Executive Summary
- In the article “A Two-Strain Tuberculosis Model with Age of Infection” by Feng, Iannelli and Milner, the authors discuss a model they developed to investigate the disease dynamics of Tuberculosis when both the latency of the disease and the development of antibiotic resistant strains are considered. Both of these characteristics have been explored individually in previous models. In this model the effects on disease dynamics of having an arbitrarily distributed delay in the latent stage of infection of individuals who have the drug-sensitive strain are analyzed. This is done through steady-state and stability analysis. Stability is explored both analytically and numerically.
Context
- Tuberculosis is a bacterial infection caused by the endoparasite, Mycobacterium tuberculosis. It is considered to be one of the world’s most deadly diseases. According to Centers for Disease Control and Prevention, roughly a third of the world’s population is infected with TB. In 2011 alone it is estimated that almost 9 million people became sick with the disease, and there were 1.4 million deaths attributed to it.
- Tuberculosis is transmitted through the air, so all that is required for the spread of the disease is to be in the same space as an infected individual. Airborne droplet nuclei containing bacilli transmit Tuberculosis. When the nuclei are inhaled, they are deposited inside an individual’s lungs. Macrophages then ingest the bacilli and then transport them to regional lymph nodes. At this point in the life cycle of the bacteria several different things can happen. The infected person’s immune system can kill the bacilli. The bacilli can multiply, developing into active Tuberculosis, or the bacilli can remain dormant within the infected person’s body without developing into the active disease. When the latter occurs it is due to the person’s immune system encapsulating the bacilli into tubercles in which they are unable to spread throughout the infected individual’s body or to other people, but they remain alive. This is referred to as the latent stage of Tuberculosis. Only about ten percent of people who become infected with TB ever move out of the latent stage of the infection, and never show symptoms. If the immune system of a person with latent TB becomes impaired often times the disease will become active.
- Active Tuberculosis is treated with antibiotics. Medication is taken every day for a minimum of two months and can be prescribed for up to nine months. Usually several different types of antibiotics are given to a patient during treatment. People who develop active TB while sick with an immune suppressing disease such as AIDS require specialized treatment plans. It is often recommended, sometimes required, that patients being treated for TB take their medication under the observation of a medical professional. Doing this either requires daily trips to a hospital or doctor’s office by the patient, or daily house calls by a nurse. This practice, referred to as directly observed therapy or DOT, is used in an attempt to slow the development of drug-resistant strains of Tuberculosis, which are becoming a very serious problem worldwide.
- Drug-resistant strains have developed as the result of patients being treated for active Tuberculosis not taking their medication correctly. When people skip doses or stop treatment early, not all of the bacteria living in their body are killed. The bacteria that survive have then been exposed to multiple antibiotics without being eradicated as a result, and this promotes the evolution of drug-resistant strains of the disease. Once there are some bacteria that have developed resistance they multiply, and the treatment originally being taken will no longer work. When a drug-resistant strain of Tuberculosis has infected a person’s lungs, the disease is incredibly contagious. Treatments for drug-resistant strains are developed based on the individual person’s medical background and their specific case of Tuberculosis. If the strain is resistant to only one kind of antibiotic a similar treatment plan to that for the nonresistant strain can sometimes be effective. For multidrug-resistant strains second-line drugs can be used, although they are incredibly expensive and therefore not available in most places where these strains are prevalent. Second-line drug treatment plans last at least twelve months, and some experts recommend continuous treatment for eighteen to twenty four months. Due to the cost and level of commitment required from both patient and physician, drug-resistant Tuberculosis has an eighty percent mortality rate.
History
Tuberculosis in the Past
- Mycobacterium tuberculosis is known to have existed 15,000 to 20,000 years ago. Evidence of the bacteria has been found in the spinal remains of ancient Egyptians. There has also been evidence of the bacteria found in the lymph nodes of people from the Middle ages in Europe. At that time the disease was known as scrofula, and it was believed in England and France that an infected person could be cured simply by being touched by the king. In the 18th century in Europe, the mortality rate of Tuberculosis was at its highest. Due to the crowded and unsanitary living conditions it is estimated that there were 900 TB caused deaths per 100,000 people. Tuberculosis has been a significant cause of death for as far back into human history as we are able to tell. It is believed that out immune systems are so apt at subduing the disease as a result of coevolution with the disease.
Previous Mathematical Models of Tuberculosis
- Two mathematical models that were previously developed were utilized in the development of Feng, Iannelli, and Milner’s model of Tuberculosis. One of which is the two-strain ODE model which was developed by Castillo-Chavex and Feng in their paper “To Treat or Not to Treat”. Using the ODE model to study the disease allows for the analysis of the effects on the prevalence of the disease by the emergence of drug-resistant strains of Tuberculosis. They analyzed a two-strain ODE model for two situations; one in which the drug-resistant strain is natural and one in which the resistance is caused by improper antibiotic usage. Analyzing these situations separately allowed them to examine the effects of antibiotic-induced resistance on the disease dynamics. They found that in the case of natural resistance the competing strains of Tuberculosis are able to coexist in a population of susceptible individuals, but their coexistence is rare. In the case of resistance due to improper drug usage they found that coexistence of the resistant and nonresistant strains is not only possible, but to be expected.
- The second model utilized was developed by Feng, Huang, and Castillo-Chavex in a paper called “On the Role of Variable Latent Periods in Mathematical Models for Tuberculosis”. In this paper they explored the effects of introducing variable periods of latency into a one-strain model of TB disease dynamics. Previous models on the disease dynamics of Tuberculosis have considered the latency period, which is such an important and unique aspect of the disease, however these models used exponentially distributed periods of latency as opposed to the variable periods explored in this model. They found that variable periods of latency have no effect on the long term dynamics of Tuberculosis. Their model showed the disease always either dying out or remaining endemic in the population despite the shape of the latent period distribution. The long and variable periods of latency which characterize Tuberculosis do not cause complex disease dynamics.
Mathematical Model
The Model
- In Feng, Iannelli, and Milner’s model of Tuberculosis discussed in “A Two-Strain Tuberculosis Model with Age of Infection” the two models discussed above were both utilized to create a more complicated model, which takes into consideration both the development of antibiotic-resistant strains of TB and the long and variable periods of latency which are characteristic of the disease. Two situations are considered and analyzed in the article. The first is when all patients being treated with active Tuberculosis finish their treatment successfully (note that this is not equivalent to saying that all treated individuals are cured). In this situation the only way new instances of TB occur are from contact with a contagious individual, and the only drug-resistant strain which prevalent is naturally resistant. The second situation considered is when not all patients complete their treatment. In this case drug-resistant strains of Tuberculosis caused by improper use of antibiotics are able to develop. In both situations the nonresistant strain is considered with variable periods of latency, although in the second case no period of latency is considered for the drug-resistant strain. This is justified due to the high mortality rate of drug-resistant strains. For most people diagnosis is shortly followed by death, and so latency is assumed to have very little, if any, influence on the disease dynamics. This model does not assume that the total human population is constant, but takes into consideration a birth rate and a natural death rate. Individuals who were infected and have been cured are assumed to return to the susceptible class. Individuals who were infected and died as a result are simply removed from the infected class since the total number of people who have died from the disease at a given time is not kept track of in this model.
- The population is divided into the following three categories:
.
- Note the relationship N(t) = S(t) + I(t) + J(t) where N(t) denotes the total population at time t. I(t) is divided into two subcategories: latent and active. The majority of people infected with drug-sensitive Tuberculosis remain in the latent category for the remainder of their lives, never developing active TB. It is estimated that in the first two years after infection five percent of infected individuals will develop the active disease, and another five percent will develop it later in life. This leads to the following definitions.
.
- p(θ) is assumed to be a constant, and it is determined based upon experimental data.
.
- The previous definition allows for I(t) to be expressed in a more useful way.
.
- Now all classes which an individual can occupy have been defined. The following definitions will be used to describe the rates at which individuals move between categories.
.
- Note that γ(θ) is the sum of three terms: (1-r)χp(θ) which is the recovery rate of treated individuals, qrχp(θ) which is the rate of developing drug-resistant TB, and νp(θ) which is the disease induced mortality rate of individuals infected with the drug-sensitive strain. The following apply to the drug-resistant class.
.
.
- Using the above information, the following system of equations is derived.
.
.
- i0(θ) is also assumed to be integrable and compactly supported in [0, ∞) which are technical assumptions which are biologically natural.
- The basic reproductive number for the drug-sensitive strain is referred to as R1 and the basic reproductive number for the drug-resistant strain is referred to as R2. They are interpreted as the average number of secondary infectious cases produced by an infected individual during the entire period in which they are infectious in an entirely susceptible population.
.
- For convenience a new variable v(t) is defined such that v(t) = i(0, t). After algebraic manipulation the following system of equations is reached.
.
Solutions
- Steady state solutions (v*, J*, N*) of the above system are found and analyzed for stability. Solutions are dependent upon the two parameters R1 and R2. The system has a disease-free equilibrium:
.
- Under specific conditions three other equilibria exist. When R1 > 1 and q = 0, then the equilibrium E1 exists.
.
- When R2 > 1, then the equilibrium E2 exists.
.
- When R1 > 1, q > 0, and R1 > R2, then the equilibrium E* exists.
.
.
.
Stability
- When R1 < 1 and R2 < 1 the only equilibrium that exists is E0, and the disease goes extinct. If R1 > 1 or if R2 > 1 then the disease-free equilibrium E0 is unstable. For q = 0 the proceeding stability properties are true. If R1 > 1 then the boundary equilibrium E1 is unstable for R2 > R1 and stable for R2 < R1 and ν = 0. If R2 > 1 then the boundary equilibrium E2 is stable for R2 > R1 and unstable for R2 < R1. For q > 0 the proceeding stability properties are true. If R2 > 1 then the boundary equilibrium E2 is stable for R2 > R1(τ) and unstable forR2 < R1(τ) where τ = qrχ + ν. If R1(0) > R2 > 1 then the interior equilibrium E* is stable for either τ small enough or τ close to τc where τc is defined such that it is greater than zero and R1(τc) = R2. These properties are illustrated in the following graphs.
.
Numerical Results
- The stability of the coexistence equilibrium E* was analyzed numerically. The values chosen for the parameters were chosen based off of actual data about the Tuberculosis when possible. The chosen values are as follows.
µ = 0.014, χ = 2, r = 0.5, ρ1 = 7, ρ2 = 7, ν = 0.14, δ = 1.8
p(θ) was expressed as the following piece wise constant.
.
- After running simulations for various values of q it was found that as q increased from very small to close to qc the prevalence of drug-sensitive Tuberculosis decreases constantly, starting at 81.6% and going to 0%. The prevalence of drug-resistant Tuberculosis increases constantly as q increases, starting at 0% and going to 74.1%. For each of the simulations the dynamics stabilized at E*, remaining consistent with the theoretical results of the model.
Analysis/interpretation
- The goal of this model was to determine if any of the results from the two-strain ODE model or the one-strain distributed-delay model upon which it was based are altered when both factors are considered together. It was shown that drug-sensitive and drug-resistant strains of Tuberculosis are able to coexist as there were several different equilibrium solutions found for various conditions in which neither of the strains goes extinct. This was the same result as in the previous ODE model, indicating that variable periods of latency do not create complex disease dynamics in the model. The article concludes that although the long and variable periods of latency which are characteristic to the disease do not seem to have a large effect upon its dynamics within a population, there are still other factors that have not been considered in this model which might. Some examples listed are exogenous reinfection, which is capable of keeping Tuberculosis endemic in a population when the basic reproductive number is less than one, and also immigration from countries in which Tuberculosis is highly prevalent to those in which it is not. Another factor which they mention is the effect on Tuberculosis dynamics of disease control programs. Since 1993 both the total number of Tuberculosis cases and the rate at which new cases arise has decreased in the United States. According to the Center for Disease Control and Prevention from 2010 to 2011 there was a 5.8% drop in the total number of cases. In fact the total numbers of cases have been decreasing steadily in the United States for the past twenty years. This is thought to be due to the increase in federal and state control programs. Taking the effects of control programs into the consideration of a TB model could influence the efforts being put forth to control Tuberculosis on a global scale.
References
C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of Tuberculosis, J. Math. Biol., 35 (1997), pp. 629-659.
Mandal, Ananya. "History of Tuberculosis." News Medical. N.p., n.d. Web. 28 Mar. 2013. <http://www.news-medical.net/health/History-of-Tuberculosis.aspx>.
"Tuberculosis: Data and Statistics." Center for Disease Control and Prevention. N.p., 26 Sept. 2012. Web. 28 Mar. 2013. <http://www.cdc.gov/tb/statistics/>.
"Tuberculosis (TB)- What Happens." WebMD. N.p., 15 Apr. 2011. Web. 28 Mar. 2013. <http://www.webmd.com/lung/tc/tuberculosis-tb-what-happens>.
Z. Feng, M. Iannelli, and F. A. Milner. A two-strain tuberculosis model with age infection. SIAM Journal on Applied Mathematics, 62(5):1634_1656, 2002.
Z. Feng, W. Huang, and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, J. Dynam. Differential Equations, 13 (2001), pp. 425-452.