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Presently, an ongoing outbreak of the monkeypox virus infection that began in Bayelsa State of Nigeria has now spread to other parts of the country including mostly States in the South-South with the Nigerian Ministry of Health confirming 4 samples out of the 43 sent for testing at WHO Regional Laboratory in Dakar, Senegal. This reminds us that apart from the eradicated smallpox, there are other poxviruses that pose potential threat to people in West and Central Africa. In this paper, we developed a mathematical model for the dynamics of the transmission of monkeypox virus infection with control strategies of combined vaccine and treatment interventions. Using standard approaches, we established two equilibria for the model namely: disease-free and endemic. The disease-free equilibrium was proved to be both locally and globally asymptotically stable if
R
_{0} < 1 using the next-generation matrix and the comparison theorem. While the endemic equilibrium point existed only when
R
_{0} > 1, was proved to be locally asymptotically stable if
R
_{0} > 1 using the linearization plus row-reduction method. The basic reproduction numbers for the humans and the non-human primates of the model are computed using parameter values to be
R
_{0,h} = 9.1304 x 10
^{-6} and
R
_{0,n} = 3.375 x 10
^{-3} respectively. Numerical simulations carried out on the model revealed that the infectious individuals in the human and non-human primates’ populations will die out in the course of the proposed interventions in this paper during the time of the study. Sensitivity analysis carried out on the model parameters shows that the basic reproduction numbers of the model which served as a threshold for measuring new infections in the host populations decrease with increase in the control parameters of vaccination and treatment.

Monkeypox is an infectious disease caused by the monkeypox virus [

In humans, the symptoms of the monkeypox virus infection are similar to but milder than the symptoms of smallpox. The infection usually begins with fever, headache, muscle ache and exhaustion [

Evidence of viral infection in humans with monkeypox virus was first identified in the Democratic Republic of Congo (DRC), formerly known as Zaire, in the town of Basankusu, Equateur Province in the year 1970 [

Between 1970 and 1986, 404 cases of monkeypox virus were reported in 7 West and Central Africa countries (DRC, Ivory Coast, Sierra Leone, Cameroon, Central African Republic, Liberia and Nigeria) among mainly children [

Currently, there is not much on the modeling aspect of the monkeypox virus infection [

Therefore, this paper is set out to review the existing work of [

The model in this paper divides the host population into two; the non-human primates and/or some wild rodents, and the humans host population. The non-human primate population was further divided into Susceptible ( S n ) , Exposed/Latent ( E n ) , Infected ( I n ) and Recovered ( R n ) subpopulations. The non-human primates and/or some wild animals are recruited into the Susceptible ( S n ) class at a constant birth rate Λ n and become exposed to the monkeypox virus after getting into contact with an infected non-human primate at a rate λ n with

λ n = β n 1 I n N n (1)

where β n 1 is the product of effective contact rate and probability of the non-human primate getting infected per contact. After incubation of the virus is achieved, the exposed primate proceeds to the infected class ( I n ) at a rate ν n . The infected animals in ( I n ) are capable of either; infecting other animals when they come into contact, die due to the disease at a rate d n or recover naturally with permanent immunity at a rate ρ n and move into ( R n ) . All non-human primates in the model experience natural mortality rate μ n .

The total human host population was also divided into Susceptible ( S h ) , Vaccinated ( V h ) , Exposed/Latent ( E h ) , Infected ( I h ) and Recovered ( R h ) human subpopulations. The Susceptible humans are recruited into ( S h ) through birth and migration at a constant rate Λ h . A susceptible individual is either vaccinated against the monkeypox virus at a rate α h and move to ( V h ) with permanent immunity or become exposed to the monkeypox virus after getting into contact with an infected human or non-human primates at a rate λ h with

λ h = β n 2 I n N n + β h I h N h (2)

where β n 2 is the product of the effective contact rate and probability of the human being infected per contact with an infectious non-human primate animal, and β h is the product of the effective contact rate and the probability of the human being infected with monkeypox virus after getting into contact with an infectious human per contact. After the incubation period, the Exposed human in ( E h ) proceeds to the infected class ( I h ) at a rate ν h . Individuals in ( I h ) either die due to the virus at a constant rate d h or recover with permanent immunity after receiving treatment at a rate ρ h into ( R h ) . All individuals in the human subpopulations suffer a natural mortality at a constant rate μ h . All parameters in the model are strictly nonnegative and will assume values presented in

From the description of the model and the schematic diagram presented in

S ′ n = Λ n − ( μ n + λ n ) S n (3)

E ′ n = λ n S n − ( μ n + ν n ) E n (4)

I ′ n = ν n E n − ( μ n + d n + ρ n ) I n (5)

Parameter | Value | Source |
---|---|---|

Λ n | 0.2 | Assumed |

Λ h | 0.029 | [ |

μ n | 0.1 | Assumed |

μ h | 0.02 | [ |

d n | 0.2 | Assumed |

d h | 0.1 | [ |

ρ n | 0.3 | Assumed |

ρ h | 0.83 | [ |

ν n | 0.3 | Assumed |

ν h | 0.095 | [ |

α h | 0.1 | Assumed |

β n 1 | 0.0027 | [ |

β n 2 | 0.00252 | [ |

β h | 0.000063 | [ |

R ′ n = ρ n I n − μ n R n (6)

S ′ h = Λ h − ( μ h + λ h + α h ) S h (7)

V ′ h = α h S h − μ h V h (8)

E ′ h = λ h S h − ( μ h + ν h ) E h (9)

I ′ h = ν h E h − ( μ h + d h + ρ h ) I h (10)

R ′ n = ρ h I h − μ h R h (11)

N n ( t ) = S n + E n + I n + R n (12)

N h ( t ) = S h + V h + E h + I h + R h (13)

Subject to the following nonnegative initial conditions:

S n ( 0 ) ≥ 0 , E n ( 0 ) ≥ 0 , I n ( 0 ) ≥ 0 , R n ( 0 ) ≥ 0 (14)

S h ( 0 ) ≥ 0 , V h ( 0 ) ≥ 0 , E h ( 0 ) ≥ 0 , I h ( 0 ) ≥ 0 , R h ( 0 ) ≥ 0 (15)

S n ( 0 ) + E n ( 0 ) + I n ( 0 ) + R n ( 0 ) ≤ N n ( 0 ) and S h ( 0 ) + V h ( 0 ) + E h ( 0 ) + I h ( 0 ) + R h ( 0 ) ≤ N h ( 0 ) (16)

The model analysis begins by showing that all feasible solutions of the model are uniformly bounded in a proper subset of Ω . Thus the feasible region

Ω = { ( S n , E n , I n , R n ) ∈ ℝ + 4 : N n ≤ Λ n μ n ( S h , V h , E h , I h , R h ) ∈ ℝ + 5 : N h ≤ Λ h μ h (17)

is considered. Therefore, after differentiation of (12) and (13), and proper substitutions, we have:

d N n ( t ) d t = Λ n − μ n N n − d n I n ≤ Λ n − μ n N n (18)

and;

d N h ( t ) d t = Λ h − μ h N h − d h I h ≤ Λ h − μ h N h (19)

Applying [

{ N n ( t ) ≤ N n ( 0 ) e − μ n t + Λ n μ n ( 1 − e − μ n t ) N h ( t ) ≤ N h ( 0 ) e − μ h t + Λ h μ h ( 1 − e − μ h t ) (20)

where N n ( 0 ) and N h ( 0 ) are the initial populations of the non-human

primates and the humans respectively. Therefore, 0 ≤ N n ≤ Λ n μ n and 0 ≤ N h ≤ Λ h μ h as t → ∞ . This implies that, Λ n μ n and Λ h μ h are upper bounds for N n ( t ) and N h ( t ) respectively, as long as N n ( 0 ) ≤ Λ n μ n and N h ( 0 ) ≤ Λ h μ h . Hence, the

feasible solution of the model equations in (3)-(13) enters the region Ω which is a positively invariant set. Thus, the system is mathematically and epidemiologically well-posed. Therefore, for an initial starting point x ∈ Ω , the trajectory lies in Ω , and so it is sufficient to restrict our analysis on Ω . Clearly, under the dynamics described by the model equations, the closed set Ω is hence a positively invariant set.

Using standard approaches, the model disease-free ε 0 and endemic ε ∗ (which existed only when R 0 > 1 ) equilibrium points are established as follows:

ε 0 = ( S h , V h , E h , I h , R h , S n , E n , I n , R n ) = ( Λ h α h + μ h , Λ h μ h α h α h + μ h , 0 , 0 , 0 , Λ n μ n , 0 , 0 , 0 ) (21)

ε ∗ = ( S h ∗ , V h ∗ , E h ∗ , I h ∗ , R h ∗ , S n ∗ , E n ∗ , I n ∗ , R n ∗ ) (22)

where:

S h ∗ = Λ h ( μ h + α h + λ h ∗ ) , V h ∗ = Λ h μ h α h ( μ h + α h + λ h ∗ ) , E h ∗ = λ h ∗ Λ h ( μ h + ν h ) ( μ h + α h + λ h ∗ ) ,

I h ∗ = ν h λ h ∗ Λ h ( μ h + d h + ρ h ) ( μ h + ν h ) ( μ h + α h + λ h ∗ ) ,

R h ∗ = Λ h μ h ρ h ν h λ h ∗ ( μ h + d h + ρ h ) ( μ h + ν h ) ( μ h + α h + λ h ∗ ) ,

S n ∗ = Λ n ( μ n + λ n ∗ ) , E n ∗ = λ n ∗ Λ n ( μ n + ν n ) ( μ n + λ n ∗ ) , I n ∗ = ν n λ n ∗ Λ n ( μ n + ν n ) ( μ n + λ n ∗ ) ( μ n + d n + ρ n ) ,

R n ∗ = Λ n μ n ρ n ν n λ n ∗ ( μ n + ν n ) ( μ n + λ n ∗ ) ( μ n + d n + ρ n ) with

λ n ∗ = β n 1 I n ∗ N n ∗ , λ h ∗ = β n 2 I n ∗ N n ∗ + β h I h ∗ N h ∗ ,

N n ∗ = Λ n − d n I n ∗ μ n and N h ∗ = Λ h − d h I h ∗ μ h .

The basic reproduction number of the model was computed using the next-generation matrix as defined in [

F = [ 0 β n 1 0 0 0 0 0 0 0 Λ h β n 2 μ n Λ n ( α h + μ h ) 0 β h μ h ( α h + μ h ) 0 0 0 0 ] (23)

and

V = [ ( μ n + ν n ) − β n 1 0 0 − ν n ( μ n + d n + ρ n ) 0 0 0 0 ( μ h + ν h ) 0 0 0 − ν h ( μ h + d h + ρ h ) ] (24)

Therefore;

F V − 1 = [ ν n β n 1 y n β n 1 ( μ n + d n + ρ n ) 0 0 0 0 0 0 Λ h β n 2 ν n μ n Λ n y n ( α h + μ h ) Λ h β n 2 μ n Λ n ( μ n + d n + ρ n ) ( α h + μ h ) β h ν h μ h y h ( α h + μ h ) β h μ h ( μ h + d h + ρ h ) ( α h + μ h ) 0 0 0 0 ] (25)

where y n = ( μ n + d n + ρ n ) ( μ n + ν n ) , y h = ( μ h + d h + ρ h ) ( μ h + ν h ) .

Hence, the basic reproduction numbers of the model are given by:

R 0 = { R 0 , n , R 0 , h }

where R 0 , n and R 0 , h are the monkeypox induced reproduction numbers for non-human primates and humans respectively and are given as:

R 0 , n = ν n β n 1 ( μ n + d n + ρ n ) ( μ n + ν n ) (26)

R 0 , h = ν h β h μ h ( μ h + d h + ρ h ) ( μ h + ν h ) ( α h + μ h ) (27)

Theorem 1: The disease-free equilibrium is locally asymptotically stable if R 0 < 1 , and unstable if R 0 > 1 with R 0 = max { R 0 , n , R 0 , h } .

The local stability will be established using linearization method. Therefore, the Jacobian matrix J of the model equations is given as:

J = [ − ( μ n + x ) 0 − m 0 0 0 0 0 0 x − ( μ n + ν n ) m 0 0 0 0 0 0 0 ν n − j n 0 0 0 0 0 0 0 0 ρ n − μ n 0 0 0 0 0 0 0 − n 0 − ( μ h + α h + w ) 0 0 − z 0 0 0 0 0 α h − μ h 0 0 0 0 0 n 0 w 0 − ( μ h + ν h ) 0 0 0 0 0 0 0 0 ν h − j h 0 0 0 0 0 0 0 0 ρ h − μ h ] (28)

where x = β n 1 I n ∗ N n ∗ , m = β n 1 S n ∗ N n ∗ , n = β n 2 S h ∗ N n ∗ , w = β n 2 I n ∗ N n ∗ + β h I h ∗ N h ∗ , z = β h S h ∗ N h ∗ ,

j h = ( μ h + d h + ρ h ) and j n = ( μ n + d n + ρ n ) .

Next, we used elementary row-operations as used by [

ψ 1 = − ( μ n + x ) (29)

ψ 2 = − ( μ n + x ) ( μ n + ν n ) (30)

ψ 3 = − r n (31)

ψ 4 = − r n μ n (32)

ψ 5 = − r n ( μ h + α h + w ) (33)

ψ 6 = − μ h r n ( μ h + α h + w ) (34)

ψ 7 = − r n 2 ( μ h + α h + w ) ( μ h + ν h ) (35)

ψ 8 = − j h r n 2 ( μ h + α h + w ) ( μ h + ν h ) (36)

ψ 9 = − μ h j h r n 2 ( μ h + α h + w ) ( μ h + ν h ) (37)

where r n = [ j n ( μ n + x ) ( μ n + ν n ) − μ n ν n m ] .

Therefore, since the real part of all the eigenvalues ψ i , for i = 1 , 2 , ⋯ , 9 are negative, the endemic equilibrium is locally asymptotically stable from the following theorem:

Theorem 2: The endemic equilibrium is locally asymptotically stable if R 0 < 1 , and unstable if R 0 > 1 with R 0 = max { R 0 , n , R 0 , h } .

Theorem 3: The disease-free equilibrium is globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1

Proof: By the comparison theorem, the rate of change of the variables representing the infectious classes in the model can be compared in the following inequality:

[ E ′ n I ′ n E ′ h I ′ h ] ≤ ( F − V ) [ E n I n E h I h ] − M 1 θ 1 [ E n I n E h I h ] 1 − M 2 θ 2 [ E n I n E h I h ] − θ 3 [ E n I n E h I h ] (38)

where F and V are defined in (23) and (24) respectively, M 1 = 1 − S h 0 N h 0 , M 2 = 1 − V h 0 N h 0 , θ 1 , θ 2 and θ 3 are nonnegative matrices. And since S h 0 ≤ N h 0 ,

then V h 0 ≤ N h 0 . Therefore, from (38) we get:

[ E ′ n I ′ n E ′ h I ′ h ] ≤ ( F − V ) [ E n I n E h I h ] (39)

Therefore the matrix ( F − V ) is obtained as:

( F − V ) = [ − ( μ n + ν n ) β n 1 0 0 ν n − ( μ n + d n + ρ n ) 0 0 0 Λ h β n 2 μ n Λ n ( α h + μ h ) − ( μ h + ν h ) β h μ h ( α h + μ h ) 0 0 ν h − ( μ h + d h + ρ h ) ] (40)

From the matrix in (40), let ψ be an eigenvalue. Then, the characteristic equation | ( F − V ) − ψ I | = 0 gives the following eigenvalues:

ψ 10 = − [ ( μ n + d n + ρ n ) ( μ n + ν n ) − β n 1 ν n ] (41)

ψ 11 = − ( μ n + d n + ρ n ) (42)

ψ 12 = − [ ( μ h + d h + ρ h ) ( μ h + ν h ) − β h ν n μ h ( α h + μ h ) ] (43)

ψ 13 = − ( μ h + d h + ρ h ) (44)

Therefore, all the four eigenvalues of the matrix in (40) have negative real part, showing that the matrix (40) is stable if R 0 < 1 . Consequently, using the model equations in (1)-(13), ( E n , I n , E h , I h ) ⇒ ( 0 , 0 , 0 , 0 ) as t ⇒ ∞ . Thus by the comparison theorem as used in [

S h 0 = Λ h ( α h + μ h ) , V h 0 = Λ h μ h α h ( α h + μ h ) , S n 0 = Λ n μ n and ( R n , R h ) ⇒ ( 0 , 0 ) as

t ⇒ ∞ for R 0 < 1 . Hence, the disease-free equilibrium is globally asymptotically stable for R 0 < 1 .

In this section, numerical simulations for the model were carried out using the parameter values in

Sensitivity indices allow us to measure the relative change in a variable when a parameter changes. The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter [

Definition: The normalized forward sensitivity index of a variable τ that depends, differentially, on a parameter p , is defined as:

ϒ p τ = ∂ τ ∂ p × p τ (45)

We computed the sensitivity index of each parameter involved in R 0 using the parameter values in

The indices with positive signs show that the value of R 0 increases when the corresponding parameters are increased and indices with negative signs indicates that, the value of R 0 decreases with increase in the corresponding parameters. This analysis is done to ascertain which parameters dominate the results of our analysis. Therefore, some parameters are deliberately excluded out of the sensitivity analysis due to their relative low importance in the actual disease transmission process. For example, the natural births, deaths in both humans and the non-human primates. The results of the analysis are presented in

Therefore, it is clear from the

The results of the analysis for the model were presented in Section 3 of this paper. The results of the numerical simulations for the model and sensitivity analysis of the model parameters using parameter values in

R 0 , n = 3.375 × 10 − 3 (46)

R 0 , h = 9.1304 × 10 − 6 (47)

Parameter Symbol | Sensitivity Index |
---|---|

d n | −0.133 |

d h | −0.089 |

ρ n | −0.15 |

ρ h | −0.105 |

ν n | +0.25 |

ν h | +0.17 |

α h | −0.83 |

β n 1 | +1.00 |

β n 2 | +0.168 |

β h | +1.00 |

Clearly, from (45) and (46), R 0 , n < 1 and R 0 , h < 1 , suggesting that the disease-free equilibrium is both locally and globally asymptotically stable while the endemic equilibrium of the model is locally asymptotically stable from our analysis.

The sensitivity indices of the model parameters in

In this paper, we studied the dynamics of the transmission of the monkeypox virus infection under the combined vaccine and treatment interventions using the work of [

The results of the numerical simulations carried out for the model using parameter values in

The treatment intervention as seen in

In this paper, we developed a mathematical model for the dynamics of transmission of the monkeypox virus infection with combined interventions of vaccination and treatment. We carried analysis on the developed model. The disease-free equilibrium was found to be both locally and globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 . Using parameter values obtained from existing literatures, we carried out numerical simulations and sensitivity analysis for the model and the parameters respectively. The simulations results revealed that, the disease will be eradicated from both humans and the non-human primates with the proposed interventions of the model in due time. Sensitivity analysis revealed that, the interventions offer an optimal control on the monkeypox virus infection in the human population with increase in the control parameter rates of vaccination and treatment.

Usman, S. and Adamu, I.I. (2017) Modeling the Transmission Dynamics of the Monkeypox Virus Infection with Treatment and Vaccination Interventions. Journal of Applied Mathematics and Physics, 5, 2335-2353. https://doi.org/10.4236/jamp.2017.512191