatic cases and the nonquarantined confirmed late symptomatic cases . Thus a confirmed late symptomatic case is either quarantined or not. The class of confirmed late symptomatic individuals may be very infectious and any bodily secretions from this class of persons can be infectious to other humans.
(6) Removed Individuals. Three types of removals are considered, but only two are related to EVD. The removals related to the EVD are confirmed individuals removed from the system through disease induced death, denoted by , or confirmed cases that recover from the infection denoted by . Now, it is known that unburied bodies or not yet cremated cadavers of EVD victims can infect other susceptible humans upon contact . Therefore, the cycle of infection really stops only when a cadaver is properly buried or cremated. Thus members from class, , representing dead bodies or cadavers of EVD victims are considered removed from the infection chain, and consequently from the system, only when they have been properly disposed of. The class, , of individuals who beat the odds and recover from their infection are considered removed because recovery is accompanied with the acquisition of immunity so that this class of individuals are then protected against further infection  and they no longer join the class of susceptible individuals. The third type of removal is obtained by considering individuals who die naturally or due to other causes other than EVD. These individuals are counted as .
The state variables are summarized in Notations.
2.2. The Mathematical Model
A compartmental framework is used to model the possible spread of EVD within a population. The model accounts for contact tracing and quarantining, in which individuals who have come in contact or have been associated with Ebola infected or Ebola-deceased humans are sought and quarantined. They are monitored for twenty-one days during which they may exhibit signs and symptoms of the Ebola Virus or are cleared and declared free. We assume that most of the quarantining occurs at designated makeshift, temporal, or permanent health facilities. However, it has been documented that others do not get quarantined, because of fear of dying without a loved one near them or fear that if quarantined they may instead get infected at the centre, as well as traditional practices and belief systems [14, 16, 22]. Thus, there may be many within communities who remain nonquarantined, and we consider these groups in our model. In all the living classes discussed, we will assume that natural death, or death due to other causes, occurs at constant rate where is approximately the life span of the human.
2.2.1. The Susceptible Individuals
The number of susceptible individuals in the population decreases when this population is exposed by having come in contact with or being associated with any of the possibly infectious cases, namely, infected probable case, confirmed case, or the cadaver of a confirmed case. The density increases when some false suspected individuals (a proportion of of nonquarantined and of quarantined) and probable cases (a proportion of of nonquarantined individuals and of quarantined individuals) are eliminated from the suspected and probable case list. We also assume a constant recruitment rate as well as natural death, or death due to other causes. Therefore the equation governing the rate of change with time within the class of susceptible individuals may be written aswhere is the force of infection and the rest of the parameters are positive and are defined in Notations. We identify two types of total populations at any time : (i) the total living population, , and (ii) the total living population including the cadavers of Ebola Virus victims that can take part in the spread of EVD, . Thus at each time we haveSince the cadavers of EVD victims that have not been properly disposed of are very infectious, the force of infection must then also take this fact into consideration and be weighted with instead of . The force of infection takes the following form:where is defined above and the parameters , , , , , , and are positive constants as defined in Notations. There are no contributions to the force of infection from the class because it is assumed that once a person recovers from EVD infection, the recovered individual acquires immunity to subsequent infection with the same strain of the virus. Although studies have suggested that recovered men can potentially transmit the Ebola Virus through seminal fluids within the first 7–12 weeks of recovery , and mothers through breast milk, we assume, here, that, with education, survivors who recover would have enough information to practice safe sexual and/or feeding habits to protect their loved ones until completely clear. Thus recovered individuals are considered not to contribute to the force of infection.
2.2.2. The Suspected Individuals
A fraction of the exposed susceptible individuals get quarantined while the remaining fraction are not. Also, a fraction (resp., ) of the nonquarantined (resp., quarantined) suspected individuals become probable cases at rate while the remainder (resp., ) do not develop into probable cases and return to the susceptible pool. For the quarantined individuals, we assume that they are being monitored, while the suspected nonquarantined individuals are not. However, as they progress to probable cases (at rates and ), a fraction of these humans will seek the health care services as symptoms commence and become quarantined while the remainder remain nonquarantined. Thus the equation governing the rate of change within the two classes of suspected persons then takes the following form:
In the context of this model we make the assumption that once quarantined, the individuals stay quarantined until clearance and are released, or they die of the infection. Notice that , so that .
2.2.3. The Probable Cases
The fractions and of suspected cases that become probable cases increase the number of individuals in the probable case class. The population of probable cases is reduced (at rates and ) when some of these are confirmed to have the Ebola Virus through laboratory tests at rates and . For some, proportions and , the laboratory tests are negative and the probable individuals revert to the susceptible class. From the proportion of nonquarantined probable cases whose tests are positive for the Ebola Virus (i.e., confirmed for EVD), a fraction, , become quarantined while the remainder, , remain nonquarantined. So . Thus the equation governing the rate of change within the classes of probable cases takes the following form:
2.2.4. The Confirmed Early Symptomatic Cases
The fractions and of the probable cases become confirmed early symptomatic cases thus increasing the number of confirmed cases with early symptoms. The population of early symptomatic individuals is reduced when some recover at rates for the nonquarantined cases and for the quarantined cases. Others may see their condition worsening and progress and become late symptomatic individuals, in which case they enter the full blown late symptomatic stages of the disease. We assume that this progression occurs at rates or , respectively, which are the reciprocal of the mean time it takes for the immune system to either be completely overwhelmed by the virus or be kept in check via supportive mechanism. A fraction of the confirmed nonquarantined early symptomatic cases will be quarantined as they become late symptomatic cases, while the remaining fraction escape quarantine due to lack of hospital space or fear and belief customs [16, 22] but become confirmed late symptomatic cases in the community. Thus the equation governing the rate of change within the two classes of confirmed early symptomatic cases takes the following form:
2.2.5. The Confirmed Late Symptomatic Cases
The fractions and of confirmed early symptomatic cases who progress to the late symptomatic stage increase the number of confirmed late symptomatic cases. The population of late symptomatic individuals is reduced when some of these individuals are removed. Removal could be as a result of recovery at rates proportional to and or as a result of death because the EVD patient’s conditions worsen and the Ebola Virus kills them. The death rates are assumed proportional to and . Additionally, as a control effort or a desperate means towards survival, some of the nonquarantined late symptomatic cases are removed and quarantined at rate . In our model, we assume that Ebola-related death only occurs at the late symptomatic stage. Additionally, we assume that the confirmed late symptomatic individuals who are eventually put into quarantine at this late period (removing them from the community) may not have long to live but may have a slightly higher chance at recovery than when in the community and nonquarantined. Since recovery confers immunity against the particular strain of the Ebola Virus, individuals who recover become refractory to further infection and hence are removed from the population of susceptible individuals. Thus the equation governing the rate of change within the two classes of confirmed late symptomatic cases takes the following form:
2.2.6. The Cadavers and the Recovered Persons
The dead bodies of EVD victims are still very infectious and can still infect susceptible individuals upon effective contact . Disease induced deaths from the class of confirmed late symptomatic individuals occur at rates and and the cadavers are disposed of via burial or cremation at rate . The recovered class contains all individuals who recover from EVD. Since recovery is assumed to confer immunity against the 2014 strain (the Zaire Virus)  of the Ebola Virus, once an individual recovers, they become removed from the population of susceptible individuals. Thus the equation governing the rate of change within the two classes of recovered persons and cadavers takes the following form:
The population of humans who die either naturally or due to other causes is represented by the variable and keeps track of all natural deaths, occurring at rate , from all the living population classes. This is a collection class. Another collection class is the class of disposed Ebola-related cadavers, disposed at rate . These collection classes satisfy the equations
Putting all the equations together we havewhere and all other parameters and state variables are as in Notations.
Suitable initial conditions are needed to completely specify the problem under consideration. We can, for example, assume that we have a completely susceptible population, and a number of infectious persons are introduced into the population at some point. We can, for example, have that
Class is used to keep count of all the dead that are properly disposed of, class is used to keep count of all the deaths due to EVD, and class is used to keep count of the deaths due to causes other than EVD infection. The rate of change equation for the two groups of total populations is obtained by using (2) and (3) and adding up the relevant equations from (11) to (23) to obtainwhere is the total living population and is the augmented total population adjusted to account for nondisposed cadavers that are known to be very infectious. On the other hand, if we keep count of all classes by adding up (11)–(23), the total human population (living and dead) will be constant if . In what follows, we will use the classes , , and , comprising classes of already dead persons, only as place holders, and study the problem containing the living humans and their possible interactions with cadavers of EVD victims as often is the case in some cultures in Africa, and so we cannot have a constant total population. Note that (26) can also be written as .
2.2.7. Infectivity of Persons Infected with EVD
Ebola is a highly infectious disease and person to person transmission is possible whenever a susceptible person comes in contact with bodily fluids from an individual infected with the Ebola Virus. We therefore define effective contact here generally to mean contact with these fluids. The level of infectivity of an infected person usually increases with duration of the infection and severity of symptoms and the cadavers of EVD victims are the most infectious . Thus we will assume in this paper that probable persons who indeed are infected with the Ebola Virus are the least infectious while confirmed late symptomatic cases are very infectious and the level of infectivity will culminate with that of the cadaver of an EVD victim. While under quarantine, it is assumed that contact between the persons in quarantine and the susceptible individuals is minimal. Thus though the potential infectivity of the corresponding class of persons in quarantine increases with disease progression, their effective transmission to members of the public is small compared to that from the nonquarantined class. It is therefore reasonable to assume that any transmission from persons under quarantine will affect mostly health care providers and use that branch of the dynamics to study the effect of the transmission of the infections to health care providers who are here considered part of the total population. In what follows we do not explicitly single out the infectivity of those in quarantine but study general dynamics as derived by the current modelling exercise.
3. Mathematical Analysis
3.1. Well-Posedness, Positivity, and Boundedness of Solution
In this subsection we discuss important properties of the model such as well-posedness, positivity, and boundedness of the solutions. We start by defining what we mean by a realistic solution.
It is evident that a solution satisfying Definition 1 is physically realizable in the sense that its values can be measured through data collection. For notational simplicity, we use vector notation as follows: let be a column vector in containing the state variables, so that, in this notation, . Let be the vector valued function defined in so that in this notation is the right-hand side of the differential equation for first variable , is the right-hand side of the equation for the second variable , and is the right side of the differential equation for the 11th variable , and so is precisely system (11)–(21) in that order with prototype initial conditions (24). We then write the system in the formwhere is a column vector of state variables and is the vector containing the right-hand sides of each of the state variables as derived from corresponding equations in (11)–(21). We can then have the following result.
Lemma 2. The function in (27) is Lipschitz continuous in .
Proof. Since all the terms in the right-hand side are linear polynomials or rational functions of nonvanishing polynomial functions, and since the state variables, , , , , , and , are continuously differentiable functions of , the components of the vector valued function of (27) are all continuously differentiable. Further, let . Then is a line segment that joins points to the point as ranges on the interval . We apply the mean value theorem to see that where is the directional derivative of the function at the mean value point in the direction of the vector . But, where is the th coordinate unit vector in . It is now a straightforward computation to verify that since is a convex set, and taking into consideration the nature of the functions , all the partial derivatives are bounded and so there exist such that and so there exist such that and hence is Lipschitz continuous.
Theorem 3 (uniqueness of solutions). The differential equation (27) has a unique solution.
Proof. We show that each trajectory of the system starting in will remain in . Assume for a contradiction that there exists a point such that , (where the prime denotes differentiation with respect to time) but for , , and , , , , , , , , , and . So, at the point , is decreasing from the value zero in which case it will go negative. If such an will satisfy the given differential equation, then we haveand so contradicting the assumption that . So no such exist. The same argument can be made for all the state variables. It is now a simple matter to verify, using techniques as explained in , that whenever we start system (27), with nonnegative initial data in , the solution will remain nonnegative for all and that if , the solution will remain , and the region is indeed positively invariant.
The last two theorems have established the fact that, from a mathematical and physical standpoint, the differential equation (27) is well-posed. We next show that the nonnegative unique solutions postulated by Theorem 3 are indeed realistic in the sense of Definition 1.
Proof. It suffices to prove that the total living population size is bounded for all . We show that the solutions lie in the bounded regionFrom the definition of given in (2), if is bounded, the rest of the state variables that add up to will also be bounded. From (25) we haveThus, from (34), we see that, whatever the size of , is bounded above by a quantity that converges to as . In particular, if , then is bounded above by , and for all initial conditions Thus is nonnegative and bounded.
Remark 6. Starting from the premise that for all , Theorem 5 establishes boundedness for the total living population and thus by extension verifies the positive invariance of the positive octant in as postulated by Theorem 4, since each of the variables functions , , , , , and , where , is a subset of .
3.2. Reparameterisation and Nondimensionalisation
The only physical dimension in our system is that of time. But we have state variables which depend on the density of humans and parameters which depend on the interactions between the different classes of humans. A state variable or parameter that measures the number of individuals of certain type has dimension-like quantity associated with it . To remove the dimension-like character on the parameters and variables, we make the following change of variables:where We then define the dimensionless parameter groupingsThe force of infection then takes the formThis leads to the equivalent system of equationsand the total populations satisfy the scaled equationwhere if it is assumed that the rate of disposal of Ebola Virus Disease victims, , is larger than the natural human death rate, . The scaled or dimensionless parameters are then as follows:
3.3. The Steady State Solutions and Linear Stability
The steady state of the system is obtained by setting the right-hand side of the scaled system to zero and solving for the scalar equations. Let be a steady state solution of the system. Then, (43), (45), and (47) indicate thatand we can use any of these as a parameter to derive the values of the other steady state variables. We use the variables and as parameters to obtain the expressionswhereHere the solution for the scaled total living () and scaled living and Ebola-deceased () populations is, respectively, obtained by equating the right-hand sides of (53) and (54) to zero, while that for is obtained by adding up (40), (41), and (42). It is easy to verify from reparameterisation (38) that the parameter groupings and are both nonnegative. In fact, showing that and .
To obtain a value for and , we substitute all computed steady state values, (56) and (57), into (41) and (42). The expression for in terms of and is obtained from (39). Performing the aforementioned procedures leads to the two equationswhereNext, we solve (60) and (61) simultaneously, which clearly differ in some of their coefficients, to obtain the expressions for and . Quickly observe that the two equations may be reduced to one such thatTwo possibilities arise: either (i) or (ii) . The first condition leads to the systemHowever, the two equations are equivalent to and (see (57)), which are unrealistic, based on our constant population recruitment model. Hence, we only consider the second possibility, which yields the relationso that substituting (65) into (60) yieldswhere with
Remark 7. It can be shown that . In fact, Thus, if , then . This will hold if . In the case where , we will require that be greater than
We identify as the unique threshold parameter of the system as follows.
Lemma 8. The parameter defined in (69) is the unique threshold parameter of the system whenever .
Proof. If , then whenever and the existence or nonexistence of a realistic solution of the form of (66) is determined solely by the size of .
The rest of the steady states are then obtained by using these values for and given by (66) in (65) and (57) to obtain the following:where , , and are as defined in (68). We have proved the following result.
Theorem 9 (on the existence of equilibrium solutions). System (40)–(52) has at least two equilibrium solutions: the disease-free equilibrium and an endemic equilibrium . The endemic equilibrium, , exists and is realistic only when the threshold parameters and , given by (69), are of appropriate magnitude.
The stability of the steady states is governed by the sign of the eigenvalues of the linearizing matrix near the steady state solutions. If is the Jacobian matrix at the steady state , then we havewhere . Thus if is an eigenvalue of the linearized system at the disease-free state, then is obtained by the solvability conditionan equation involving a polynomial of degree 12 in , where is a polynomial of degree in , given by