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Abstract
For a two-patch transmission of tuberculosis (TB), the disease-free equilibrium and the basic reproduction rate ℛ0 are computed. The disease-free equilibrium is globally asymptotically stable when the basic reproduction rate is less than one. The model can have one or more endemic equilibria. The increased progression rate from latent to active TB in one population may play a significant role in the rising prevalence of TB in the other population. The increased migration from the first to the second population increases the prevalence level of TB in the second population and decreases the TB prevalence in the first population.
1. INTRODUCTION
Tuberculosis—abbreviated as TB for tubercle bacillus—is a common, deadly, infectious disease caused mainly by Mycobacterium tuberculosis. It attacks the lungs (pulmonary TB), but can also affect the central nervous system, the circulatory system, the genital–urinary system, bones, joints, and even the skin. Persons infected with active pulmonary TB can spread the disease through germs transmitted by coughing, sneezing, speaking, kissing, or by spit (Snider et al., 1994; Dye, 2000; Bleed et al., 2001; World Health Organization, 2009). According to a 2009 report, 95% of the estimated 8 million people newly diagnosed with TB each year live in developing countries, and 80% of them are between the ages of 15 and 59 years of age (World Health Organization, 2009). In sub-Saharan Africa, TB is the leading cause of mortality and in developing countries it accounts for an estimated 2 million deaths, with a quarter of avoidable adult deaths (World Health Organization, 2009).
Spatial heterogeneity contributes to host-parasite relationships. In modelling spatial or geographic effects on the spread of a disease, a distinction is usually made between diffusion and dispersal models. In diffusion models, the spread is to adjacent zones, and travelling waves can appear. These models traditionally use partial differential equations.
When the considered space is discrete, situations appear that cannot be modelled by partial differential equations, notably, when the human population is located in patches, such as in families, villages, or cities. When humans or vectors can travel a long distance in a short period of time, dispersal models are more appropriate. For the metapopulation concept, the population is divided into discrete patches. In each patch, the population is divided into compartments corresponding to different epidemic statuses. This leads to a multipatch, multicompartment system.
In the first view, an infective in one patch can infect susceptible individuals in another patch (Lajmanovich and Yorke, 1976; Lloyd and Jansen, 2004; Arino et al., 2006). There is a spatial coupling between patches; vectors or hosts do not migrate between patches. They make short “visits” from their home patches to other patches.
In the second view, the infection does not take place during migration. The situation corresponds to a directed graph, where vertices represent patches and arcs represent links between patches.
We compute the basic reproduction rate ℛ0 and the disease-free equilibrium in a two-patch population. We prove that when ℛ0 is less than 1, the disease-free equilibrium is globally asymptotically stable. We state conditions for the existence of endemic equilibria when the disease persists in the two populations.
2. MODEL
Each of the two populations is divided into three classes: susceptible, latently infected, or infectious. The sizes of these groups are S i , E i , and I i , with i = 1, 2. The subscript i stands for population i. We assume that the transmission does not occur during migration. All recruitment is done into the susceptible class, and occurs at a constant rate Λ i . The force of mortality is a constant μ i . The additional death rate due to disease affects only the class I i and has a constant rate d i .
Mycobacterium tuberculosis is transmitted through contacts between the susceptible and infectious. Latently infected individuals are not infectious. The rate at which the susceptible are infected is β i S i I i . A fraction p i of newly infected individuals of population i becomes infectious (I i ), while the remainder is latently infected and enters the latent class (E i ). The fraction of individuals receiving effective chemoprophylaxis is r i E i , and γ i is the rate of the effective therapy of the infectious. We assume that chemoprophylaxis of latently infected individuals reduces their reactivation and that the initiation of therapeutics immediately removes individuals from the active status I i and places them into a latent state E i . The time before latently infected individuals who did not receive effective chemoprophylaxis become infectious follows an exponential distribution, with a mean waiting time of 1/k i . Individuals leave the class (E i ) to the class (I i ) at the rate of k i (1 − r i ). After receiving an effective therapy, individuals leave the class (I i ) to the class (E i ) at the rate of γ i . The migration rates of the susceptible, latently infected, and infectious between the two populations are respectively a i , b i , and c i , i = 1, 2. The transfer diagram is shown in Figure 1.
Published online:
01 August 2011![]({"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gmps20/2011/gmps20.v018.i03/08898480.2011.596757/production/images/medium/gmps_a_596757_o_f0001g.gif"})
FIGURE 1 Flow chart of the two-patch model for the transmission of tuberculosis.
Putting the above formulations and assumptions together yields the following system of differential equations. 
System (1) is: 
In Eq. (2), ⟨a | b⟩ = a T b is the usual inner scalar product.
The matrices A x and A y are Metzler matrices, that is, matrices with all off-diagonal entries nonnegative (Jacquez and Simon, 1993; Berman and Plemmons, 1994). A linear Metzler system lets the nonnegative orthant invariant (Jacquez and Simon, 1993; Berman and Plemmons, 1994).
3. PROPERTIES
3.1. Positive Invariance of the Nonnegative Orthant
System (2) is: 
and A x are Metzler matrices because x(t) ≥0. ℝ2+ is positively invariant by
3.2. Boundedness of the Trajectories
N(t) = S 1(t) + E 1(t) + I 1(t) + S 2(t) + E 2(t) + I 2(t) is the total population size of Eq. (1).
Lemma 1
The closed set 
Proof
Adding the six equations into differential Eq. (1), 
From Eq. (6), 
. Then,
. This implies that the trajectories of Eq. (1) are bounded. In particular,
if
. Thus, Ωϵ is positively invariant. If
, either the solution enters Ωϵ in finite time or N(t) approaches (Λ1 + Λ2)/μ, and the infected E 1(t), E 2(t), I 1(t), and I 2(t), approach zero. Hence, Ωϵ is attracting (all solutions in ℝ6+ eventually enter Ωϵ).
In Ωϵ, Eq. (1) is well-posed mathematically. Hence, it is sufficient to study the dynamics of Eq. (1) in Ωϵ for ϵ ≥0.
3.3. Local Stability of the Disease-Free Equilibrium (DFE)
Eq. (1) has a DFE, obtained by setting the right-hand sides of Eq. (1) to zero, given by Q* = (x*, 0) where x* = (−A x )−1Λ. With the notations of Eq. (2), the explicit expression of the disease-free equilibrium is 
where
and
are defined as:
The linear stability of Q* is established using the next generation operator on Eq. (2). The matrix F for the new infection terms and V for the remaining transfer terms are given by 
Thus, 
(with i = 1, 2).
is the basic reproduction number of population i.
Consequently, from Theorem 2 of van den Driessche and Watmough (2002),
Lemma 2
The disease-free equilibrium Q* of Eq. (1) is locally asymptotically stable whenever ℛ0 < 1, and instable if ℛ0 > 1.
ℛ0 is the average total number of secondary cases generated by a single infectious individual in a completely susceptible population. Lemma 2 shows that if ℛ0 < 1, a small flow of infectious individuals will not generate large outbreaks of the disease. To eradicate the disease independently of the initial total number of infectious individuals, a global asymptotic stability property has to be established for the DFE (for the case when ℛ0 < 1).
3.4. Global Stability of the Disease-Free Equilibrium
Theorem 1
When ℛ0 ≤ 1, the DFE is globally asymptotically stable in Ωϵ. This implies the global asymptotic stability of the disease-free equilibrium on the nonnegative orthant ℝ6+. The disease dies out.
Proof
The LaSalle function: 
As x(t) ≤ x* in Ωϵ, 
3.5. Existence of Endemic Equilibria
Let 
be the positive endemic equilibrium of Eq. (2). The positive endemic equilibrium (steady state with y(t) > 0) is obtained by setting the right-hand side of Eq. (2) to zero, giving:
Multiplying the second equation of Eq. (13) by (−A y )−1: 
From Eq. (14): 
From the second equation of Eq. (15): 
Replacing this expression in the first equation of Eq. (15) yields: 
Case 1: 
and
.
From the inequality ,
. For the second condition
, we distinguish two cases:
| 1. | Assume that | ||||
| 2. | Assume that This yields the condition: | ||||
Case 2: 
and
. The inequality
yields
.
The inequality 
is satisfied when
. Then:
In sum, 
exists when
satisfies conditions (21), (22), and (24).
Using the first equation of Eq. (13): 
After putting Eq. (25) into the first equation of Eq. (15) and using Eq. (18): 
The positive endemic equilibria 
are obtained by solving Eq. (26) for
. The coefficient m 3 of Eq. (26) is always positive. The number of possible real roots of the polynomial of Eq. (26) depends on the signs of m 0, m 1, m 2, and m 3. The various possibilities for the roots of
are tabulated in Table 1.
TABLE 1 Total Number of Possible Real Roots of 
From Table 1:
Lemma 3
System (1)
| i. | has no endemic equilibrium when Case 4 is satisfied. | ||||
| ii. | has a unique endemic equilibrium when Cases 1, 2, and conditions (21), (22), and (24) are satisfied. | ||||
| iii. | can have more than one endemic equilibrium when Cases 2, 3, 4, and conditions (21), (22), and (24) are satisfied. | ||||
| iv. | can have one or more endemic equilibria when Case 2 and conditions (21), (22), and (24) are satisfied. | ||||
4. SIMULATIONS
System (1) is simulated with parameter values presented in Table 2 using real data of the cities of Yaounde and Bafia in Cameroon (National Committee of Fight Against Tuberculosis, 2001; National Institute of Statistics, 2007; Bowong and Tewa, 2009).
TABLE 2 Numerical Values for the Parameters of the Model
Figure 2 shows the convergence of the total number of infected individuals of populations 1 and 2 to the disease-free equilibrium Q* of Eq. (1) when β1 = 0.003 and β2 = 0.004 (so that 
,
, and ℛ0 = 0.8511). This shows that the disease disappears in the host population when ℛ0 ≤ 1.
Published online:
01 August 2011FIGURE 2 Simulation of Eq. (1) for different initial conditions when β1 = 0.003 and β2 = 0.004 (so that and
). All other parameters are as in Table 2.
![]({"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gmps20/2011/gmps20.v018.i03/08898480.2011.596757/production/images/medium/gmps_a_596757_o_f0002g.gif"})
FIGURE 2 Simulation of Eq. (1) for different initial conditions when β1 = 0.003 and β2 = 0.004 (so that and
). All other parameters are as in Table 2.
Figure 3 shows (E 1(t) + I 1(t))/N(t) (fraction of infected individuals in population 1) and (E 2(t) + I 2(t))/N(t) (fraction of infected individuals in population 2) for a fixed value of 
and two different values of
. In Figures 3(a) and 3(b), the basic reproduction rate of population 1 is less than 1 (
or β1 = 0.006), and the basic reproduction rate of population 2 is
(or β2 = 0.007) in (a) and
(or β2 = 0.015) in (b). The solid and dashed lines stand for populations 1 and 2, respectively. In Figure 3(a), TB cannot persist in the two populations when
and
, and even if ℛ0 > 1. If
, TB can prevail in the two populations even if
(Figure 3(b)).
Published online:
01 August 2011FIGURE 3 Prevalence of TB in the two populations. In Figures (a) and (b), the basic reproduction rate of population 1 is fixed and less then 1 (
or β1 = 0.006), and the basic reproduction rate of population 2 is either less than 1 (Figure (a),
and equivalently β2 = 0.007) or greater than 1 (Figure (b),
and β2 = 0.015). In Figures (c) and (d), the basic reproduction rate of population 1 is greater than 1 (
, or β1 = 0.02), the basic reproduction rate of population 2 is less than 1 (in Figure 3(c),
or β2 = 0.006) or greater than 1 (Figure 3(d),
or β2 = 0.04). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
![]({"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gmps20/2011/gmps20.v018.i03/08898480.2011.596757/production/images/medium/gmps_a_596757_o_f0003g.gif"})
FIGURE 3 Prevalence of TB in the two populations. In Figures (a) and (b), the basic reproduction rate of population 1 is fixed and less then 1 ( or β1 = 0.006), and the basic reproduction rate of population 2 is either less than 1 (Figure (a),
and equivalently β2 = 0.007) or greater than 1 (Figure (b),
and β2 = 0.015). In Figures (c) and (d), the basic reproduction rate of population 1 is greater than 1 (
, or β1 = 0.02), the basic reproduction rate of population 2 is less than 1 (in Figure 3(c),
or β2 = 0.006) or greater than 1 (Figure 3(d),
or β2 = 0.04). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
In Figures 3(c) and 3(d), the basic reproduction rate of population 1 is greater than 1 ( or β1 = 0.02), and
(or β2 = 0.006) in (c) and
(or β2 = 0.04) in (d). TB persists in the two populations even if
. An increase in
leads to increase the prevalence of TB in the two populations.
Figure 4 shows the time series of (E 1(t) + I 1(t))/N(t) (fraction of infected individuals in population 1) and (E 2(t) + I 2(t))/N(t) (fraction of infected individuals in population 2) for different values of k i , i = 1, 2 (progression from latent to active TB in the two populations). The solid and dashed lines stand for populations 1 and 2, respectively. Figure 4(a) presents the case k 1 = k 2, when the progression from latent to active TB is the same for individuals in the two populations. Figure 4(b) presents the case k 2 = 5k 1, when the progression from latent to active TB for individuals in population 2 is five times faster than for individuals in population 1. TB is favored by k 2 > k 1. Other factors such as the migration rates (a i , b i , and c i , i = 1, 2) of susceptible, latently infected, and infectious individuals in the two populations may also play a role.
Published online:
01 August 2011FIGURE 4 Prevalence for different values of k i , i = 1, 2. (a) k 1 = k 2 (progression from latent to active TB is the same for the two populations). (b) k 2 = 5k 1 (the progression from latent to active TB in population 2 is five times faster than for individuals in population 1). In both plots β1 = 0.015 and β2 = 0.025, 
and
for (a) and
and
for (b). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
![]({"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gmps20/2011/gmps20.v018.i03/08898480.2011.596757/production/images/medium/gmps_a_596757_o_f0004g.gif"})
FIGURE 4 Prevalence for different values of k i , i = 1, 2. (a) k 1 = k 2 (progression from latent to active TB is the same for the two populations). (b) k 2 = 5k 1 (the progression from latent to active TB in population 2 is five times faster than for individuals in population 1). In both plots β1 = 0.015 and β2 = 0.025, and
for (a) and
and
for (b). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
Our simulations suggest that migration changes the prevalence of TB in the two populations. Figure 5 shows (E 1(t) + I 1(t))/N(t) (fraction of infected individuals in population 1) and (E 2(t) + I 2(t))/N(t) (fraction of infected individuals in population 2) for different values of a i , b i , and c i , i = 1, 2 (rates of migration of susceptible, latently infected individuals and infectious individuals between the two populations). The solid and dashed lines stand for populations 1 and 2, respectively. Figure 5(a) shows the case a 1 = a 2, b 2 = b 1, and c 2 = c 1 (where migration rates from population 1 to population 2 and from population 2 to population 1 are the same), while Figure 5(b) is for the case of a 2 = 0.5a 1, b 2 = 0.5b 1, and c 2 = 0.5c 1 (migration rates of individuals from population 1 to population 2 are twice those from population 2 to population 1). Transmission rates are the same for both plots β1 = 0.02 and β2 = 0.04 with 
and
for Figure 5(a) and
and
for Figure 5(b). All other parameters are the same as in Table 2. As expected, the increased migration rates of individuals in population 1 have increased the prevalence of TB in population 2 while decreasing the prevalence of TB in population 1.
Published online:
01 August 2011FIGURE 5 (E 1(t) + I 1(t))/N(t) and (E 2(t) + I 2(t))/N(t) for different values of a i , b i , and c i , i = 1, 2. (a) the case a 1 = a 2, b 2 = b 1, and c 2 = c 1 (migration rates of individuals from population 1 to population 2 and from population 2 to population 1 are the same). (b) case a 2 = 0.5a 1, b 2 = 0.5b 1 and c 2 = 0.5c 1 (migration rates of individuals from population 1 to population 2 are twice the migration rates from population 2 to population 1). In both plots β1 = 0.02 and β2 = 0.04 with and
for (a) and
and
for (b). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
![]({"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gmps20/2011/gmps20.v018.i03/08898480.2011.596757/production/images/medium/gmps_a_596757_o_f0005g.gif"})
FIGURE 5 (E 1(t) + I 1(t))/N(t) and (E 2(t) + I 2(t))/N(t) for different values of a i , b i , and c i , i = 1, 2. (a) the case a 1 = a 2, b 2 = b 1, and c 2 = c 1 (migration rates of individuals from population 1 to population 2 and from population 2 to population 1 are the same). (b) case a 2 = 0.5a 1, b 2 = 0.5b 1 and c 2 = 0.5c 1 (migration rates of individuals from population 1 to population 2 are twice the migration rates from population 2 to population 1). In both plots β1 = 0.02 and β2 = 0.04 with and
for (a) and
and
for (b). The solid and dashed lines stand for populations 1 and 2, respectively. All other parameters are as in Table 2.
5. CONCLUSION
In our model of transmission of TB, our expression of ℛ0 allows us to explore the effects of the parameters on the basic reproduction rate ℛ0.
The model has a globally asymptotically stable disease-free equilibrium whenever the associated reproduction rate is less than 1. The model can have more than one endemic equilibrium under certain conditions, whenever the associated reproduction number exceeds 1.
The coexistence of TB in the two populations is possible even if the basic reproduction rate in one population is less than 1, provided the basic reproduction rate in the other population is greater than 1. If k 2 ≫ k 1, the progression of latent to active TB is faster for individuals in population 1 than for individuals in population 2. Then, k 2 has a significant role in the rising prevalence of TB in population 1. Simulation results presented in Figure 5 suggest that when a 1 = 2a 2, b 1 = 2b 2, and c 1 = 2c 2, the propagation of TB in population 1 has a significant influence on TB dynamics in population 2.
| [2pt] Cases | m 0 | m 1 | m 2 | m 3 | Number of sign changes | Number of possible positive real roots |
|---|---|---|---|---|---|---|
| 1 | − | − | − | + | 1 | 1 |
| − | − | + | + | 1 | 1 | |
| 2 | − | + | − | + | 3 | 3 |
| − | + | + | + | 1 | 1 | |
| 3 | + | − | − | + | 2 | 2 |
| + | − | + | + | 2 | 2 | |
| 4 | + | + | − | + | 2 | 2 |
| + | + | + | + | 0 | 0 |
| [2pt] Parameters | Description | Estimated value/range | Reference |
|---|---|---|---|
| Λ1 | Recruitment rate into the S 1 class | 100/yr | Assumed |
| Λ2 | Recruitment rate into the S 2 class | 110/yr | Assumed |
| β1 | Transmission coefficient of infectious in the first subpopulation | variable | Assumed |
| β2 | Transmission coefficient of infectious in the second subpopulation | variable | Assumed |
| μ1 | Force of mortality in the first subpopulation | 0.019896/yr | Estimated |
| μ2 | Force of mortality in the second subpopulation | 0.019897/yr | Estimated |
| k 1 | Rate of progression from the E 1 class to the I 1 class | 0.00013/yr | Assumed |
| k 2 | Rate of progression from the E 2 class to the I 2 class | 0.00023/yr | Assumed |
| r 1 | Rate of effective chemoprophylaxis in the E 1 class | 0/yr | Estimated |
| r 2 | Rate of effective chemoprophylaxis in the E 2 class | 0/yr | Estimated |
| γ1 | Rate of effective therapy in the I 1 class | 0.8182/yr | Estimated |
| γ2 | Rate of effective therapy of in the I 2 class | 0.8183/yr | Assumed |
| a 1 | Rate of migration of individuals from the susceptible class S 1 to the susceptible class S 2 | 0.07/yr | Assumed |
| a 2 | Rate of migration of individuals from the susceptible class S 2 to the susceptible class S 1 | 0.0701/yr | Assumed |
| b 1 | Rate of migration of individuals from the latent class E 1 to the latent class E 2 | 0.05/yr | Assumed |
| b 2 | Rate of migration of individuals from the latent class E 2 to the latent class E 1 | 0.0501/yr | Assumed |
| c 1 | Rate of migration of individuals from the infectious class I 1 to the infectious class I 2 | 0.02/yr | Assumed |
| c 2 | Rate of migration from the infectious class I 2 to the infectious class I 1 | 0.0201/yr | Assumed |
| d 1 | Additional death rate in the I 1 class | 0.0575/yr | Assumed |
| d 2 | Additional death rate in the I 2 class | 0.05751/yr | Assumed |
| p 1 | Fast progression to the I 1 class | 0.015/yr | Assumed |
| p 2 | Fast progression to the I 2 class | 0.016/yr | Assumed |
ACKNOWLEDGMENTS
We are grateful to the anonymous reviewers. Samuel Bowong acknowledges the support of the Alexander von Humboldt Foundation, Germany.
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