Citation: Surabhi Pandey, Ezio Venturino. A TB model: Is disease eradication possible in India?[J]. Mathematical Biosciences and Engineering, 2018, 15(1): 233-254. doi: 10.3934/mbe.2018010
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Tuberculosis (TB) is an infectious disease caused by the bacterium Mycobacterium Tuberculosis, which typically affects lungs (known as pulmonary TB of PTB) and that mostly occurs in adults, i.e. in individuals above 14 years of age, but could also affect other parts of body (known as extra-pulmonary TB or EPTB). Tuberculosis is one of the leading causes of death worldwide after HIV and remains a major public threat in many countries. An estimate shows that globally about one-third of the population is infected with TB bacteria. The global incidence of all forms of TB cases during 2008 was estimated to be 9.4 million, at the rate of 139/100,000 population and in 2014 there were an estimated 9.6 million incident cases of TB, which is equivalent to 130/100,000 population.
In this paper we present a model for the disease situation in India, one of the countries in which the disease is endemic. This work is an extension of previous results, [3,15,18] and in particular of the work in progress [19], in that we account for the diagnostic system to be partioned into the public and private sectors, contrary to what was assumed in the earlier works.
In India, the tuberculosis situation is characterized by high prevalence (total number of TB cases over a period of one year) and incidence (new TB cases over a period of one year) of disease or active TB (when the individual is infectious and transmits bacteria to others) and high rate of transmission of infection (the latent TB-individual is infected with TB bacteria but cannot transmit them to others; also, in this situation, bacteria remain in the dormant state). Primary surveys show that about 30% -50% of India population is latently infected. This means that people have been infected by TB bacteria but are not ill with the disease and do not transmit it but may become infectious in the future.
The countrywide National Tuberculosis Program (NTP) to control TB was originally undertaken in 1962, but it did not achieve the goal of disease burden reduction. The Government of India has intensified anti-tuberculosis activities by implementing the DOTS strategy under the Revised National Tuberculosis Control Program (RNTCP) since 1998. DOTS is the WHO recommended treatment strategy to cure TB. The 2015 TB statistics show that the incidence rate is 217 [112-315] per thousand individuals in the population [16] and the prevalence rate estimate for the year 2014 is 195 [131-271] per 100,000 individuals in the population, [17]. The implementation of DOTS across the world has shown a decline of 1.5% in incidence over the past decade. In the new 2016-2020 Global Plan to end TB, "The Paradigm Shift", one of the targets is to achieve a 10% annual decline in TB incidence. The question of resistance to treatment and reactivation has been addressed in [13,14].
By formulating a model and analysing it, the objective of this paper is to give some estimates of the crucial parameters so that acting on these parameters may help in achieving the eradication of TB. These results could help in defining polices to bring down the TB burden in India. To achieve these goals the need for modelling is evident.
As it will become clear in the following sections, there is a discrepancy between the rates at which TB is diagnosed between the public and the private sectors. While one would expect the latter to be more efficient, and therefore to achieve a higher diagnosis rate, in fact exactly the opposite occurs. One highly improbable possibility is that the equipment or the doctors in the private sector is less efficient than in the public one. Another alternative considers instead psychologic reasons: doctors would be reluctant to notify paying patients that they have been exposed to the disease or maybe are even asymptomatic disease-carriers. At this point, independently of the reason, the question arises whether this diagnosis failure rate contributes in a substantial way to the endemicity of the disease.
On the basis of the above discussion, we would like then to address two main questions: namely whether the somewhat surprising difference in the diagnosis rates of the public and private sector makes a relevant difference for the disease eradication, and whether this eradication is at all possible.
The flow diagram in Figure 1 below corresponds to the understanding among epidemiologists in India of how a susceptible person in a population may become infected and infectious, move through treatment, recover and then possibly become infectious again. This provides the road map for writing the mathematical model for TB. In India, the healthcare sector is segregated into two sectors: the public one (i.e. government run hospitals) and the private one (clinics and/or hospitals run by private practitioners). Research shows that diagnosis and treatment in the public sector has a larger success rate in comparison to seeking care in the private sector. But, unfortunately, due to different reasons TB patients tend to seek care in private sector clinics/hospitals. TB spread is determined largely by the nature of interaction of patients with active TB with the rest of the population.
We assume that the total human population is
N(t)=S(t)+L1(t)+L2(t)+D(t)+T1(t)+T2(t). | (1) |
For convenience we omit the explicit dependence on time in these population classes.
The model, given by equations (2) below, assumes standard incidence between the diseased and susceptible population and mass action interaction between the latent but recovered
Further, note that the individuals under treatment cannot infect the suceptibles and recovered populations, since the TB bacterium becomes inactive already in the first few weeks of the treatment. Also, other safety measures, like wearing masks, reduce the chances of transmitting disease early in the treatment. Therefore, in the model we have not assumed transmission of disease from individuals undergoing treatment i.e.
The parameter
Recalling (1) and using standard incidence for modeling the disease spread among susceptibles and diseased, as explained in the previous section, we have
dSdt=A−βSDN−α0SdL1dt=(1−σ)βSDN−α0L1−ϕ1L1dDdt=σβSDN−α2D+ϕ1L1+ϕ2L2D+(μ12T1+μ22T2)−(ν1+ν2)DdT1dt=ν1D−(α3+μ11+μ12)T1dT2dt=ν2D−(α4+μ21+μ22)T2dL2dt=μ11T1+μ21T2−α1L2−ϕ2L2D | (2) |
For later purposes, we give here the Jacobian of (2)
J=(J11βSDN2βSN(DN−1)βSDN2βSDN2βSDN2J21J22J23J24J25J26J31J32J33J34J35J3600ν1J440000ν20J55000−ϕ2Dμ11μ21J66) | (3) |
with
J11=βDN(SN−1)−α0,J21=(1−σ)βDN(1−SN),J22=−(1−σ)βSDN2−(α0+ϕ1),J23=J24=J25=J26=−(1−σ)βSDN2,J31=σβDN(1−SN),J32=ϕ1−σβSDN2,J33=σβSN(1−DN)+ϕ2L2−(ν1+ν2+α2),J34=μ12−σβSDN2,J35=μ22−σβSDN2,J36=ϕ2D−σβSDN2,J44=−(α3+μ11+μ12),J55=−(α4+μ21+μ22),J66=−(α1+ϕ2D). |
It is easily seen that there are only two possible equilibria, as all other combinations of population values lead to some inconsistency in the solution of the equilibrium system.
The disease-free equilibrium (DFE)
Note that presently there are no available data on the value of
There are two infected classes
[New infections, i.e. Gains to L1New infections, i.e. Gains to DLosses from L1Losses from D]=[(1−σ)βSDNσβSDN+ϕ2L2D+ϕ1L1(α0+ϕ1)L1(α2+ν1+ν2)D−(μ12T1+μ22T2)] |
By suitably taking partial derivatives, we find
F=[∂∂L1[(1−σ)βSDN]∂∂D[(1−σ)βSDN]∂∂L1(σβSDN+ϕ2L2D)∂∂D(σβSDN+ϕ2L2D)] =[−(1−σ)βSDN2(1−σ)βSN−(1−σ)βSDN2ϕ1−σβSDN2σβSN−σβSDN2+ϕ2L2],V=[∂∂L1((α0+ϕ1)L1)∂∂D((α0+ϕ1)L1)V21V22]=[α0+α100α2+ν1+ν2], |
with
V21=∂∂L1((α2+ν1+ν2)D−(μ12T1+μ22T2))V22=∂∂D((α2+ν1+ν2)D−(μ12T1+μ22T2)). |
Evaluation at
FE0=[0(1−σ)βϕ1σβ],VE0=[α0+α100α2+ν1+ν2] |
so that
V−1E0=[1α0+ϕ1001α2+ν1+ν2],G=FE0V−1E0=[0(1−σ)βα2+ν1+ν2 ϕ1α0+ϕ1σβα2+ν1+ν2]. |
It follows that the dominant eigenvalue of
R0=12{σβα2+ν1+ν2+[σ2β2(α2+ν1+ν2)2+4(1−σ)βϕ1(α0+ϕ1)(α2+ν1+ν2)]12}. |
With the values reported in Table 1, it appears that
Description | Symbol | Value | Unit | Reference |
Immigration rate | [21] | |||
transmission rate | [1,4,5,7,8] [11,24,25,31] | |||
Proportion of infectious rapidly progressing to active disease | pure number | [6] | ||
Progression from latent to diseased class | year |
[9] | ||
Diagnosis and treatment rate in the public sector | year |
[16,17,22] | ||
Diagnosis and treatment rate in the private sector | year |
[16,17,22] | ||
Recovery (cure) rate after treatment in the public sector | year |
[16] | ||
Recovery (cure) rate after treatment in the public sector | year |
[10,27,28] | ||
Failure rate after treatment in the private sector | year |
[16] | ||
Failure rate after treatment in the private sector | year |
[27] | ||
Relapse from treatment | year |
[23,26] | ||
Natural death rate | person |
[21] | ||
Latently infected population |
year |
[23] | ||
Diseased population death rate (Case fatality rate in untreated) | year |
[17] | ||
Population under treatment death rate in public sector | year |
[16] | ||
Population under treatment death rate in private sector | year |
[27] |
To gain more insight, we provide a sensitivity analysis of
Recall that to achieve disease eradication, a value of
From Figure 2 it appears that from the current situation, variations in
Sensitivity with respect to the diagnosis and treatment rates is shown in Figures 4 and 5. The same result is obtained in Figure 7, by an increased diseased-related mortality, which however goes in the opposite direction of the goal of saving lives. Incidentally, this is an example of what in ecology and epidemiology is sometimes found, that something harmful at the individual level is beneficial for the community, and vice versa. Such an observation is found for instance in beehives, [2]. It appears that an increase in the identification of the cases and their cure will help in reducing the endemicity range of the disease. Clearly Figure 6 shows that an increase in the natural death rate will contribute also to disease eradication, but this is certainly not a practical recommendation to follow.
In Figure 8 we observe that changes in
Finally, in Figure 9 we plot the value of the basic reproduction number as function of the treatment rates in both public and private sectors. The findings indicate that the difference in the public and private sectors diagnosis rate is not essential for eradicating the disease. In fact, even if they were 100%, the plot indicates that the surface for
We can assess the asymptotic stability of the disease-free equilibrium
L1=aS0[S−S0S0−ln(1+S−S0S0)]+bL1+cD+eT1+fT2+gL2. |
Note that
L′1=−aA(S−S0)2SS0+β(S−S0)DN[b(1−σ)−a+cσ]+βDN(N−N0)S0[b(1−σ)+cσ]+L1[cϕ1−b(α0+ϕ1)]+DL2(c−g)ϕ2+D[βS0N0[b(1−σ)+cσ]+eν1+fν2−c(α2+ν1+ν2)]−α1gL2+T1[cμ12−e(α3+μ11+μ12)+gμ11]+T2[cμ22−f(α4+μ21+μ12)+gμ21]. |
Taking then
L′1≤−aA(S−S0)2SS0+L1[cϕ1−b(α0+ϕ1)] +D[aβS0+aβS0N0+eν1+fν2−c(α2+ν1+ν2)]−α1cL2+T1[cμ12−e(α3+μ11+μ12)+cμ11]+T2[cμ22−f(α4+μ21+μ12)+cμ21]. |
Finally imposing the following inequalities
cϕ1<b(α0+ϕ1),aβS0(1+1N0)+eν1+fν2<c(α2+ν1+ν2),c(μ12+μ11)<e(α3+μ11+μ12),c(μ22+μ21)<f(α4+μ21+μ22), |
it follows that
L′1≤−aA(S−S0)2SS0−α1cL2≤0 |
showing that it is nonpositive definite. This is not enough to ensure global stability. Rather, it could be the starting point to assess the largest domain of attraction of the DFE. Following [20], one can search for the largest invariant subset in
At the disease-free equilibrium the Jacobian (3) has immediately three easy eigenvalues
J0=J(E0)=(σβ−(ν1+ν2+α2)μ12μ22ν1−(α3+μ11+μ12)0ν20−(α4+μ21+μ22)) | (4) |
We now investigate numerically the disease-free equilibrium and the endemic one. Departing from the Table values only for the new individuals recruitment rate, setting it to the value
−tr(J0)=3.3184>0, −det(J0)=1.1772>0,−tr(J0)M2(J0)−det(J0)=10.4481>0. |
This shows that the DFE is stable, therefore giving rise to a bistability situation and to a backward, or subcritical, bifurcation, see [29] and [12] page 28. We investigated numerically the possibility of attaining the DFE, searching the initial values space for values that upon integration would lead to the DFE. In fact, this occurs, but for unrealistically small values of the diseased classes, namely
T1(0)=0.49D(0), T2(0)=0.41D(0), L2(0)=0.85T1(0)+0.51T2(0). |
Any increase in the values of
We then tried to study this situation as
A further decrease,
S(0)=S0+180000=194084.5070, D(0)=0.3S(0)=58225.3521,T1(0)=28530.4225, T2(0)=23872.3944,L2(0)=36425.7803, L1(0)=.8S(0)=155267.6056. | (5) |
It is plotted in Figure 10.
A similar situation arises for
For
Although the result of the previous section indicates that the disease could in principle be eradicated, in addition, we could now pursue an alternative road for trying to curb it.
First of all, we investigate whether the the coexistence equilibrium can be assessed analytically. We solve the fourth and the fifth equilibrium equations in terms of
η1=ν1α3+μ11+μ12, η2=ν2α4+μ21+μ22, L2=μ11T1+μ21T2α1+ϕ2D. | (6) |
Using the first two above equations in the last one, we find
L2=Dα1+ϕ2D[μ11ν1α3+μ11+μ12+μ21ν2α4+μ21+μ22]=D(μ11η1+μ21η2)α1+ϕ2D. | (7) |
Taking the linear combination of the first two equilibrium equations with weights
L1=(1−σ)A−α0Sα0+ϕ1. | (8) |
Adding the first, with weight
σA−α0σS+ϕ1L1+ϕ2D2μ11η1+μ21η2α1+ϕ2D+DΩ=0. | (9) |
Rearranging (9) and letting
W=Ω+μ11η1+μ21η2∈R, θ=σα0+ϕ1α0+ϕ1>0, |
leads to
σAα1+ϕ1α1L1+D(σAϕ2+Ωα1)+ϕ1ϕ2L1D+ϕ2D2W=α0σS(α1+ϕ2D). | (10) |
Use of (8) into (10) gives
Φ(S,D):=θAα1−α0α1θS+D(θAϕ2+Ωα1)−α0ϕ2θSD+ϕ2D2W=0. | (11) |
Finally, the first equilibrium equation can be rewritten as
Ψ(S,D):=−βSD+(A−α0S)[S+(1−σ)A−α0Sα0+ϕ1+D(η1+η2+1)+Dμ11η1+μ21η2α1+ϕ2D]=0. | (12) |
The curve obtained by taking the common denominator in (12) and setting the numerator to zero is a third order implicit function and therefore very difficult to study, even numerically. The values of the diseased and susceptible populations at the coexistence equilibrium would be obtained by the intersection of
The mathematical problem appears to be a hard task. For this reason, in order to gain anyway some insight into the actual situation, we make a very strong simplifying assumption. The mathematical difficulty arises from the denominator in the last fraction of (12). To simplify it, we assume
In view of the simplification, the equations for
Φs(S,D):=θA−α0θS+DΩ=0, | (13) |
Ψs(S,D):=ρA2+νAS+πAD−(α0π+β)SD−α0θS2=0, | (14) |
where
ρ=1−σα0+ϕ1>0, ν=2α0σ−α0+ϕ1α0+ϕ1∈R, π=η1+η2+1+μ11η1+μ21η2α1>0. |
Note that the straight line
14A2[α0θπ2−(α0π+β)πν−(α0π+β)2ρ]≠0. |
In particular it is a hyperbola, since its second invariant is negative,
D=α0θk2−kνA−ρA2Aπ−(α0π+β)k. |
The latter are feasible when positive, which shows that there is a feasible branch of
dθdϕ1=α01−σ(α0+ϕ1)2>0, |
implying that we must reduce
Disease eradication can in fact be achieved for an almost extreme case, taking
The model that has been introduced here was meant to compare the TB treatments performed in the public and private sectors of health care in India. The bottom line of the results of our investigation are the considerations that can be inferred from Fig. 9, namely that the low rate of diagnosis actually found in the private sector, whether it be due to malpractice, poorer diagnostic means or simply ascribed to the fact that doctors may be reluctant to let paying patients know that they are infected, seems not to constitute the main problem in the disease endemicity. It appears thus that even achieving
From the extensive simulations that we ran, see Figures 2-8, it appears that apart from the disease contact rate, the effect of the other parameters affecting
Alternatively, for values of
A different approach has also been attempted, namely to try to render the endemic equilibrium unfeasible. This approach appears to be analytically untractable, except for the unrealistic case of no disease relapses, i.e.
For the particular case
The rater sad conclusion that we must draw from all these considerations is therefore that eradicating the disease in the present state of affairs is rather difficult if not at all impossible.
This work was undertaken when the first author visited the University of Torino, with a WWS2 grant, which is thankfully acknowledged. The research has also been partially supported by the project "Metodi numerici nelle scienze applicate" of the Dipartimento di Matematica "Giuseppe Peano". EV gratefully acknowledges very useful discussions with Antoine Perasso and Rafael Bravo de la Parra and the referees for their constructive comments.
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1. | M. Marvá, E. Venturino, M.C. Vera, A Minimal Model Coupling Communicable and Non-Communicable Diseases, 2023, 18, 0973-5348, 23, 10.1051/mmnp/2023026 |
Description | Symbol | Value | Unit | Reference |
Immigration rate | [21] | |||
transmission rate | [1,4,5,7,8] [11,24,25,31] | |||
Proportion of infectious rapidly progressing to active disease | pure number | [6] | ||
Progression from latent to diseased class | year |
[9] | ||
Diagnosis and treatment rate in the public sector | year |
[16,17,22] | ||
Diagnosis and treatment rate in the private sector | year |
[16,17,22] | ||
Recovery (cure) rate after treatment in the public sector | year |
[16] | ||
Recovery (cure) rate after treatment in the public sector | year |
[10,27,28] | ||
Failure rate after treatment in the private sector | year |
[16] | ||
Failure rate after treatment in the private sector | year |
[27] | ||
Relapse from treatment | year |
[23,26] | ||
Natural death rate | person |
[21] | ||
Latently infected population |
year |
[23] | ||
Diseased population death rate (Case fatality rate in untreated) | year |
[17] | ||
Population under treatment death rate in public sector | year |
[16] | ||
Population under treatment death rate in private sector | year |
[27] |
Description | Symbol | Value | Unit | Reference |
Immigration rate | [21] | |||
transmission rate | [1,4,5,7,8] [11,24,25,31] | |||
Proportion of infectious rapidly progressing to active disease | pure number | [6] | ||
Progression from latent to diseased class | year |
[9] | ||
Diagnosis and treatment rate in the public sector | year |
[16,17,22] | ||
Diagnosis and treatment rate in the private sector | year |
[16,17,22] | ||
Recovery (cure) rate after treatment in the public sector | year |
[16] | ||
Recovery (cure) rate after treatment in the public sector | year |
[10,27,28] | ||
Failure rate after treatment in the private sector | year |
[16] | ||
Failure rate after treatment in the private sector | year |
[27] | ||
Relapse from treatment | year |
[23,26] | ||
Natural death rate | person |
[21] | ||
Latently infected population |
year |
[23] | ||
Diseased population death rate (Case fatality rate in untreated) | year |
[17] | ||
Population under treatment death rate in public sector | year |
[16] | ||
Population under treatment death rate in private sector | year |
[27] |