Research Article | Open Access
Folashade B. Agusto, Shamise Easley, Kenneth Freeman, Madison Thomas, "Mathematical Model of Three Age-Structured Transmission Dynamics of Chikungunya Virus", Computational and Mathematical Methods in Medicine, vol. 2016, Article ID 4320514, 31 pages, 2016. https://doi.org/10.1155/2016/4320514
Mathematical Model of Three Age-Structured Transmission Dynamics of Chikungunya Virus
We developed a new age-structured deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the qualitative dynamics of the model are not altered by the inclusion of age structure. This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure. However, the numerical simulations show the flaw of the exclusion of age in the transmission dynamics of chikungunya with regard to control implementations. The exclusion of age structure fails to show the age distribution needed for an effective age based control strategy, leading to a one size fits all blanket control for the entire population.
Chikungunya is a viral disease that is transmitted to humans from an infected mosquito of the Aedes genus (particularly the Aedes aegypti and Aedes albopictus mosquitoes [1, 2]). It is an RNA virus that belongs to Alphavirus genus of the family Togaviridae . It was first described about 1952 during an outbreak in southern Tanzania . Chikungunya in the Kimakonde language (the language from where the name was derived) means to become contorted or “bend over” . There have been numerous cases of reemergence of chikungunya in Africa, Asia, Europe, and more recently the Caribbean . The virus was isolated in 1960s in Bangkok and in 1964, the virus resurfaced in parts of India including Vellore, Calcutta, and Maharastha . Other outbreaks include Sri Lanka in 1969, Vietnam in 1975, Myanmar in 1975, and Indonesia in 1982 . A large outbreak occurred in the Democratic Republic of the Congo in 1999-2000 . In the years 2005–2007, an outbreak occurred in the islands of the Indian Ocean. Gabon was hit with an outbreak in 2007 . Since 2005, India, Indonesia, Thailand, Maldives, and Myanmar have encountered over 1.9 million cases . The disease spread to Europe by 2007 with 197 cases being recorded . More recently, in December 2013, the French part of the Caribbean island of St. Martin reported two laboratory-confirmed autochthonous (native) cases [3, 4]. Since then, local transmission have been confirmed in the Dutch part of Saint Martin (St Maarten), Anguilla, British Virgin Islands, Dominica, French Guiana, Guadeloupe, Martinique, and St Barthelemy . As of October 2014, over 776,000 suspected cases of chikungunya have been recorded in the Caribbean islands, Latin American countries, and some south American countries . About 152 deaths have also been attributed to the disease during the same period. Mexico and USA have also recorded imported cases. On October 21, 2014, France confirmed four cases of chikungunya locally acquired infection in Montpellier, France .
In 2005-2006, a major chikungunya outbreak involving numerous islands in the Indian Ocean (notably La Reunion Island) occurred; one-third of the population were infected . According to Schuffenecker et al.  and Vazeille et al. , in two concurrent studies, the chikungunya virus strains in the Reunion Island outbreak mutated to facilitate the disease transmission by Aedes albopictus (Tiger mosquito) [6, 9]. The mutation was a point mutation in one of the viral envelope genes (E1 glycoprotein gene (E1-226V)) [10, 11]. Dubrulle et al.  found that this mutation allowed the virus to be present in the mosquito saliva only two days after the infection, instead of approximately seven days in the Aedes aegypti mosquitoes. This shows that Aedes albopictus is a slightly more efficient host than Aedes aegypti in transmitting the variant E1-226V of chikungunya virus. Hence, this result indicates that other areas where the tiger mosquitoes are present could be at greater risk of outbreak with an enhanced transmission of chikungunya virus by Aedes albopictus.
Following an effect bite (i.e., a bite leading to an infection) from infected mosquitoes [1, 2], the incubation period is usually within 3–7 days; symptoms include fever, headache, nausea, fatigue, rash, and severe joint pain (including lower back, ankle, knees, wrists, or phalanges) [1, 2]. There is no antiviral medicine to treat the disease [1, 2]; all the treatments are directed at relieving the disease symptoms . There is no preventative vaccine for chikungunya ; however, findings of an experimental vaccine in an early-stage clinical trial are promising; it prompted an immune response in all 25 volunteers .
Chikungunya rarely results in death and infected individuals are expected to make full recovery with life-long immunity . However there are some cases where individuals experience joint pains for several months or years after the initial infection . There have also been reports of eye, neurological, and heart complications and gastrointestinal complaints . The disease symptoms, generally, are mild and the infection may go unrecognized; however, several studies [13–16] have shown that children (especially neonates), the elderly (≥65 years), and people with medical conditions (such as high blood pressure, diabetes, or heart disease) have more severe clinical manifestation of chikungunya than in older children and adults (<65 years). The severity of the symptoms can be described by a U-shaped curve, with a maximum occurring in young infants and the elderly and a minimum in older children . Furthermore, the rate of asymptomatic infection among children varies according to different outbreak reports (range 35–40%) ; overall, approximately 3–28% of infected individuals will remain asymptomatic [17, 18]. In the study of the 2006-2007 chikungunya epidemic in Kerala, South India, Vijayakumar et al.  showed an age distribution of people affected with chikungunya. Their study indicates that the adult group (ages 15–59) were the most affected age group; they consist of about 73.4% of the entire study population; this is followed by the elderly group (ages > 60); this group make up about 15.6% of the study population. Finally, 11% of the cases occurred in persons ages <15 years. Similar age distribution was reported in other epidemics in India , Thailand , and Reunion Islands [5, 22] and across Europe  (including Austria, Czech Republic, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Poland, Romania, Slovakia, Slovenia, Spain, Sweden, and United Kingdom).
A number of studies have been carried out to study the chikungunya virus, considering different factors that effect the outbreak of the disease (see [21, 24–30]). Ruiz-Moreno et al.  analyzed the potential risk of chikungunya introduction into the US; their study combines a climate-based mosquito population dynamics stochastic model with an epidemiological model to identify temporal windows that have epidemic risk. Dumont et al.  propose a model, including human and mosquito compartments, that is associated with the time course of the first epidemic of chikungunya in Reunion Island. Using entomological results, they investigated the links between the episode of 2005 and the outbreak of 2006. Manore et al.  investigated, via an adapted mathematical model, the differences in transient and endemic behavior of chikungunya and dengue, risk of emergence for different virus-vector assemblages, and the role that virus evolution plays in disease dynamics and risk. Poletti et al.  developed a chikungunya transmission model for the spread of the epidemic in both humans and mosquitoes; the model involves a temporal dynamics of vector (Aedes albopictus), depending on climatic factors. In the study, they provided estimates of the transmission potential of the virus and assessed the efficacy of the measures undertaken by public health authorities to control the epidemic spread in Italy. Yakob and Clements  developed a simple, deterministic mathematical model for the transmission of the virus between humans and mosquitoes. They fitted the model to the large Reunion epidemic data and estimated the type reproduction number for chikungunya; their model provided a close approximation of both the peak incidence of the outbreak and the final epidemic size. Pongsumpun and Sangsawang  developed and studied theoretically an age-structured model for chikungunya involving juvenile and adult human populations, giving conditions for the disease-free and endemic states, respectively. They also suggested alternative way for controlling the disease.
The aim of this study is to develop a new deterministic transmission model to gain qualitative insight into the effects of age on chikungunya transmission dynamics and to determine the importance or otherwise of the inclusion of age in the transmission dynamics. A notable feature of the model is the incorporation of three different human age classes involving juvenile, adult, and senior human populations; the model also involves two infectious human classes, notably the asymptomatic and symptomatic classes. The paper is organized as follows; the model is formulated in Section 2, the analysis of the mathematical properties of the model is stated in Section 3. The effect of the age structure on the disease transmission is explored in Section 4. The sensitivity analysis of the model is investigated in Section 4.1. Following the result obtained from the sensitivity analysis, various control strategies are implemented in Section 5. The key theoretical and epidemiological results from this study are discussed and summarized in Section 6.
2. Model Formulation
The model is formulated as follows with human and mosquito subgroups. The human population is divided into juvenile, adult, and senior subpopulations. The human subgroup is further divided into susceptible (), exposed (), symptomatic (), asymptomatic (), and recovered (), where for the juvenile, adult, and senior subpopulations. Thus, the total human population + . The mosquito population is divided into three classes consisting of susceptible mosquitoes (), exposed mosquitoes (), and infected mosquitoes (). Hence, the total mosquito population .
Individuals move from one class to the other as their status evolves with respect to the disease. The population of susceptible juvenile () is generated at the rate via birth or immigration. It is assumed that there is no vertical transmission or immigration of infectious humans; thus there is no inflow into the infectious classes. The population is reduced by the juvenile maturation at the rate and by natural death at the rate . The infection rate of susceptible juveniles is given asThe parameter in (1) is the probability that a bite from an infectious mosquito leads to infection of the susceptible juvenile and the parameter is the mosquito biting rate. The derivation of (1) is given in Appendix A.
Similarly, it can be shown that the rate at which mosquitoes acquire infection from infectious (asymptomatic and symptomatic) human hosts is given byThe parameter is the probability that a bite from a susceptible mosquito to a human leads to infection of the mosquito.
Susceptible juveniles are infected by the chikungunya virus at a rate and move into the exposed class. Thus, the susceptible population is given asThe exposed juvenile population is generated following infection of the susceptible juveniles by infected mosquitoes. A fraction of exposed juveniles enter the asymptomatic class at the rate and the remaining fraction () goes into the symptomatic class () at the rate . The population of the exposed juvenile is reduced by the juvenile maturation at the rate . The exposed juvenile population is further reduced by natural death at the rate . Thus, the exposed population is given asMembers of the juvenile asymptomatic class are generated from the fraction that moved from the juvenile exposed class. This class is reduced by maturation to the adult asymptomatic class () at the rate , by recovery (either naturally or via the use of treatment) at a rate to the recovered class. Similarly, members of the juvenile symptomatic class are populated from the fraction that moved from the juvenile exposed class. The class is reduced due to maturation to the adult symptomatic class () at the rate and by progression to the recovered class recovery at a rate . These populations are further reduced by natural death at the rate ; chikungunya rarely results in death [1, 2]; as such we have ignored the disease induced death rate. Thus, the equations for these classes are given as follows:The juvenile recovered class is populated from the juvenile asymptomatic and symptomatic classes; the class is reduced by maturation to the adult class at a rate and by natural death at a rate . The equation for this class is given as follows: The corresponding equations (susceptible, exposed, asymptomatic, symptomatic, and recovered) for the adult and senior classes are similarly obtained; additionally there is a maturation rate from the adult population into the senior class. We assume that the recovery rates from adults asymptomatic and symptomatic classes are greater than those from juvenile classes which in turn are greater than those from the senior classes (i.e., ) . Furthermore, we assume that seniors progress more quickly to the asymptomatic and symptomatic classes than juveniles and adults (i.e., ) .
The population of the susceptible mosquitoes () is generated by the recruitment rate and reduced following effective contact with an infected human. All mosquitoes classes are reduced by natural death at a rate . The equation for this class is given as follows:Mosquitoes in the exposed class are generated following the infection of the susceptible mosquitoes. They progress to the infected class at a rate . The equation for the exposed mosquitoes dynamics is given as follows:The infected mosquitoes class are populated from the exposed mosquitoes. The equation for this class is given as follows:Combining the aforementioned derivations and assumptions the model for the transmission dynamics of chikungunya virus in a population is given by the following deterministic system of nonlinear differential equations:The flow diagram of the age-structured chikungunya model (10) is depicted in Figure 1 and the associated variables and parameters are described in Table 1. Model (10) is an extension of some of the chikungunya transmission models (e.g., those in [21, 25–30]) by (inter alia): (a)Including a compartment for the exposed humans and mosquitoes (this was not considered in [21, 27, 28]).(b)Adding a compartment for asymptomatic and symptomatic individuals (these were not considered in [25, 26, 30]).(c)Including an age structure for humans (this was not included in [26, 29]).(d)Adding compartments for seniors (these were not included in [21, 25–28, 30]).
3. Analysis of the Model
3.1. Basic Qualitative Properties
3.1.1. Positivity and Boundedness of Solutions
For the age-structured chikungunya transmission model (10) to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time. In other words, solutions of the model system (10) with non-negative initial data will remain non-negative for all time .
Lemma 1. Let the initial data , where . Then the solutions of the age-structured chikungunya model (10) are nonnegative for all . Furthermorewith
3.1.2. Invariant Regions
The age-structured chikungunya model (10) will be analyzed in a biologically feasible region as follows. Consider the feasible region with
Lemma 2. The region is positively invariant for the age-structured chikungunya model (10) with nonnegative initial conditions in .
In the next section, the conditions for the existence and stability of the equilibria of the age-structured chikungunya model (10) are explored.
3.2. Stability of Disease-Free Equilibrium (DFE)
The age-structured chikungunya model (10) has a disease-free equilibrium (DFE), obtained by setting the right-hand sides of the equations in the model to zero, given byThe linear stability of can be established using the next generation operator method on system (10). Taking , , , , , , , , , , as the infected compartments and then using the notation in , the Jacobian matrices and for the new infection terms and the remaining transfer terms are, respectively, given bywhere , , , , andwhere , , , , , , , , , , and .
It follows that the reproduction number of the age-structured chikungunya model (10) is given by where is the spectral radius andFurthermore, the expression is the number of secondary infections in juveniles by one infectious mosquito, is the number of secondary infections in adults by one introduced infectious mosquito, is the number of secondary infections in seniors as a result of one infectious mosquito, and lastly is the number of secondary infections in mosquitoes resulting from a newly introduced infectious juvenile, adult, and senior. Further, using Theorem 2 in , the following result is established.
Lemma 3. The disease-free equilibrium (DFE) of the age-structured chikungunya model (10) is locally asymptotically stable (LAS) if and unstable if .
The basic reproduction number is defined as the average number of new infections that result from one infectious individual in a population that is fully susceptible [42–45]. The epidemiological significance of Lemma 3 is that chikungunya will be eliminated from the community if the reproduction number () can be brought to (and maintained at) a value less than unity. Figure 2 shows convergence of the solutions of the age-structured chikungunya model (10) to the DFE () for the case when (in accordance with Lemma 3).
3.3. Global Asymptotic Stability of the DFE
Consider the feasible region where .
Lemma 4. The region is positively invariant for the age-structured chikungunya model (10).
Theorem 5. The DFE, , of the age-structured chikungunya model (10), is globally asymptotically stable (GAS) in whenever .
3.4. Existence of Endemic Equilibrium Point (EEP)
In this section, we will investigate conditions for the existence of endemic equilibria (i.e., equilibria where the infected components of the age-structured model (10) are nonzero).
Let be an arbitrary endemic equilibrium of age-structured chikungunya model (10). Also, let be the forces of infection for susceptible juveniles, adults, and seniors and susceptible mosquitoes at steady state, respectively. Components of the steady-state solution of the equations of the age-structured chikungunya model (10) are given in Appendix F. Substituting the expressions for , , , , and into (22) for and simplifying giveswhereSubstituting expression for into the force of infection in (22) gives and then solving for givesSubstituting this result into (23), and simplifying, leads to the following cubic equation:where Thus, the number of possible positive real roots polynomial (27) can have depends on the signs of and . This can be analyzed using the Descartes Rule of Signs on the cubic polynomial , given in (27) (with , , , , ). The various possibilities for the roots of are tabulated in Table 2.
Theorem 6. The age-structured chikungunya model (10) has a unique endemic equilibrium if .
Numerical simulations of the age-structured chikungunya model (10), depicted in Figure 3, show convergence to a unique endemic equilibrium when (suggesting that the unique EEP of the age-structured chikungunya model (10) is asymptotically stable when it exists).
3.5. Backward Bifurcation Analysis: Special Case
Chikungunya rarely leads to death of the infected individuals [1, 2]; however a number of deaths have been reported as a result of the infection [3, 5, 30, 46, 47]. We introduced into the asymptomatic and symptomatic human compartments disease-induced mortality parameters (, , , , , ) and study the implication on the dynamics of the disease transmission. Thus, the asymptomatic and symptomatic juveniles, adults, and seniors compartments of age-structured chikungunya model (10) can be written as follows:It can be shown that the reproduction number for the age-structured chikungunya model (10) with the asymptomatic and symptomatic human compartments stated in (29) is given bywhere