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Computational and Mathematical Methods in Medicine
Volume 2017 (2017), Article ID 1473287, 29 pages
https://doi.org/10.1155/2017/1473287
Research Article

A Multiscale Model for the World’s First Parasitic Disease Targeted for Eradication: Guinea Worm Disease

Modelling Health and Environmental Linkages Research Group (MHELRG), Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Correspondence should be addressed to Winston Garira; az.ca.nevinu@arirag.notsniw

Received 13 October 2016; Revised 8 April 2017; Accepted 15 May 2017; Published 20 July 2017

Academic Editor: José Siri

Copyright © 2017 Rendani Netshikweta and Winston Garira. 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.

Abstract

Guinea worm disease (GWD) is both a neglected tropical disease and an environmentally driven infectious disease. Environmentally driven infectious diseases remain one of the biggest health threats for human welfare in developing countries and the threat is increased by the looming danger of climate change. In this paper we present a multiscale model of GWD that integrates the within-host scale and the between-host scale. The model is used to concurrently examine the interactions between the three organisms that are implicated in natural cases of GWD transmission, the copepod vector, the human host, and the protozoan worm parasite (Dracunculus medinensis), and identify their epidemiological roles. The results of the study (through sensitivity analysis of ) show that the most efficient elimination strategy for GWD at between-host scale is to give highest priority to copepod vector control by killing the copepods in drinking water (the intermediate host) by applying chemical treatments (e.g., temephos, an organophosphate). This strategy should be complemented by health education to ensure that greater numbers of individuals and communities adopt behavioural practices such as voluntary reporting of GWD cases, prevention of GWD patients from entering drinking water bodies, regular use of water from safe water sources, and, in the absence of such water sources, filtering or boiling water before drinking. Taking into account the fact that there is no drug or vaccine for GWD (interventions which operate at within-host scale), the results of our study show that the development of a drug that kills female worms at within-host scale would have the highest impact at this scale domain with possible population level benefits that include prevention of morbidity and prevention of transmission.

1. Introduction

Guinea worm disease (GWD), sometimes known as Dracunculiasis or dracontiasis [1], is a nematode infection transmitted to humans exclusively through contaminated drinking water. People become infected when they drink water contaminated with copepods or cyclopoids (tiny aquatic crustaceans) harbouring infective Dracunculus larvae also known as Dracunculus medinensis. The larvae of Dracunculus medinensis are released into the stomach, when the copepods are digested by the effect of the gastric juice and get killed by the acid environment. Although the disease has low mortality, its morbidity is considerably high causing huge economic losses and devastating disabilities [2]. There is no vaccine or drug for the disease. Our ability to eliminate GWD rests partly on gaining better insights into the functioning of the immune system, especially its interaction with Guinea worm parasite and partly on development of drugs to treat the disease together with implementation of preventive measures. Currently, the only therapy for GWD is to physically extract the worm from the human body. Humans are the sole definitive host for GWD parasite. Efforts to eradicate the disease are focused on preventive measures which include the following:(a)Parasite control in the physical water environment. This may involve chlorination of drinking water, or boiling the water before drinking, or applying a larvicide, all of which have the effect of killing the parasite and thereby reduce parasite population in the physical water environment.(b)Parasite control within the human host. This involves physically extracting the worm from the human body by rolling it over an ordinary stick or matchstick [1, 3] and ensuring that the patient receives care by cleaning and bandaging the wound until all the worms are extracted from the patient. This process may take up to two months to complete, as the worm can grow up to a meter in length and only 1-2 centimeters can be removed per day [4, 5].(c)Vector control. This consists of killing the copepods in water (the intermediate host) by applying a chemical called temephos, an organophosphate, to unsafe drinking water sources every month during the transmission season, thus reducing vector population and reducing the chances of individuals contracting the disease [2, 6, 7]. The adult vector may also be removed from drinking water by filtering the water using a nylon cloth or by boiling the water.(d)Health education. This is disseminated through poster, radio and television broadcast, village criers and markets, face-to-face communication (social mobilization and house-to-house visits) by health workers and volunteers to ensure that greater numbers of individuals and communities adopt behavioural practices aimed at preventing transmission of GWD [8]. These behavioural practices include voluntary reporting of GWD cases, prevention of GWD patients from entering drinking water bodies, regular use of water from safe water sources, and, in the absence of such water sources, filtering or boiling water before drinking [6].(e)Provision of safe water sources. This involves providing safe drinking water supplies through protecting hand-dug wells and sinking deep bore wells, improving existing surface water sources by constructing barriers to prevent humans from entering water, and filtering the water through sand-filters [4].

To date, these preventive measures have reduced the incidence of GWD by over 99% [6], making GWD the most likely parasitic disease that will soon be eradicated without the use of any drug or vaccine. Most countries, including the whole of Asia, are now declared free from GWD and transmission of the disease is now limited to African countries, especially Sudan, Ghana, Mali, Niger, and Nigeria [8]. GWD is one of the neglected tropical diseases. It is also an environmentally driven infectious disease. Therefore, its transmission depends on the parasite’s survival in the environment and finding new hosts (humans and copepod vectors) in order to replicate and sustain parasite population. Because this process is complex, it has hampered eradication efforts. During the parasite’s movement through the environment to the human and copepod vector hosts, many environmental factors influence both the parasite’s population and the vector population.

For infectious diseases, including environmentally driven infectious diseases such as GWD, mathematical models have a long history of being used to study their transmission and also to compare and evaluate the effectiveness and affordability of intervention strategies that can be used to control or eliminate them [9, 10]. Currently, the predominant focus of modelling of infectious diseases is centered on concepts of epidemiological modelling and immunological modelling being considered as separate disease processes even for the same infectious disease. In epidemiological or between-host modelling of infectious diseases, the focus is on studying of transmission of infectious diseases between hosts, be they animals or humans or even both in the case of multiple host infections. In the immunological or within-host modelling of infectious diseases, the focus is on studying the interaction of pathogen and the immune system together with other within-host processes in order to elucidate outcomes of infection within a single host [11, 12]. To the best of our knowledge there has been no mathematical model to study the multiscale nature of GWD transmission by integrating between-host scale and within-host scale disease processes. Such models are sometimes called immunoepidemiological models [13]. Most of the mathematical models that have been developed so far are focused on the study of GWD at the epidemiological scale [1416]. The purpose of this study is to develop an immunoepidemiological model of GWD. Immunoepidemiological modelling of infectious diseases is the quantitative approach which assists in developing a systems approach to understanding infectious disease transmission dynamics with regard to the interdependences between epidemiological (between-host scale) and immunological (within-host scale) processes [17, 18]. The immunoepidemiological model of GWD presented in this paper is based on a modelling framework of the immunoepidemiology of environmentally driven infectious diseases developed recently by the authors [13]. This new and innovative immunoepidemiological modelling framework, while maintaining the limits of a mathematical model, offers a solid platform to bring the separate modelling efforts (immunological modelling and epidemiological modelling) that focus on different aspects of the disease processes together to cover a broad range of disease aspects and time-scales in an integrated systems approach. It bridges host, environmental, and parasitic disease phenomena using mathematical modelling of parasite-host-environment-vector interactions and epidemiology to illuminate the fundamental processes of disease transmission in changing environments. For GWD there are three distinct time-scales associated with its transmission cycle which are as follows.(i)The epidemiological time-scale, which is associated with the infection between hosts (human and copepod vector hosts).(ii)The within-host time-scale, which is related to the replication and developmental stages of Guinea worm parasite within an individual human host and the individual copepod vector host.(iii)The environmental time-scale, which is associated with the abundance and survival of Guinea worm parasite population and vector population in the physical water environment.

In order to try and integrate these different processes and the associated time-scales of GWD, the immunoepidemiological model of GWD presented here incorporates the actual parasite load of the human host and copepod vector, rather than simply tracking the total number of infected humans. It also incorporates the various stages of the parasite’s life cycle as well as the within-host effects such as the effect of gastric juice within an infected human host and describes how the life stages in the definitive human host, environment, and intermediate vector are interconnected with the parasite’s life cycle through contact, establishment, and parasite fecundity. The paper is organized as follows. In Section 2 we present brief discussion of the life cycle of Guinea worm parasite and use this information to develop the immunoepidemiological model of GWD in the same section. In Sections 3, 4, and 5, we derive the analytical results associated with the immunoepidemiological model and show that the model is mathematically and epidemiologically well-posed. We also show the reciprocal influence between the within-host scale and between-host scale of GWD transmission dynamics. The results of the sensitivity analysis of the reproductive number are given in Section 6 while the numerical results of the model are presented in Section 7. The paper ends with conclusions in Section 8.

2. The Mathematical Model

We develop a multiscale model of Guinea worm disease that traces the parasite’s life cycle of Guinea worm disease. The life cycle of GWD involves three different environments: physical water environment, biological human host environment, and biological copepod host environment. For more details on the life cycle of GWD see the published works [6, 19]. We only give a brief description in this section. The transmission cycle of Guinea worm disease begins when the human individual drinks contaminated water with copepods that are infected with Guinea mature worm larvae (L3 larvae). After ingestion, gastric juice in the human stomach kills the infected copepods and mature worm larvae are released. Then the released mature worm larvae penetrate the human stomach and intestinal wall and move to abdominal tissues where they grow and mate. After mating the male worms die soon and fertilized female worms migrate towards the skin surface (usually on the lower limbs or feet). After a year of infection, the fertilized female worm makes a blister on the infected individual’s skin causing burning and itching, which forces an infected individual to immerse his or her feet into water (which is the only source of drinking water) to seek relief from pain. At that point the female worm emerges and releases thousands of worm eggs. The worm eggs then hatch Guinea worm larvae (L1 larvae stage) which are then consumed by copepods and take approximately two weeks to develop and become infective mature larvae (L3 larvae) within the copepods. Then ingestion of the infected copepods by human closes the life cycle. The multiscale model which we now present explicitly traces this life cycle of Dracunculus medinensis in three different environments, which are physical water environment, biological human environment, and biological copepod environment. The model flow diagram is shown in Figure 1.

Figure 1: A conceptual diagram of the multiscale model of Guinea worm disease transmission dynamics.

The full multiscale model presented in this paper is based on monitoring the dynamics of ten populations at any time , which are susceptible humans and infected humans in the behavioural human environment; infected copepods in the human biological environment; mature Guinea worms and fertilized female Guinea worms in the biological human environment (within-host parasite dynamics); Guinea worm eggs and Guinea worm larvae in the physical water environment; susceptible copepods and infected copepods in the physical water environment; and gastric juice in the human biological environment. We make the following assumptions for the model:(a)There is no vertical transmission of the disease.(b)The transmission of the disease in the human population is only through drinking contaminated water with infected copepods, , harbouring infective free-living pathogens (first-stage larvae), , in the physical water environment.(c)For an infected individual, more than one Guinea worm can emerge simultaneously or sequentially over the course of weeks, depending on the number and intensity of infection the preceding year.(d)Humans do not develop temporary or permanent immunity.(e)Copepods do not recover from infection.(f)The total population of humans and copepods is constant.(g)Except for the effects of gastric juice in the stomach, there is no immune response in the human host.(h)Copepods die in the human stomach due to the effects of gastric juice at a rate before their larvae undergoes two molts in the copepod to become L3 larvae and therefore are nonviable and noninfectious larvae.

From the model flow diagram presented in Figure 1 and the assumptions that we have now made, we have the following system of ordinary differential equations as our multiscale model for GWD transmission dynamics:where

Equations () and () of the model system (1) describe the evolution with time of susceptible and infected human hosts, respectively. At any time , new susceptible humans are recruited at a constant rate and we assume that the recruited humans are all susceptible. Susceptible individuals leave the susceptible class either through infection at rate by drinking contaminated water with infected copepods to join infected group or through natural death at a rate . The infected group is generated through infection when susceptible humans acquire the disease at a rate through drinking water contaminated with copepods infected with Dracunculus medinensis. Infected humans leave the infected group either through recovery at a rate to join the susceptible group or through natural death at a rate , or through disease induced death at a rate . Equation () of the model system (1) represents the evolution with time of infected copepods within an infected human host. The infected copepods within a human host are generated following uptake of infected copepods in the physical water environment through drinking contaminated water. In the human population, this uptake of infected copepods, which harbour Guinea worm larvae, is the transmission of Guinea worm parasite from the physical water environment to susceptible humans who become infected humans. Following the methodology described in [13] for modelling reinfection (superinfection) for environmentally transmitted infectious disease systems (because GWD and schistosomiasis are both water-borne and vector-borne infections), we model the average rate at which a single susceptible human host uptakes the infected copepods in the physical water environment through drinking contaminated water and becomes an infected human host by the expressionwhere , , and are as defined previously. This is because, in our case, we define such a single infection by a single transition

Therefore, the average number of infected copepods, , within a single infected human host increases at a mean rate and decreases through death due to digestion by human gastric acid at a rate after their larvae undergo two molts in the copepod to become L3 larvae and release viable and infectious larvae or naturally at a rate before their larvae undergo two molts in the copepod to become L3 larvae and release nonviable and noninfectious larvae.

Equations (4–6) of the model system (1) represent changes with time of the average population of mature worms , fertilized female worms , and the amount of gastric acid within a single infected human host, respectively. The average mature worm population in a single infected human host is generated following the digestion of infected copepods in the human stomach by gastric acid and then mature worms are released. We assume that mature worms die naturally at a rate and they exit the human stomach to the abdominal tissues at a rate , where they grow and mate. The population of fertilized female worms, within an infected human host, is generated following the developmental changes undergone by mature fertilized female worms. These developmental changes result in mature worms reaching sexual maturity and mating and all male worms die soon after mating. We assume that fertilized female worms die naturally at a rate and emerge out through an infected human individual’s skin (usually the lower limbs) to release Guinea worm eggs into a water source at a rate , when an infected human comes into contact with water. The average amount of gastric acid inside a human stomach is generated following copepod vector induced proliferation at a rate , which is proportional to the density of infected copepods within an infected human host. We assume that the amount of gastric acid is also increased by the spontaneous production of gastric acid by the human body at a rate and diluted or degraded at a rate . Equation () of model system (1) describes the evolution with time of the Guinea worm eggs in the physical water environment. We note that the population of Guinea worm eggs increases when each infected human host excretes eggs at a rate . Therefore the rate at which infected humans contaminate the physical water environment by excreting Guinea worm eggs is modelled by . The last three equations of the model system (1) describe the evolution with time of Guinea worm larvae , susceptible copepods , and infected copepods in the physical water environment, respectively. The population of Guinea worm larvae is generated through each egg hatching an average of worms larvae with eggs hatching at an average rate of . Therefore the total Guinea worm larvae in the physical water environment are modelled by . We assume that worm larvae in the physical water environment die naturally at a constant rate . Similar to human population, at any time , new susceptible copepods are recruited at a constant . Susceptible copepods leave the susceptible group to join the infected copepods group through infection at a rate when they consume first-stage Guinea worm larvae in the physical water environment. We assume that the population of copepods die naturally at a constant rate and further, we also assume that infected copepods have an additional mortality rate due to infection. The model state variables are summarized in Table 1.

Table 1: Description of the state variables of the model system (1).

3. Invariant Region of the Model

The model system (1) can be analysed in a region of biological interest. Now assume that all parameters and state variables for model system (1) are positive for all and further suppose that is bounded above by . It can be shown that all solutions for the model system (1) with positive initial conditions remain bounded.

Letting and adding () and () of model system (1) we obtain

This implies that

Similarly, letting and adding () and () of model system (1) we obtain

This also implies that

Now considering the third equation of model system (1), given bywe obtain

This implies that

Using (6), (8), and (11) similar expression can be derived for the remaining model variables. Hence, all feasible solutions of the model system (1) are positive and enter a region defined by

which is positively invariant and attracting for all , where

Therefore it is sufficient to consider solutions of the model system (1) in , since all solutions starting in remain there for all . Hence, the model system is mathematically and epidemiologically well-posed and it is sufficient to consider the dynamics of the flow generated by model system (1) in whenever . We shall assume in all that follows (unless stated otherwise) that .

4. Determination of Disease-Free Equilibrium and Its Stability

To obtain the disease-free equilibrium point of system (1), we set the left-hand side of the equations equal to zero and further we assume that . This means that all the populations are free from the disease. Thus we getas the disease-free equilibrium of the model system (1).

4.1. The Basic Reproduction Number of the Model System (1)

The basic reproduction number of the system model (1) is calculated in this section using next generation operator approach described in [20]. Thus the model system (1) can also be written in the formwhere(i) represents all compartments of individuals who are not infected,(ii) represents all compartments of infected individuals who are not capable of infecting others,(iii) represents all compartments of infected individuals who are capable of infecting others.

We also let the disease-free equilibrium of the model (1) be denoted by the following expression:

Following [20], we let

withwhere

We deduce thatwith

where

matrix

can be written in the form , so that

The basic reproductive number is the spectral radius (dominant eigenvalue) of the matrix . Hence, the basic reproduction number of the immumoepidemiological model (1) is expressed by the following quantity.with

The expression, , in (27) represents GWD’s partial reproductive number associated with the between-host transmission of the disease while the expression, , in (28) represents GWD’s partial reproductive number associated with the within-host transmission of the disease. From the above two expressions in (27) and (28), respectively, we therefore make the following deductions.(i)The epidemiological (between-host) transmission parameters such as the rate at which susceptible humans come into contact with water contaminated with infected copepods (through drinking contaminated water with infected copepods) and the rate at which susceptible copepods come into contact with Guinea worm larvae ; the supply rate of susceptible humans and copepods (through birth); the rate at which worms emerge from infected humans to contaminate the physical water environment , by laying eggs every time infected humans come into contact with water sources; the rate at which eggs in physical water environment hatch to produce worm larvae all contribute to the transmission of Guinea worm disease. Therefore control measures such as reducing the rate at which infected human hosts visit water sources when an individual is infected, reducing contact rate between susceptible humans with contaminated water through educating the public, and treating water bodies with chemicals that kill worm eggs, worm larvae, and copepods may help to reduce the transmission risk of GWD.(ii)The immunological (within-host) transmission parameters such as the rate at which infected copepods within an infected human host release mature worms after digestion by human gastric juice; the rate at which mature worms become fertilized females worms ; and the rate at which mature worms and females worms die all contribute to the transmission of Guinea worm disease. Therefore immune mechanisms that kill infected copepods and worms within infected human host and also treatment intend to kill both mature worms and fertilized female worm population may help to reduce the transmission risk of GWD.

Therefore, both the epidemiological and immunological factors affect the transmission cycle of GWD in both humans and copepod population.

4.2. Local Stability of DFE

In this section we determine the local stability of DFE of the model system (1). We linearize equations of the model system (1) in order to obtain a Jacobian matrix. Then we evaluate the Jacobian matrix of the system at the disease-free equilibrium (DFE),

The Jacobian matrix of the model system (1) evaluated at the disease-free equilibrium state (DFE) is given by

where

We consider stability of DFE by calculating the eigenvalues () of the Jacobian matrix given by (30). The characteristic equation for the eigenvalues is given by

where

It is clear from (32) that there are four negative eigenvalues (, , , and ). Now in order to make conclusions about the stability of the DFE, we use the Routh-Hurwitz criteria to determine the sign of the remaining eigenvalues of the polynomial

where

Using the Routh-Hurwitz stability criterion, the equilibrium state associated with the model system (1) is stable if and only if the determinants of all the Hurwitz matrices associated with the characteristic equation (34) are positive; that is,

where

The Routh-Hurwitz criterion applied to (37) requires that the following conditions (H1)–(H6) be satisfied, in order to guarantee the local stability of the disease-free equilibrium point of the model system (1).(H1).(H2).(H3).(H4).(H5).(H6).

From (37) we note that all the coefficients , , , , , and of the polynomial are greater than zero whenever . And we also noted that the conditions above are satisfied if and only if . Hence all the roots of the polynomial either are negative or have negative real parts. The results are summarized in the following theorem.

Theorem 1. The disease-free equilibrium point of the model system (1) is locally asymptotically stable whenever .

4.3. Global Stability of DFE

To determine the global stability of DFE of the model system (1), we use Theorem in [21] to establish that the disease-free equilibrium is globally asymptotically stable whenever and unstable when . In this section, we list two conditions that if met, also guarantee the global asymptotic stability of the disease-free state. We write the model system (1) in the formwhere(i) represents all uninfected components.(ii) represents all compartments of infected and infectious components.

We letdenote the disease-free equilibrium (DFE) of the system. To guarantee global asymptotic stability of the disease-free equilibrium, conditions (H1) and (H2) below must be met [20].(H1) is globally asymptotically stable,(H2) and for , where is an -matrix and is the region where the model makes biological sense.

In our case

Matrix is given by

where

Assume that and . It is clear that for all , since , , and provided that . It is also clear that is an -matrix, since the off diagonal elements of are nonnegative. We state a theorem which summarizes the above result.

Theorem 2. The disease-free equilibrium of model system (1) is globally asymptotically stable if and the assumptions (H1) and (H2) are satisfied.

5. The Endemic Equilibrium State and Its Stability

At the endemic equilibrium humans are infected by copepods that have been infected by first-stage larvae . The endemic equilibrium point of the model system (1) given bysatisfies

for all . We therefore obtain the following endemic values. The endemic value of susceptible humans is given by

From (45) we note that the susceptible human population at endemic equilibrium is proportional to the average time of stay in the susceptible class and the rate at which new susceptible individuals are entering the susceptible class either through birth or through infected individuals who recover from the disease. Individuals leave the susceptible class through either infection or death. The endemic value of infected humans is given by

We note from (46) that the population of infected humans at the endemic equilibrium point is proportional to the average time of stay in the infected class, the rate at which susceptible individuals become infected, and the density of susceptible individuals. The endemic value of infected copepods population within a single infected human at the equilibrium point is given bywhere . From (47) we note that the average infected copepod population within a single infected human is proportional to the average life-span of infected copepods within a single infected human host and the rate of infection of a single susceptible individual to become infected. We also note that this expression provides a link between the dynamics of the infected copepods within-host and human population dynamics. The endemic value of mature worm population within a single infected human is given by

We note from (48) that the population of mature worms within a single infected human at endemic equilibrium point is proportional to the average life-span of mature worms and the rate at which mature worms are released after infected copepods within human host have been killed by human gastric juice. The endemic value of fertilized female worm population within a single infected human is given by

The average population of fertilized female worms within an infected human at endemic equilibrium point is equal to the average life-span of female worms and the rate at which mature worms become fertilized female worms. The endemic value of a single human gastric juice is given bywhere . The endemic value of Guinea worm eggs population in the physical water environment is given by

We note from (51) that the worm egg population at equilibrium point is proportional to the average life-span of eggs, the rate at which each infected human host excretes Guinea worm eggs, and the total number of infected humans. The endemic value of Guinea worm larva population in the physical water environment is given by

We note from (52) that the larvae population at equilibrium point is proportional to the rate at which Guinea worm eggs hatch, the number of larvae generated by each egg, and the average life-span of larvae. The value of susceptible copepod population at equilibrium point is given by

From (53) we note that susceptible copepod population at endemic equilibrium is proportional to the average time of stay in susceptible copepod class and the rate at which new susceptible copepods are entering the susceptible copepod class through birth. The endemic value of infected copepod population is given by

We note from (54) that infected copepod population at the endemic equilibrium point is proportional to the average time of stay in the infected copepod class, the rate at which susceptible copepods become infected, and the density of susceptible copepods. We also make the endemic equilibrium of the model system (1) given by expressions (45)–(54) depend on both within-host and between-host disease parameters.

5.1. Existence of the Endemic Equilibrium State

In this section we present some results concerning the existence of an endemic equilibrium solution for the model system (1). To determine the existence and uniqueness of the endemic equilibrium point (EEP) of the model system (1), we can easily express , , , , , , and in terms of in the formwhere

Substituting the expression and into (25) we getwhere

We can easily note that (57) gives , which corresponds to the disease-free equilibrium and

which corresponds to the existence of endemic equilibria. Solving for in , the roots of are determined by using Descartes’s rule of sign. The various possibilities are tabulated in Table 2.

Table 2: Number of possible positive roots of .

We summarize the results in Table 2 in the following Theorem 3.

Theorem 3. The model system (1)(1)has a unique endemic equilibrium whenever Cases  1, 2, 3, 4, 5, 6, 7, and 8 are satisfied and if ,(2)could have more than one endemic equilibrium if Case  8 is satisfied and ,(3)could have two endemic equilibria if Cases  3, 5, and 7 are satisfied.

We now employ the center manifold theory [22] to establish the local asymptotic stability of the endemic equilibrium of model system (1).

5.2. Local Stability of the Endemic Equilibrium

We determine the local asymptotic stability of the endemic steady state of the model system (1) by using the center manifold theory described in [22]. In our case, we use center manifold theory by making the following change of variables. Let , , , , , , , , , and . We also use the vector notation so that the model system (1) can be written in the formwhere

Therefore, model system (1) can be rewritten aswhere

The method involves evaluating the Jacobian matrix of system (62) at the disease-free equilibrium denoted by . The Jacobian matrix associated with the system of (62) evaluated at the disease-free equilibrium () is given by where

By using the similar approach from Section 4.1, the basic reproductive number of model system (62) is

Now let us consider , regardless of whether or , and let . Taking as the bifurcation parameter and if we consider and solve for in (66), we obtain

Note that the linearized system of the transformed equations (62) with bifurcation point has a simple zero eigenvalue. Hence, the center manifold theory [22] can be used to analyse the dynamics of (62) near .

In particular, Theorem  4.1 in Castillo-Chavez and Song [23], reproduced below as Theorem 4 for convenience, will be used to show the local asymptotic stability of the endemic equilibrium point of (62) (which is the same as the endemic equilibrium point of the original system (1), for ).

Theorem 4. Consider the following general system of ordinary differential equations with parameter :where is an equilibrium of the system, that is, for all , and assume that(A1) is a linearization matrix of the model system (68) around the equilibrium with evaluated at . Zero is a simple eigenvalue of , and other eigenvalues of have negative real parts,(A2)matrix has a right eigenvector and a left eigenvector corresponding to the zero eigenvalue.Let be the th component of andThe local dynamics of (68) around are totally governed by and and are summarized as follows.(i) and . When with , is locally asymptotically stable, and there exists a positive unstable equilibrium; when , is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii) and . When with , is unstable; when , is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii) and . When with , is unstable, and there exists a locally asymptotically stable negative equilibrium; when , is stable and a positive unstable equilibrium appears.(iv) and . When changes from negative to positive, changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

In order to apply Theorem 4, the following computations are necessary (it should be noted that we are using as the bifurcation parameter, in place of in Theorem 4).

Eigenvectors of . For the case when , it can be shown that the Jacobian matrix of (62) at (denoted by ) has a right eigenvector associated with the zero eigenvalue given by