Citation: Oying Doso, Sarsing Gao. An overview of small hydro power development in India[J]. AIMS Energy, 2020, 8(5): 896-917. doi: 10.3934/energy.2020.5.896
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Tuberculosis (TB) is the second leading cause of death from an infectious disease worldwide [32]. Active TB refers to disease that occurs in someone infected with Mycobacterium tuberculosis. It is characterized by signs or symptoms of active disease, or both, and is distinct from latent tuberculosis infection, which occurs without signs or symptoms of active disease. Only individuals with active TB can transmit the infection. Many people with active TB do not experience typical TB symptoms in the early stages of the disease. These individuals are unlikely to seek care early, and may not be properly diagnosed when seeking care [31].
Delays to diagnosis of active TB present a major obstacle to the control of a TB epidemic [27], it may worsen the disease, increase the risk of death and enhance tuberculosis transmission to the community [24,26]. Both patient and the health system may be responsible for the diagnosis delay [24]. Efforts should be done in patient knowledge/awareness about TB, and health care systems should improve case finding strategies to reduce the delay in diagnosis of active TB [15,24,25].
Mathematical models are an important tool in analyzing the spread and control of infectious diseases [13]. There are several mathematical dynamic models for TB, see, e.g., [5,6,11,18]. In this paper we consider the mathematical model for TB proposed in [11]. We introduce a discrete time delay which represents the delay on the diagnosis of individuals with active TB and commencement of treatment. The stability of the disease free and endemic equilibriums is analyzed for any time delay.
Optimal control theory has been successfully applied to TB mathematical models (see, e.g., [19,22,23] and references cited therein). We propose and analyze an optimal control problem where the control system is the mathematical model from [11], but with a time delay in the state variable that represents individuals with active TB, and introduce two control functions. The control functions represent the fraction of early and persistent latent individuals that are treated for TB. Treatment of latent TB infection greatly reduces the risk that TB infection will progress to active TB disease. Certain groups are at very high risk of developing active TB disease once infected. Every effort should be made to begin appropriate treatment and to ensure completion of the entire course of treatment for latent TB infection [33]. Treatment of latent TB infection should be initiated after the possibility of TB disease has been excluded. It can take 2 to 8 weeks after TB infection for the body's immune system to react to tuberculin and for the infected to be detected, which justifies the introduction of a time delay on the control associated to treatment of early latent individuals. On the other hand, delays in the treatment of latent TB may also occur due to clinical and demographic patient and health care services characteristics. For these reasons, we consider discrete time delays in both control functions. To our knowledge, this work is the first to apply optimal control theory to a TB model with time delay in state and control variables.
The paper is organized as follows. In Section 2 we formulate the TB model with state delay. The stability of the disease free equilibrium is analyzed in Section 3 while stability of the endemic equilibrium is investigated in Section 4. Optimal control of TB with state and control delays is carried out in Section 5 and some numerical results given in Section 6. We end with Section 7 of conclusions.
In this section we consider a TB mathematical model proposed in [11], where reinfection and post-exposure interventions for tuberculosis are considered. The model divides the total population into five categories: susceptible (
{˙S(t)=μN−βNI(t)S(t)−μS(t),˙L1(t)=βNI(t)(S(t)+σL2(t)+σRR(t))−(δ+τ1+μ)L1(t),˙I(t)=ϕδL1(t)+ωL2(t)+ωRR(t)−τ0I(t−dI)−μI(t),˙L2(t)=(1−ϕ)δL1(t)−σβNI(t)L2(t)−(ω+τ2+μ)L2(t),˙R(t)=τ0I(t−dI)+τ1L1(t)+τ2L2(t)−σRβNI(t)R(t)−(ωR+μ)R(t). | (1) |
The initial conditions for system (1) are
S(θ)=φ1(θ),L1(θ)=φ2(θ),I(θ)=φ3(θ),L2(θ)=φ4(θ),R(θ)=φ5(θ), | (2) |
Throughout this paper, we focus on the dynamics of the solutions of (1) in the restricted region
Ω={(S,L1,I,L2,R)∈R5+0|0≤S+L1+I+L2+R=N}. |
In this region, the usual local existence, uniqueness and continuation results apply [12,14]. Hence, a unique solution
A mathematical model has a disease free equilibrium if it has an equilibrium point at which the population remains in the absence of the disease [28]. The model (1) has a disease free equilibrium given by
The basic reproduction number
R0=βμωR(ω+τ2+μ)τ1+δ[(ωR+μ)(ϕμ+ω)+(ωR+ϕμ)τ2](τ0+μ+ωR)(ω+τ2+μ)(δ+τ1+μ)=ND. | (3) |
Note that in [11] the basic reproduction number is deduced under the assumption that
It is important to analyze the stability of the disease free equilibrium, as it indicates whether the population will remain in the absence of the disease, or the disease will persist for all time [28,29]. System (1) is equivalent to
{˙S(t)=μN−βNI(t)S(t)−μS(t),˙L1(t)=βNI(t)(S(t)+σL2(t)+σR(N−S(t)−L1(t)−I(t)−L2(t)))−(δ+τ1+μ)L1(t),˙I(t)=ϕδL1(t)+ωL2(t)+ωR(N−S(t)−L1(t)−I(t)−L2(t))−τ0I(t−dI)−μI(t),˙L2(t)=(1−ϕ)δL1(t)−σβNI(t)L2(t)−(ω+τ2+μ)L2(t), | (4) |
where the equation for
{˙s(t)=−β¯I+μNNs(t)−βN¯Si(t)˙l1(t)=−βN¯I(σR−1)s(t)−βN(σR¯I+(δ+τ1+μ)N)l1(t)−βN(−¯S−σ¯L2−σR(N+¯S+¯L1+2¯I+¯L2))i(t)+β¯I(σ−σR)Nl2(t)˙i(t)=−ωRs(t)+(ϕδ−ωR)l1(t)−(ωR−μ)i(t)+(ω−ωR)l2(t)−τ0i(t−dI)˙l2(t)=(1−ϕ)δl1(t)−βNσ¯L2i(t)−βN(σ¯I+(ω+τ2+μ)N)l2(t). | (5) |
We then express system (5) in matrix form as follows:
ddt(s(t)l1(t)i(t)l2(t))=A1(s(t)l1(t)i(t)l2(t))+A2(s(t−dI)l1(t−dI)i(t−dI))l2(t−dI)) |
with
A1=[−β¯I+μNN0−β¯SN0β¯I(1−σR)N−β¯IσR+c1NNβ(¯S+σ¯L2+σR(N+¯S+¯L1+2¯I+¯L2))Nβ¯I(σ−σR)N−ωRϕδ−ωR−ωR−μω−ωR0−(−1+ϕ)δ−σβ¯L2N−β¯Iσ+c2NN], |
where
Δ(λ)=P(λ)+Q(λ)=0, | (6) |
where
P(λ)=λ4+a3λ3+a2λ2+a1λ+a0,Q(λ)=τ0(λ+μ)(λ+c1)(λ+c2)(e−λdI−1), |
with
a1=2μD+μ2(c1+c2+c4)−c4c5c6−β(τ1ωR+ωδ+δϕ(ωR+τ2+2μ)),a2=c4c5+3μ(c1+c2+c4)+c6(c4+c5)−βϕδ,a3=c1+c2+c3+μ, |
and
Remark 1. For any
Recall that an equilibrium point is asymptotically stable if all roots of the corresponding characteristic equation have negative real parts [1].
Lemma 3.1 If
Proof. The characteristic equation (6) satisfies
Lemma 3.2. If (ⅰ)
Proof. When
In the case
a3b3+τ0bc1c2−a1b−τ0b3+τ0μc1b+τ0μbc2=Acos(bdI)−Bsin(bdI)−b4+a2b2−τ0b2c1−a0−τ0b2c2−μτ0b2+τ0μc1c2=Asin(bdI)+B(cos(bdI)) |
with
b8+α3b6+α2b4+α1b2+α0=0, | (7) |
where
α1=2τ0(μ(a0+a2c1c2−a1(c1+c2))+a0(c1+c2)−a1c1c2)−2a2a0+a21,α2=2τ0(μ(a3(c1+c2)−a2−c1c2)−a2(c1+c2)+a3c1c2+a1)+2a0+a22−2a3a1,α3=2τ0(μ+c1+c2)+a32−2(a3τ0+a2). |
Let
z4+α3z3+α2z2+α1z+α0=0. | (8) |
By the Routh--Hurwitz criterion, (8) has no positive real roots if
For the parameter values of Table 1 and
z4+241.429794z3+31.065028z2−221.270089z−0.037233=0 | (9) |
Symbol | Description | Value |
Transmission coefficient | ||
Death and birth rate | ||
Rate at which individuals leave | ||
Proportion of individuals going to | ||
Endogenous reactivation rate for persistent latent infections | ||
Endogenous reactivation rate for treated individuals | ||
Factor reducing the risk of infection as a result of acquired immunity to a previous infection for | ||
Rate of exogenous reinfection of treated patients | 0.25 | |
Rate of recovery under treatment of active TB | ||
Rate of recovery under treatment of early latent individuals | ||
Rate of recovery under treatment of persistent latent individuals | ||
Total population | ||
Total simulation duration | ||
Efficacy of treatment of early latent | ||
Efficacy of treatment of persistent latent TB |
and we immediately see that the coefficient
Lemma 3.3 Let
Remark 2.Observe that there may exist specific time delays for which the disease free equilibrium
χ(λ)=λ4+17.057363λ3+20.733305λ2+4.489748λ+0.048755+2(λ+170)(λ+98170)(λ+1.014486)(e−λ0.1−1). |
The derivative
In this paper we assume that the time delay
System (1) has an unique endemic equilibrium such that
A1=[−0.0509740−28.02556100.027516−14.02345845.9707340−0.0000200.599980−0.0143060.000180011.400000−0.335130−1.023658] |
and
λ4+15.112395λ3−12.243801λ2−28.331139λ−0.966336+(30.196179λ2+30.244462λ+2λ3+1.463482)e−λdI=0. | (10) |
When
λ4+17.112395λ3+17.952378λ2+1.913323λ+0.497146=0. | (11) |
The roots of (11) are
b8+281.828573b6−51.906667b4−1.236501b2−0.000246=0. | (12) |
It is easy to verify that
λ4+15.112395λ3−12.243801λ2−28.331139λ−0.966336+(2λ3+30.196179λ2+30.244462λ+1.463482)e−0.1λ=0. | (13) |
Similarly to Remark 2, it follows from Bolzano's theorem and the monotonicity of the characteristic function associated to (13) that all roots of equation (13) have a negative real part. Therefore, the endemic equilibrium
We now consider the TB model (1) with a time delay in the state variable
dI=0.1,du1,du2∈[0.05,0.2]. | (14) |
The resulting model is given by the following system of nonlinear ordinary delay differential equations:
{˙S(t)=μN−βNI(t)S(t)−μS(t),˙L1(t)=βNI(t)(S(t)+σL2(t)+σRR(t))−(δ+τ1+ϵ1u1(t−du1)+μ)L1(t),˙I(t)=ϕδL1(t)+ωL2(t)+ωRR(t)−τ0I(t−dI)−μI(t),˙L2(t)=(1−ϕ)δL1(t)−σβNI(t)L2(t)−(ω+ϵ2u2(t−du2)+τ2+μ)L2(t). | (15) |
Recall that the recovered population is determined by
˙R(t)=τ0I(t−dI)+(τ1+ϵ1u1(t−du1))L1(t)+(τ2+ϵ2u2(t−du2))L2(t)−σRβNI(t)R(t)−(ωR+μ)R(t). |
Note, however, that this equation is not needed in the subsequent optimal control computations. We prescribe the following initial conditions for the state variables
S(0)=(76/120)N,L1(0)=(36/120)N,L2(0)=(2/120)N,R(0)=(1/120)N,I(t)=(5/120)Nfor−dI≤t≤0,uk(t)=0for−duk≤t<0(k=1,2). | (16) |
In the case
0≤uk(t)≤1∀t∈[0,T](k=1,2). | (17) |
Let us denote the state and control variable of the control system (15), respectively, by
J1(x,u)=∫T0(I(t)+L2(t)+W1u1(t)+W2u2(t))dt, | (18) |
which is linear in the control variable
J2(x,u)=∫T0(I(t)+L2(t)+W1u21(t)+W2u22(t))dt, | (19) |
which is quadratic in the control variable. In both objectives,
The optimal control problem then is defined as follows: determine a control function
H(x,y3,λ,u1,u2,v1,v2)=−(I+L2+W1u1+W2u2)+λS(μN−βNIS−μS)+λL1(βN(S+σL2+σRR)−(δ+τ1+ϵ1v1+μ)L1)+λI(ϕδL1+ωL2+ωRR−τ0y3−μI)+λL2((1−ϕ)δL1−σβNIL2−(ω+ϵ2v2+τ2+μ)L2). |
We obtain the adjoint equations
λS(T)=λL1(T)=λI(T)=λL2(T)=0. | (20) |
To characterize the optimal controls
ϕk(t)=Huk[t]+χ[0,T−duk](t+duk)Hvk[t+duk]={−Wk−ϵkλLk(t+duk)Lk(t+duk)for0≤t≤T−duk,−WkforT−duk≤t≤T. | (21) |
Then the maximum condition for the optimal controls
uk(t)={1ifϕk(t)>0,0ifϕk(t)<0,singularifϕk(t)=0onIs⊂[0,T],k=1,2. | (22) |
We do not discuss singular controls further, since both in the non-delayed and the delayed control problem we did not find singular arcs. In view of the transversality conditions (20), the terminal value of the switching function is
We choose the numerical approach "First Discretize then Optimize" to solve both the non-delayed and delayed optimal control problem. The discretization of the control problem on a fine grid leads to a large-scale nonlinear programming problem (NLP) that can be conveniently formulated with the help of the Applied Modeling Programming Language AMPL [9]. AMPL can be linked to several powerful optimization solvers. We use the Interior-Point optimization solver IPOPT developed by Wächter and Biegler [30]. Details of discretization methods for delayed control problems may be found in [10]. The subsequent computations for the terminal time
Also, the control package NUDOCCCS developed by Büskens [3] (cf. also [4]) provides a highly efficient method for solving the discretized control problem, because it allows to implement higher order integration methods. However, so far NUDOCCCS can only be implemented for non-delayed control problems. For the non-delayed TB control problem, we obtained only bang-bang controls. An important feature of NUDOCCCS is the fact that it provides an efficient method for optimizing the switching times of bang-bang controls using the arc-parametrization method [16]. This approach is called the Induced Optimization Problem (IOP) for bang-bang controls. NUDOCCCS then allows for a check of second-order sufficient conditions of the IOP, whereby the second-order sufficient conditions for bang-bang controls can be verified with high accuracy; cf. [16,17].
First, we consider the optimal control of non-delayed TB model, where formally we put
uk(t)={1for0≤t≤tk,0fortk<t≤T,k=1,2. | (23) |
To obtain a refined solution, we solve the IOP with respect to the switching times
J1(x,u)=28390.73,t1=3.677250,t2=4.866993,S(T)=1034.634,L1(T)=53.59586,I(T)=25.89556,L2(T)=780.7667,R(T)=28105.11. | (24) |
The initial value of the adjoint variable
λ(0)=(0.376159,0.452761,4.03059,0.394839). |
The control and state trajectories are displayed in Figure 2. The Hessian
(u1(t),u2(t))={(1,0)for0≤t≤t1,(1,1)fort1<t≤t2,(0,1)fort2<t≤t3,(0,0)fort3<t≤T,k=1,2. |
The objective value and the switching times are computed as
To see more distinctively the difference between delayed and non-delayed solutions, we consider state and control delays with values at their upper bounds in (14), that is,
uk(t)={1for0≤t≤tk,0fortk<t≤T,k=1,2. | (25) |
We obtain the numerical results
ϕk(t)>0for0≤t<tk,˙ϕk(tk)<0,ϕk(t)<0fortk<t≤T(k=1,2). |
However, we are not aware in the literature of any type of sufficient conditions which could be applied to the extremal solution shown in Figure 5.
We also compared the extremal solutions for the
The most significant influence on the optimal controls is exerted by the transmission coefficient
We introduced a discrete time delay
When a time delay is introduced into a mathematical model, the stability of its disease free and endemic equilibriums may change. We proved that the disease free equilibrium (DFE) of the TB model with delay in the state variable
We proposed an optimal control problem where the control system is the mathematical model for TB with time delay in the state variable
Firstly, we considered the non-delayed case (
This work was partially supported by FCT within project TOCCATA, reference PTDC/EEI-AUT/2933/2014. Silva and Torres were also supported by CIDMA and project UID/MAT/04106/2013; Silva by the post-doc grant SFRH/BPD/72061/2010. The authors are very grateful to anonymous reviewers for their careful reading and helpful comments.
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Symbol | Description | Value |
Transmission coefficient | ||
Death and birth rate | ||
Rate at which individuals leave | ||
Proportion of individuals going to | ||
Endogenous reactivation rate for persistent latent infections | ||
Endogenous reactivation rate for treated individuals | ||
Factor reducing the risk of infection as a result of acquired immunity to a previous infection for | ||
Rate of exogenous reinfection of treated patients | 0.25 | |
Rate of recovery under treatment of active TB | ||
Rate of recovery under treatment of early latent individuals | ||
Rate of recovery under treatment of persistent latent individuals | ||
Total population | ||
Total simulation duration | ||
Efficacy of treatment of early latent | ||
Efficacy of treatment of persistent latent TB |
Symbol | Description | Value |
Transmission coefficient | ||
Death and birth rate | ||
Rate at which individuals leave | ||
Proportion of individuals going to | ||
Endogenous reactivation rate for persistent latent infections | ||
Endogenous reactivation rate for treated individuals | ||
Factor reducing the risk of infection as a result of acquired immunity to a previous infection for | ||
Rate of exogenous reinfection of treated patients | 0.25 | |
Rate of recovery under treatment of active TB | ||
Rate of recovery under treatment of early latent individuals | ||
Rate of recovery under treatment of persistent latent individuals | ||
Total population | ||
Total simulation duration | ||
Efficacy of treatment of early latent | ||
Efficacy of treatment of persistent latent TB |