Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Problem statement: The fight Against Tuberculosis (TB) has mainly foc used on the adult population because children were perceived to pose a very low risk in TB transmission. This assumption ignores the potential risk children had as reservoirs of latent infections from which futur e cases evolve when they become adults. It was there for important to investigate the dynamics of TB taking into consideration, children. Approach: We formulated a compartmental model for TB with two age classes, children and adults. Qualitative a nalysis of the model was done to investigate the stability of the model equilibria in terms of the m odel reproduction number R 0 Numerical simulations were also done to investigate the role played by so me key epidemiological parameters in the dynamics of the disease. Results: The model had two equilibria: The disease free equ ilibri m which was globally stable for R0<1 and the endemic equilibrium which was locally as ymptotically stable for R0>1, for R0 near 1. The study showed increased latent infecti ons in the adult population as a result of increased latently infected children who mature to adulthood with latent infections. Conclusion/Recommendations: Progression to active TB among adults is epidemiol og cally significant and interventions should focus on the a dult population. Anti-tuberculosis, treatment of adults is crucial in controlling the epidemic and s hould interventions be proposed, they should target progression to active TB for those latently infecte d. The fight against TB should also take into consideration tuberculosis among children.


INTRODUCTION
Primary tuberculosis is the fountainhead of tuberculosis disease and when acquired during childhood, it may develop into serious tuberculosis disease within a short period of time or remain latent during childhood only to be reactivated in adulthood (Wood et al., 2010). Children usually progress to active disease within 12 months of primary infection because children have a high risk of progression to disease following infection (Brent et al., 2008;Sharomi et al., 2008). There has been limited interest in childhood tuberculosis because more than 95% of children with active disease are sputum smear negative and therefore not infectious. Infected children represent a pool from which a large proportion of future cases of adult TB will arise and thus perpetuating the TB epidemic (Bloch and Snider, 1985;Brent et al., 2008;Vynnycky et al., 2001). The burden of childhood tuberculosis is a clear indication of the TB severity in the adult population Warren et al., 2004). The contribution of children to TB caseload is not well documented in poor-resource settings with high disease burden, but research has shown that in high disease burdened areas, children less than 13 years of age, contribute about 13.7% of the total TB caseload in a particular community in South Africa (Murray and Salomon, 1998). The result compared well with that determined for low income countries for children less than 15 years of age, which was 15% (Schaaf et al., 2002). Children usually develop TB as a direct complication of the initial infection. Children with household or community exposure to TB are highly likely to be infected with the disease. Children acquire TB infection from an adult who is in their immediate environment Wood et al., 2010).
Lack of contagiousness is one of the reasons that public health programs on TB prevention have excluded children. They are regarded as the end of the transmission chain and therefore pose no threat to the evolution of the TB epidemic in the population. This view could have contributed significantly to early childhood TB.
Mathematical models for the dynamics of TB have been extensively developed and have been used in designing control programs and as predictive tools (Bhunu et al., 2008;Blower et al., 1995;Blower and Daley, 2002;Castillo-Chavez and Feng, 1997;Castillo-Chavez and Song, 2004;Gomes et al., 2004;Starke, 2003). Models that consider the potential impact of TB vaccines were studied in (Mairais et al., 2005). Recently, Bhunu et al. (2008) considered a model that studied the impact of treatment and chemoprophylaxis in combating TB.
Most of the studies reviewed concentrated on adult TB and ignored TB in children. It is upon this background that we propose a simple deterministic model for TB that incorporates children and adults, with the goal of investigating the dynamics of interaction of pediatric and adult TB epidemics. We develop a mathematical model with the object of quantifying the underlying factors that drive the epidemic in children.
This study aims to challenge public health policies regarding childhood TB. We endeavor to answer the following questions: What is the role of increased adult TB control on the growth of childhood TB? What is the proportion of adults that need to be targeted in order to control TB in children? To the best of our knowledge, there is a limited number of theoretical models of childhood TB.

MATERIALS AND METHODS
A two class age-structured model: The model considers a constant population N, subdivided into three different subgroups: susceptible (S) latently infected (L) and infected with active TB (I). Individuals move from one subgroup to the other as their status with respect to the disease changes. We again divide the population into two age classes, children and adults. Individuals below 5 years belong to the "children's" age class while those above 5 years belong to the "adult's" age-class. These age groups are separated primarily to distinguish between pediatric TB and adult TB. While the exact age demarcation is somewhat arbitrary, the main goal here is to divide the population into two reasonable age classes. Data from medical institutions also shows different phenomena in TB development for the two classes. We use the following notation for individuals in the two age classes: we use the subscript c and a to represent children and adults respectively, so that S c , LC, I c and SA, L a , I a respectively represent susceptible, latently infected and infects with active TB in children and adults. The model is built on the following assumptions. All subgroups are subjected the natural mortality rateµ. The effective contact rate of infected adults is assumed to be higher with susceptible adults than children (c 2 > c 1 ). An effective contact is one that is sufficient to result in an infection if the contacted individual has never been infected (Walls and Shingadia, 2004 The rate at which children join the adult classes is given by f. However, infected children are assumed not to graduate into adulthood being infective. We acknowledge here that the consideration of being constant is merely for mathematical convenience. We shall assume that f is less than or equal to the birth rate. Figure 1 depicts the interaction of the two ageclasses.
The model is thus described by the following set of ordinary differential Eq. 1: By setting: dx The model has initial conditions given by x i (0)≥0, = 1, 2,…, 6. Biological considerations entail that we study systematically (2) in the following region: Proposition: Solutions of system (2) are positive for all t ≥ 0 and are bounded. The region G is thus positively invariant and all solutions in G remain in G for all time.
Suppose there exists a first time t 1 such that x 1 (t 1 ) = 0 and the derivative 1 dx 0 dt < at t = t 1 and x i (t) >0, (i = 1,…,6) for 0<t <t 1 . The first equation of the system (2) gives 1 1 dx (t ) 0, dt = µ > which is a contradiction? Thus x 1 (t) will remain positive for all t. Similar steps can be followed for the remaining variables. We conclude that all solutions of the system (2) are positive for all time. Our population is also bounded in G thus all solutions starting with G will remain in the G. Thus G is positively invariant and attracting and our model is thus epidemiologically well posed.
Determination of equilibria: We analyze the system (2) by finding the model equilibria and carrying out their stability analysis. At the steady state we set the right hand side of equations of the system (2) to zero and determine the state variables.
From the first five equations of a system (2), we have: Solving for * 2 x and * 3 x simultaneously, we have Eq. 4: Substituting for * 1 x in * 4 x , we have Eq. 5: Substituting Eq. 3 into the expression for * 5 x , we have: Substituting Eq. 6 into the last equation of system (2) Where: Here, R a represents the contribution of adults to secondary infections while R c is the contribution of children to secondary infection. The threshold parameter R 0 defines the mean number of secondary cases generated by introducing a single infected individual into a wholly susceptible population in which children are involved in the dynamics of the disease.
The case * 6 x 0 = gives the disease free equilibrium point: The endemic equilibrium is thus determined from the solutions of (7). As regards the sign of a 1 in (7), we have the following proposition.
We note that if R 0 < 1 then K ≺ 1 since: This implies that a 1 >0. We thus have the following theorem on the existence of the endemic equilibrium.
Theorem 1: If R 0 <1 then P (x 6 ) has no positive solution. However, if R 0 >1, then P (x 6 ) has one positive solution. We therefore conclude that system (2) has a unique endemic equilibrium point E 1 whenever R 0 >1.

Proof:
The solution of (7) is given by: Here a 2 >0 and a 0 >0 if R 0 <1 and a 0 < 0 if R 0 >1and from (8), if R 0 <0, then we have two negative solutions and for R 0 >1, we have one positive solution for all a 1 .

RESULTS AND DISCUSSION
Qualitative results: Analysis of the Model Reproduction Number: We consider first, the effects of increasing the recovery rates σ a and σ c . We note that as σ a → ∝, R 0 → 0 the implications of this result is that emphasis should be placed on preventing adult individuals from progressing to active TB. A high rate of recovery of individuals with active TB plays a crucial role in decreasing the pool of infected individuals. However, as σ a → ∝, This result shows that the recovery rate of children does not play a crucial role in the dynamic of TB as compared to that of adults. The contour plot in Fig. 2 shows the relationship between R 0 , σ a and σ c . Figure 2 represents a set of R 0 contours for which, for a chosen set of parameter values, changes in the values of σ a and σ a would give. We note that as the value of σ a increases, the value of R 0 decreases. In fact, this result is similar to the one proven analytically, that as σ a → ∝, R 0 → 0. Recovery in this case may be due to treatment or natural recovery. Since the fight against TB has been restricted to, vaccination, identifying infectious cases and treating them, one can consider this result to imply that the fight against TB must be directed mostly to the adult population since they are the source of infection. It is clear from Fig. 2 that increasing the recovery rate σ c for children does not play any significant role in the transmission dynamics of the disease.
We now investigate the role of the progression rates r a and r c . As r a → 0, . Thus a reduction in the progression rates leads to a reduction in the number of secondary infections generated by an infected individual.
We also note that as r c →0, we have . Reduction in the progression of latently infected children to active pediatric TB leads to increased secondary infections in the adult population. The implications of this scenario are that there are increased latent infections in the adult population as a result of increased latently infected children who mature to adulthood with latent infections. Children become a reservoir of latent infections from which reactivation is highly likely to occur when they become adults. Figure 3 illustrates the theoretical results relating variations of progression r a and r c to secondary infections.
Global stability of E 0 : Theorem 2: The disease free equilibrium point E 0 is globally asymptotically stable in G whenever R 0 < 1.

Proof:
We consider the following Lyapunov function:  (Lietman and Blower, 2000), that every solution of (2) with initial conditions in G approaches E 0 as t→∝ whenever R 0 <1.
Following Theorem 4.1 in (Feng et al., 2001), we thus compute a and b to determine the stability of E 1 around R 0 = 0, where: And β * the bifurcation parameter. The non-zero partial derivatives of F at the disease free equilibrium point are given by: To determine a we substitute the above expressions of the partial derivatives into Eq. 10. We thus have: Since y 1 <0 and y 4 <0, the right hand side of a is always negative. For the sign of b the following are the non-zero partial derivatives of F: 2 2 2 2 * * * * 2 3 5 6 1 1 1 1 2 4 2 4 * * * * 6 6 6 6 An evaluation of b shows that b is always greater than 0. We thus have the following results based on Theorem 4.1 in (Feng et al., 2001).

Theorem 3:
The endemic equilibrium point E 1 is locally asymptotically stable for R 0 >1 but with R 0 near 1.

Numerical results:
We now give a discussion of the numerical solutions of systems (2), the parameter values and interpretations of the cases that arise thereof. The dynamics of the system (2) are studied numerically using the fourth order Runge-Kutta numerical scheme in MATLAB.
Parameter values: Some of the model parameters were decided upon based on the following assumptions: • Since the birth and death rates are taken to be the same, we assume that the average lifespans of human beings are between 40 and 80 years. We use the natural mortality rate in the range 0.0125≤µ≤ • The rate at which children join the adult class f is assumed to be less than or equal to the birth rate. We try to accommodate, the fact that not every child survives to adulthood. So we set f≤µ • Susceptible children and adults are infected with probability β upon contact. Contact patterns that can result in infection can be random, associative, age-specific and sex-specific and so on, but in this study we assume random mixing. This means that every susceptible is equally likely to be infected by an infective should contact occur • While the probability of infection is taken to be the same for children and adults, we assume that adults have increased chances of mixing. We thus consider different contact rates • Estimates have shown that 10% (proportions p, q) of TB infection progression fast to active TB (Jung et al., 2002). The risk for young children to develop active TB if the infection is not treated is up to 43% in children less than a year old, approximately 24% for children between 1 and 5 years of age and much lower for those between 6 and 13 years (Schaaf et al., 2003). The proportion of children that develop active TB fast can thus be assumed reasonably to be higher than 10% • Active TB has always been taken as a continued development of the primary infection first acquired or due to endogenous reactivation or exogenous reinfection with a second or the same strain. Separation and quantification of these mechanisms especially (endogenous reactivation and exogenous re-infection) requires technology that can differentiate between strains (Zhang et al., 2007). We crudely lump the two processes (Jung et al., 2002) in this study to represent a progression to active TB, denoted by r a and r c • Recovery, may be due to treatment or natural recovery In which case individuals revert back to the exposed class at rates σ c and σ a . The rate at which individuals recover usually depends on the immune status of the individual, types of drugs, genetic and socioeconomic factors. We acknowledge that quantification of these rates of recovery is difficult as evidenced by variations in the parameter values in research publications The estimated parameter values used are given in Table 1.
Numerical plots: Figure 4a shows the changes in the proportion of the susceptible populations for children and adults, with increasing time.   (Song et al., 2002) β (0, 1) Variable p, q (0, 1) Variable σc, σa (0.5, 1) (Bhunu et al., 2008;LaSalle, 1976;Nelson and Wells, 2004) rc 0.03 (Nelson and Wells, 2004) c1, c2 (1, 100) Estimated F f≤µ Estimated ra (0.5,1) (LaSalle, 1976) The susceptible population of adults initially falls as the infective among adults increase, but later increases to a steady state value over time. The susceptible children increase to a steady state. This is reasonable since in most population structures, the proportion of children is higher than that of adults. In Fig. 4b the latently infected population for the two subgroups decreases over time to zero. In Fig. 4c the graphs for the two sub-populations show a similar trend in which their proportions fall over time to zero We observe that infection in the adult population clear later than that of children mainly because of the latent reservoir of infection from children. Overall, the Fig. 4a-c, represent the disease free equilibrium point E 0 over a longer period of time than the one presented in the simulations. In Fig. 5a shows the changes in the proportion of the susceptible populations for children and adults, with increasing time. Both sub-populations decrease to a steady state over time and again we have the proportion of children being higher than that of adults as the epidemic evolves. In Fig. 5b, the latently infected population for the two subgroups initially decreases and then rises to a steady state over time. Figure 5c shows the graphs for the proportions of infectives. The adult infects increased rapidly and settled in an endemic steady state over time. This scenario represents the endemic steady state E 1 , which we deduced analytically to be asymptotically stable for R 0 >1.

CONCLUSION
The study presented in this study is a very simple model showing the dynamics of TB in children and adults in the absence of any intervention. Many deterministic models of TB have been developed but a few, to the best of our knowledge, have been developed this way without using the age structures. We presented a model in which there are no interventions. A more realistic approach would be to include the current intervention strategies. This forms our research that is in progress. Although some of the model assumptions are realistic, many of them over simplifies the natural evolution of TB. In particular, the ageing function f, was chosen so that it does not exceed the birth rate of the population based on the assumption that the individuals that graduate into adulthood survive with some probability less than or equal to one. We acknowledge that it is difficult to estimate and an age-structured model would have eliminated such a difficulty. The lumping up of progression to active TB to include reinfection and reactivation and recovery to include both natural recovery and recovery due to the treatment represent significant short comings to the model. The agreement between the model output and epidemiological data has been restricted to very broad TB epidemiological trends. Despite these inadequacies, this study provides some unique insights in the way progression and recovery parameters influence the dynamics of the disease. Any form of intervention implemented should influence these parameters and quantify their changes and influence on the reproduction number.
Our analysis shows that progression to active TB among adults is epidemiologically significant and interventions should focus on the adult population. This perhaps explains why TB among children has not been a major subject for research. Similar trends can be observed in the recovery rates r c and r a . The relationship between the progression rates to active TB for children and adults and the model reproduction number draws out important information that can help in prophylactic treatment such as the use of Isoniazid in preventing progression to active TB. Figure 3, also shows that change in the progression rate for adults, results in significant changes in the model reproduction number. Attempts to use chemoprophylaxis for active TB prevention should be targeted to adults.
The global stability of the model shows that if R 0 <1 then every solution tends to the disease free equilibrium and the disease clears from the population. We showed that there exists a unique endemic equilibrium point whenever R 0 >1. If the TB is present in a population then, as long as R 0 >1, it will always persist.
In conclusion, despite all our results pointing out the need to focus interventions on adults, pediatric TB forms an important indicator of how the TB epidemic is evolving in adults. Infected children act as a reservoir of latent infections that can reactivate into active TB when they become adults. One of the most important factors to consider with regards to pediatric TB is that, it is possible to clearly distinguish between disease progression stages, exposure, infection and disease. This is particularly important to public health policy formulation, designing interventions and modeling in general.