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This paper focuses on the study and control of a non-linear mathematical epidemic model (
*SS*
_{vih}
*VELI *) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with HIV/AIDS coinfection. Existence of both disease free equilibrium and endemic equilibrium is discussed. Reproduction number R
_{0} is determined. Using Lyapunov-Lasalle methods, we analyze the stability of epidemic system around the equilibriums (disease free and endemic equilibrium). The global asymptotic stability of the disease free equilibrium whenever R
_{vac < 1} is proved, where R
_{0} is the reproduction number. We prove also that when R
_{0} is less than one, tuberculosis can be eradicated. Numerical simulations are conducted to approve analytic results. To achieve control of the disease, seeking to reduce the infectious group by the minimum vaccine coverage, a control problem is formulated. The Pontryagin’s maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically using the Runge Kutta fourth procedure.

Tuberculosis is one of the top 10 causes of death in the world [

The theme “End the global TB” has been decided by the world health organization and it covers the period 2016-2035 and the overall goal is to end the global TB epidemic. As part of the necessary multidisciplinary research approach, mathematical models have been extensively used to provide a framework for understanding tuberculosis transmission dynamics and control strategies of the infection spread in the host population [

Some mathematical models have been used to control tuberculosis. In [

This paper deals with the stability analysis of an S V S v i h , E L I transmission model and uses optimal control technique to find and evaluate the impact of a mass vaccination schedule in the spread of TB/VIH coinfection. Individuals are classified as one of susceptible (S) V vaccinated, AIDS patients ( S v i h ), earlier latent (E) late latent (L) infectious or tuberculous (I), but allow that susceptible individuals may be given an imperfect vaccine that reduces their susceptibility to the disease, the V-compartment of vaccinated individuals is considered [

Since V and S v i h are considered as the susceptible compartments, thus we are dealing with a differential susceptibility system with bilinear mass action as in Hyman and Li [

For numerical simulation, the programs used in this paper are designed so that no knowledge of MATLAB is required. For the control problem, there is a user-friendly interface that will guide you through. We have two different MATLAB programs, plotTB.m and codeTB.m. Here, .m is the extension given to all files intended for use in MATLAB.The file codeTB.m is the Runge-Kutta based, forward-backward sweep solver. It takes as input the values of the various parameters in the problem and outputs the solution to the optimality system. The ?le plotTB.m is the user friendly interface. It will ask you to enter the values of the parameters one by one, compile codeTB.m with these values, and plot the resulting solutions. All the files must be in the directory that MATLAB treats as the home directory. This is usually the Work directory. This paper is organized as follows.

In next section, model is described, in Section 3 we investigate stability analysis for the ( S S v i h V E L I ) epidemic model in this section We followed the methods of Nkamba, Leontine Nkague et al. 2019 [

When first infected with TB bacteria, a person typically goes through a latent, asymptomatic and non-infectious period during which the body’s immune system fights the TB bacteria. There are two distinct stages of the latent TB infection. During the first two years, the risk of developing active disease is much higher, whereas during the later stage, the progression to active disease is much slower.

Compartmental modeling is used among epidemiologists to simulate disease dynamics. These models treat each disease state as a different compartment that contains a homogeneous population of individuals. Using a compartmental approach, the total host population can be partitioned into seven compartments: susceptible individuals (S), susceptible infected with AIDS ( S v i h ), vaccinated individuals (V) early latent (E) late latent (L) individuals, individuals with active TB disease (I) and recovered individuals (R). S ( t ) , S v i h ( t ) , V ( t ) , E ( t ) , L ( t ) and I ( t ) denote the density of populations in the four corresponding compartments at time t. Only individuals in compartment I are infectious, and new infections result on the one hand from contacts between a susceptible and an infectious individual, with an incidence rate β S ( t ) I ( t ) ; from contacts between a HIV patient and Tuberculous, with an incidence ( 1 + σ ) β S v i h ( t ) I ( t ) ,and on the other hand from contacts between a vaccinated and infectious individual, with an incidence θ 1 β V ( t ) I ( t ) ,due to the fact that the vaccine does not confer a total immunity, but Vaccination reduces the risk of infection by a factor θ 1 ∈ [ 0,1 ] and the efficacy of the vaccine is 1 − θ 1 . AIDS increases the risk of infection by a factor, the immune deficiency rate σ ∈ [ 0,1 ] σ = 1 − T C D 4 T C D 4 is the rate of CD4 cells. Let us pose θ 2 = ( 1 + σ ) the cost induced by the immune deficiency status in the transmission of tuberculosis. The per capita death rates for susceptibles, HIV patients, early latents, late latents and infectious individuals are μ S , μ v i h , μ E , μ L and μ I respectively. Once infected, individuals progress through the early latent stage with an average rate ω . A fraction p ; 0 < p ≤ 1 ; of these individuals progress directly to the active TB stage, and the remaining 1 − p fraction progresses to the late latent stage. Once there, the rate of progression to active disease is at a lower rate ν . The recruitment makes respectively into the susceptible class, the vaccinated class, the VIH/patient class with the constant rate π 1 π 2 and π 3 . α is the vaccination coverage rate. π 2 is the recruitment of vaccinated a few day after birth, so we suppose that immunity is passed during the birth. π 3 is VIH/AIDS vertical transmission recruitment, it’s means some peoples born with VIH/AIDS infection.

The dynamical transfer among the seven compartments is depicted in the transfer diagram (

All parameters described in

Our model consists of the following system of ordinary differential equations:

(S) | Susceptible: Health and non immune individuals |
---|---|

(V) | Vaccinated: immunized individuals |

(S_{vih}) | AIDS patient: people infected with HIV |

(E) | Earlier latent: people infected but without clinic signs of disease |

(L) | Later latent: People who remains without clinic signs |

( α ) | Vaccination coverage |

η S | Recruitment rate of susceptible |

π 1 = η S S | Number of susceptible rescued at birth |

π 2 = η V V | Number of vaccinated rescued at birth |

π 3 = η v i h S v i h | Number of VIH/AIDS patients rescued at birth |

(I) | People infected with Tuberculosis and are infectious |

( ε ) | AIDS-HIV prevalence |

( ω ) | Progression rate to latent earlier stage |

( β S I ) | Incidence rate: number of new infected cases |

( σ = 1 − T CD4 ) | Immune deficiency rate |

( θ 2 = 1 + σ ) | Cost of immune deficiency rate |

( θ 1 ) | Vaccine efficacity |

d X ≥ 0, d E ≥ 0, d E ≥ 0, d T ≥ 0 | Removal rate, include death due to the TB |

( β ) | Effective contact rate |

μ v i h | HIV specific death rate |

p | Fraction of individuals who progress directly in TB active stage |

ν | Progression rate to active disease |

1 − p | Progression fraction to the late latent stage |

{ S ˙ = π 1 − β S I − ( μ S + ε + α ) S , V ˙ = π 2 + α S − θ 1 β V I − μ V V S ˙ v i h = π 3 + ε S − μ v i h S v i h − θ 2 β S v i h I , E ˙ = β S I + θ 2 β S v i h I + θ 1 β V I − ( μ E + ω ) E , L ˙ = ( 1 − p ) ω E − ( μ L + ν ) L , I ˙ = p ω E + ν L − ( μ I + τ ) I , R ˙ = τ I − μ R N = S + S v i h + V + E + L + I + R (1)

with initial conditions ( S ( 0 ) , S v i h ( 0 ) , V ( 0 ) E ( 0 ) , L ( 0 ) , I ( 0 ) R ( 0 ) ) ∈ ℝ + 5 . We have also θ 2 = 1 + σ and σ = 1 − T c d 4 .

In order that the model be well-posed, it is necessary that the state variables S ( t ) , S v i h ( t ) , V ( t ) E ( t ) , R ( t ) I ( t ) and I ( t ) remain nonnegative for all t ≥ 0 . That is, the nonnegative orthant ℝ + 5 must be positively invariant.

Let

Γ = { ( S , S v i h , V , E , L , I ) ∈ ℝ + 5 : 0 ≤ S ≤ S 0 ,0 ≤ S v i h ≤ S v i h 0 ,0 ≤ V ≤ V 0 ,0 ≤ N ≤ Λ μ } (2)

where μ = min { μ S , μ v i h μ V , μ E , μ L , μ I } and Λ = π 1 + π 2 + π 3 with ( S 0 V 0 S v i h 0 0000 ) the disease free equilibrium.

Lemma 1. The compact set Γ is a positively invariant and attracting.

It is easy to check that model 1 always has the disease-free equilibrium

P 0 = ( S 0 , S v i h 0 , V 0 , 0 , 0 , 0 , 0 )

where

S 0 = π 1 μ S + ε + α , S v i h 0 = ε S 0 + π 3 μ v i h and V 0 = α S 0 + π 2 μ V (3)

In order to assume that vaccinated people don’t produced infected more than susceptible people we should have 0 ≤ θ 1 ≤ S 0 V 0 .

Additionally, an endemic equilibrium P * = ( S * , S v i h * , V * E * , L * , I * ) may also exist.

To consider the existence and uniqueness of endemic equilibrium P * = ( S * , S v i h * , V * E * , L * , I * ) ,we firstly study the basic reproductive number induced by vaccine R v a c of model.

Using the method of James Watmouth and all the next generation matrix [

R 0 = β ω ( S 0 + θ 1 V 0 + θ 2 S v i h 0 ) [ ν ( 1 − p ) + p ( μ L + ν ) ] ( μ I + τ ) ( μ E + ω ) ( μ L + ν ) . (4)

We will see in the Section 3.4 theorem 1 that, when R 0 is less than unity, infection can disappear in the population. Numerical simulations will confirm our results.

In this section, we show that the disease-free equilibrium P 0 is globally asymptotically stable with respect to if R 0 ≤ 1 ; and P 0 is unstable if R 0 > 1 :

Theorem 1. If R 0 ≤ 1 ,then the disease-free equilibrium is globally asymptotically stable.

Proof. Consider a Lyapunov function,

V = V ( S , S v i h , V , E , L , I ) = p ω ( μ L + ν ) E + ν ( μ E + ω ) L + ( μ E + ω ) ( μ L + ν ) I . (5)

Direct calculation leads to

V ˙ = p ω ( μ L + ν ) E ˙ + ν ( μ E + ω ) L ˙ + ( μ E + ω ) ( μ L + ν ) I ˙ = p ω ( μ L + ν ) ( β S I + θ 1 β V I + θ 2 β S v i h ) I + ν ( 1 − p ) ( μ E + ω ) E + ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) I (6)

at equilibrium we have the relation

β S I + θ 2 β S v i h I + θ 1 β V I = ( μ E + ω ) E

then

V ˙ = ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) × [ ( β S + θ 2 β S v i h + θ 1 β V ) [ p ω ( μ L + ν ) + ν ω ( 1 − p ) ] ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) − 1 ] I (7)

Because S , V , S v i h ∈ Γ we have:

V ˙ ≤ ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) × [ ( β S 0 + θ 2 β S v i h 0 + θ 1 β V 0 ) [ p ω ( μ L + ν ) + ν ω ( 1 − p ) ] ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) − 1 ] I V ˙ ≤ ( μ E + ω ) ( μ L + ν ) ( μ I + τ ) [ R v a c − 1 ] I . (8)

Furthermore

V ˙ = 0 ⇔ I = 0 ⇒ I ˙ = 0 ⇒ L = E = 0

Therefore, the largest compact invariant set in G = { ( S ; S v i h ; V ; E ; L ; I ) ∈ Γ : V ˙ = 0 } ; when R v a c ≤ 1 ,is the singleton { P 0 } : LaSalle’s Invariance Principle implies that all solutions in Γ converges to P 0 . This establishes the theorem.

Theorem 1 completely determines the global dynamics of (1) in Γ when R 0 ≤ 1 . It establishes the basic reproduction number R 0 as a sharp threshold parameter. Namely, if R 0 < 1 ; all solutions in the feasible region converge to the disease-free equilibrium P 0 ; and the TB will die out from the population irrespective of the initial conditions. If R 0 > 1 ; P 0 is unstable, it could exist and endemic equilibrium and the system is uniformly persistent, TB epidemic will always become endemic.

In matlab code, the parameters are named as followed:

pi1 = π_{1} pi2 = π_{2} pi3 = π_{1}; beta = β varepsilon = ε; muvih = μ_{vih}; sigma = σ; theta1 = θ_{1}; theta2 = θ_{2} p; muS = μ_{S} muV = μ_{V} muE = μ_{E} omega = ω muL = μ_{L} nu = ν muI = μ_{I} tau = τ A; alpha = α S0 = S_{0} V0 = S_{0} Svih0 = S_{0} E0 = E_{0} L0 = L_{0} I0 = I_{0}.

Let us take the following set of parameters

pi1 = 15; pi2 = 0.1; pi3 = 0.1; beta = 0.085; varepsilon = 0.2; muvih = 0.05; sigma = 0.01; theta1 = 0.001; theta2 = 1.01; p = 0.2; muS = 0.01; muV = 0.01; muE = 0.02; omega = 0.0645; muL = 0.02; nu = 0.00375; muI = 0.3; tau = 0.5; alpha = 0.2; S0 = 80; V0 = 50; Svih0 = 10; E0 = 20; L0 = 15; I0 = 25.

When β = 0.01 and R 0 = 0.57 R 0 is less than unity, the trajectory of TB patients (I) reach axis axe by 10 years and remains at that position with time (

We remark also that when β = 0.085 R 0 = 4.58 greater than unity, the TB patients remains in the community, so it could exist an endemic equilibrium stable in

In this section, an optimal control is formulated and it examined to study properties of optimal control strategies.

Let u 1 ( t ) = α ( t ) (vaccine coverage), the first control, be the percentage of susceptible individuals being vaccinated per unit of time. As vaccination of the entire susceptible population is impossible, we bound the control with 0 ≤ u ( t ) ≤ 0.9 . In cash we seek to minimise the infectious group with the minimum possible of vaccine coverage. We consider an optimal control problem

to minimize the objective functional

min u ∫ 1 T I + A u ( t ) 2 d x (9)

subject to S ˙ ( t ) = π 1 − β S ( t ) I ( t ) − ( μ S + u ( t ) + ε ) S , S ( 0 ) = S 0 V ˙ ( t ) = π 2 + u 1 ( t ) S ( t ) − θ 1 β V ( t ) I ( t ) − μ V V ( t ) , V ( 0 ) = S 0 S ˙ v i h ( t ) = ( π 3 + ε ) S − μ v i h S v i h ( t ) − θ 2 β S v i h ( t ) I ( t ) , S v i h ( 0 ) = S v i h 0

E ˙ ( t ) = β S ( t ) I ( t ) + θ 2 β S v i h ( t ) I ( t ) + θ 1 β V ( t ) I ( t ) − ( μ E + ω ) E ( t ) , E ( 0 ) = E 0 L ˙ ( t ) = ( 1 − p ) ω E ( t ) − ( μ L + ν ) L ( t ) , L ( 0 ) = L 0 I ˙ ( t ) = p ω E ( t ) + ν L ( t ) − ( μ I + τ ) I ( t ) , I ( 0 ) = I 0 0 ≤ u ( t ) ≤ 0.8 (10)

Let us pose

λ ( t ) = ( λ i ( t ) ) with i = 1 , 2 , ⋯ , 6 X ( t ) = ( X 1 , X 2 , X 3 , ⋯ , X 6 ) (11)

Hamiltonian of our control problem is

H ( t X ( t ) u ( t ) λ ( t ) ) = I + A u 1 ( t ) 2 + ∑ i = 1 6 λ i ( t ) X ˙ i (t)

H ( t X ( t ) u ( t ) λ ( t ) ) = I + A u ( t ) 2 + λ 1 [ π 1 − β S ( t ) I ( t ) − ( μ S + u ( t ) + ε ) S ] + λ 2 [ π 2 + u ( t ) S ( t ) − θ 1 β V ( t ) I ( t ) − μ V V ( t ) ] + λ 3 [ ( π 3 + ε ) S − μ v i h S v i h ( t ) − θ 2 β S v i h ( t ) I ( t ) ]

+ λ 4 [ β S ( t ) I ( t ) + θ 2 β S v i h ( t ) I ( t ) + θ 1 β V ( t ) I ( t ) − ( μ E + ω ) E ( t ) ] + λ 5 [ ( 1 − p ) ω E ( t ) − ( μ L + ν ) L ( t ) ] + λ 6 [ p ω E ( t ) + ν L ( t ) − ( μ I + τ ) I ( t ) ] (12)

The adjoint equations and transversality conditions can be obtained by using Pontryagin’s Maximum Principle such that

− ∂ H ∂ S = λ ′ 1 = ( λ 1 − λ 4 ) β I * + λ 1 ( μ S + u + ε ) − λ 2 u − λ 3 ε , λ 1 ( t f ) = 0 − ∂ H ∂ V = λ ′ 2 = ( λ 2 − λ 4 ) θ 1 β I * + λ 2 μ V , λ 2 ( t f ) = 0 − ∂ H ∂ S v i h = λ ′ 3 = ( λ 3 − λ 4 ) θ 2 β I * + λ 3 μ v i h , λ 3 ( t f ) = 0 − ∂ H ∂ E = λ ′ 4 = λ 4 ( μ E + ω ) , λ 4 ( t f ) = 0

− ∂ H ∂ L = λ ′ 5 = λ 5 ( μ L + ν ) , λ 5 ( t f ) = 0 − ∂ H ∂ I = λ ′ 6 = − 1 + λ 6 ( μ I + τ ) + β ( λ 1 − λ 4 ) S + β θ 1 ( λ 2 − λ 4 ) V + β θ 2 ( λ 3 − λ 4 ) S v i h , λ 6 ( t f ) = 0 (13)

Taking into account the bounds on control 0 ≤ u ≤ 0.8 Optimal control u * ( t ) ,is derived using the following optimality conditions:

{ u = 0 if ∂ H ∂ u ≥ 0 0 ≤ u ≤ 0.8 if ∂ H ∂ u = 0 u = 0.8 if ∂ H ∂ u ≤ 0 (14)

From relations 14 we have:

∂ H ∂ u = 2 A u * + S * ( λ 2 − λ 1 ) = 0 when 0 ≤ u * ≤ 0.8 (15)

∂ H ∂ u = 0 ⇔ 2 A u * + S * ( λ 2 − λ 1 ) = 0 (16)

From relations 16 we have:

u * ( t ) = S * ( λ 1 − λ 2 ) 2 A where 0 ≤ u * ≤ 0.8 (17)

Taking account the optimality conditions 14 induce by the bounds conditions of control u we have

u * ( t ) = min ( 0.8 , max ( 0 , S * ( λ 1 − λ 2 ) 2 A ) ) (18)

The numerical algorithm presented below is a classical Rung -Kutta 4 method.

We discretize the interval [t0,T = tf] at the points t i = t 0 + i h ( i = 0 , 1 , ⋯ , n ), where h is the time step such that t n = T = M , h 2 = h / 2 and j = M + 2 − i .

Next, we define the state and adjoint variables respectivily S ( t ) , S v i h ( t ) , V ( t ) E ( t ) , R ( t ) I ( t ) and λ 1 ( t ) , λ 2 ( t ) , λ 3 ( t ) , λ 4 ( t ) , λ 5 ( t ) , λ 6 (t))

The control u in terms of nodal points S ( i ) , S v i h ( i ) , V ( i ) , E ( i ) , R ( i ) , I ( i ) and λ 1 ( j ) , λ 2 ( j ) , λ 3 ( j ) , λ 4 ( j ) , λ 5 ( j ) , λ 6 ( j ) u ( i ) .

Most of parameters values are from Cameroon, like natural rate of mortality and rate of birth according to the World Health organisation report 2017. Other parameters are extracted to the data collected at the Hospital Jamot Center, where is housed the screening center of tuberculosis. This center receive approximatively one thousand new cases of tuberculosis each year. Data have been collected during one year (31 March 2016 to 31 March 2017). Those data concerned new cases of pulmonary and extrapulmonary TB patients; number of TB patients tested HIV; Number of TB patients co-infected by VIH; The distribution of TB patients by 11 age and sex. Helped by theses data we found out that, about three new cases are detected by day, the mortality date is 0.1; the percentage of TB patients tested for HIV is 90% and about 35% of them are HIV positive.The mortality rate due to the infection is 0.1% Recovery rate and rate of apparition of clinical symptoms are coming from WHO. See

Some key parameters like the effective contact rate β medical coverage rate of VIH/Patients σ ,HIV vertical transmission rate π 3 and HIV prevalence ε have a great impact in the spread of TB infectious. we are going to simulate five different scenarios and observed the mass vaccination optimal strategy induced respectively by the low effective contact rate, the hight effective contact rate, the absence of HIV medical coverage, the hight HIV medical coverage and the combination (hight HIV medical coverage, low HIV vertical transmission rate, and the low HIV prevalence rate) which assure the eradication of TB infectious.

Symbols | parameters | values | source |
---|---|---|---|

fixed disease parameters | |||

ω | Early progression rate | 0.0645 years | Diel et al. 2011 |

ν | Reactivation | 0.00375 years | Blower et al. 1995 |

fixed epidemiological parameters | |||

η | Recruitment rate of susceptible | 0.038/year | Cameroon |

μ J J ∈ { S L } | specific death rate in population J | 0.0098 | Cameroon |

μ v i h | VIH specific death rate | [0.1] | UNAIDS 2016 |

μ T | TB specific death rate | [0.1] | OMS 2015 |

( α ) | vaccination coverage | variable | |

π 1 = η S S | number of susceptible rescued at birth | variable | |

π 2 = η V V | number of vaccinated rescued at birth | variable | |

π 3 = η v i h S v i h | number of VIH/AIDS patients rescued at birth | variable | |

1 − p | proportion to the late latent stage | 0.4105 over 23 months | James M, Justin T |

p | infectious proportion fraction | 0.5895 | |

modifiable parameters | |||

σ = 1 − C D 4 ( r a t e ) | immune deficiency status rate | variable | Cameroon |

Λ = η S | variable | Cameroon | |

β | Effective contact rate | variable | Cameroon |

ε | AIDS prevalence rate | variable |

For simulations we have built two matlab file:

The first one is tuberculosesida.m the main file and the second one is val_{t} uberculosesida.m in order to plot some figures.

In matlab code, the parameters are named as followed.

pi1 = π_{1}; pi2 = π_{2} pi3 = π_{1}; beta = β varepsilon = ε; muvih = μ_{vih}; sigma = σ; theta1 = θ_{1}; theta2 = θ_{2} p; muS = μ_{S}; muV = μ_{V}; muE = μ_{E}; omega = ω; muL = μ_{L}; nu = ν; muI = μ_{I}; tau = τ A; alpha = α; S0 = S_{0}; V0 = S_{0}; Svih0 = S_{0}; E0 = E_{0}; L0 = L_{0}; I0 = I_{0}.

Set of parameters values

pi1 = 50; pi2 = 1; pi3 = 0.1; beta = 0.08; varepsilon = 0.00001; muvih = 0.05; sigma = 0.0001; theta1 = 0.3; theta2 = 1.0001; p = 0.6; muS = 0.01; muV = 0.01; muE = 0.02; omega = 0.0605; muL = 0.02; nu = 0.00375; muI = 0.12; tau = 0.1; A = 1; alpha = 0.2; S0 = 80;V0 = 50; Svih0 = 10; E0 = 8; L0 = 7; I0 = 5.

With the precedents values of parameters we obtain a Reproduction number ( R 0 ) bigger than unity. In this case

Set of parameters values

pi1 = 50; pi2 = 1; pi3 = 0.1; beta = 0.6; varepsilon = 0.00001; muvih = 0.05; sigma = 0.0001; theta1 = 0.3; theta2 = 1.0001; p = 0.6; muS = 0.01; muV = 0.01; muE = 0.02; omega = 0.0605; muL = 0.02; nu = 0.00375; muI = 0.12; tau = 0.1; A = 1; alpha = 0.2; S0 = 80; V0 = 50; Svih0 = 10; E0 = 8; L0 = 7; I0 = 5.

Here in

reached more later abscise axis and TB patients leave abscise axis later also, this means that spread of epidemic is retarded. we could conclude that more earlier system is controlling more later epidemic occurs in the population.

Set of parameters values

pi1 = 50; pi2 = 1; pi3 = 0.1; beta = 0.4; varepsilon = 0.00001; muvih = 0.05; sigma = 1; theta1 = 0.3; theta2 = 2; p = 0.6; muS = 0.01; muV = 0.01; muE = 0.02; omega = 0.0605; muL = 0.02; nu = 0.00375; muI = 0.12; tau = 0.1; A = 1; alpha = 0.2; S0 = 80; V0 = 50; Svih0 = 10; E0 = 8; L0 = 7; I0 = 5.

In this case 5 T c d 4 is nul, that’s suppose HIV patients are not cured. In

When VIH patients are cured, optimal control appears later after 40 years and stay more than five years

Set of parameters values

pi1 = 50; pi2 = 1; pi3 = 0.1; beta = 0.4; varepsilon = 0.00001; muvih = 0.05; sigma = 0.0001; theta1 = 0.3; theta2 = 1.0001; p = 0.6; muS = 0.01; muV = 0.01;

muE = 0.02; omega = 0.0605; muL = 0.02; nu = 0.00375; muI = 0.12; tau = 0.1; A = 1; alpha = 0.2; S0 = 80; V0 = 50; Svih0 = 10; E0 = 8; L0 = 7; I0 = 5.

When HIV medical coverage is hight, control optimal appears more later step 80 and stay until step 200. In

Set of parameters values

pi1 = 50; pi2 = 1; pi3 = 0.01; beta = 0.4; varepsilon = 0.00001; muvih = 0.05; sigma = 0.0001; theta1 = 0.3; theta2 = 1.0001; p = 0.6; muS = 0.01; muV = 0.01; muE = 0.02; omega = 0.0605; muL = 0.02; nu = 0.00375; muI = 0.12; tau = 0.1; A = 1; alpha = 0.2; S0 = 80; V0 = 50; Svih0 = 10; E0 = 8; L0 = 7; I0 = 5.

Here we have reproduction number R 0 less than unity, and the system 1 reach a disease free equilibrium (

prevalence rate ε = 0.00001 and the low VIH vertical transmission π 3 = 0.1 ) assures the eradication of TB infectious in community. We can conclude that, well controlling the spread of VIH infectious (hight medical coverage of HIV patients, voluntary HIV screening campaign and HIV awareness campaigns) has positive effects in the propagation of TB infectious. Mass vaccination is not necessary when we have at least a good percentage of peoples who are vaccinated at birth.

All the scenarios are resumed in the following

See

The goals of this paper were to study the overall and asymptotic stability of the system around the point of equilibrium on one hand and, to use optimal control techniques to find mass vaccination strategies for each situation and assess impact on the second hand. We simulated the spread of tuberculosis/HIV coinfection and mass vaccination schedule. The database used was essentially made of the World Health organisation report 2017 and the data collected at the Centre Jamot Hospital where is housed the screening center of tuberculosis. The

approach of optimal control used in various cases with adapted tools, leads to some lessons learnt. The higher the effective contact rate will be, the earlier we should start the mass vaccination that could economically expensive (

In the area of HIV and TB co-infection, to reach the target of eradication of the TB propagation, we need to control the HIV propagation and make an emphasis on the immunization against the TB infection, the early screening and treatment of TB patients.

The first author acknowledges with thanks, the Chair of Computational Mathematics at Deusto Tech Laboratory in the University of Deusto, Bilbao (Basque Country, Spain), the Center for Tuberculosis Screening and Treatment Jamot Hospital, Yaounde, Cameroon.

No potental conflict of interest was reported by the authors.

The data used to support the findings of this study are available from the corresponding author upon request.

This work was supported by Women for Africa Foundation in collaboration with Diputacion Foral de Bizkaia and Deustotech-Deusto Foundation.

Nkamba, L.N., Manga, T.T. and Sakamoto,^{ }N. (2019) Stability and Optimal Control of Tuberculosis Spread with an Imperfect Vaccine in the Case of Co-Infection with HIV. Open Journal of Modelling and Simulation, 7, 97-114. https://doi.org/10.4236/ojmsi.2019.72005