Development and exploration of a mathematical model for transmission of monkey-pox disease in humans

1. Introduction

Monkey-pox is known as pathogens, affecting livestock animals and humans and belongs to the orthopox virus, which included small pox cause infection in humans and cow pox viruses [1]. The monkey pox causes lymph nodes to swell. The symptoms are fever, headache, muscle aches, backache, swollen lymph nodes, a feeling of discomfort and exhaustion. The virus is similar to small pox in human. The virus can spread both from animal to human and from human to human. The increased transmission risk associated with factors involving introduction of virus to the oral mucosa [2].

The incubation period range from 7 to 14 days. After 1 to 3 days of the appearance of fever, the patient develops raised bumps and the illness lasted for 2 to 4 weeks [5]. Thus, the reported case fatality ratio is range from 1 % to 10 % [3]. Currently, there are no vaccines for monkey pox, however, evidence showed that small pox vaccine reduced the risk of monkey pox among previously vaccinated persons in Africa [4, 5]. Centre for disease control and prevention also recommended that persons investigating monkey pox outbreaks and caring for infected individuals should vaccinated with small pox vaccine, to protect against the monkey pox. Individuals who have had close contact with persons confirmed to have monkey pox should be vaccinated up to 14 days after exposure were recommended [4].

Mathematical models have played a central role to capture the dynamics of different Disease transmission [6]. The aim of this paper is therefore to the development and exploration of a mathematical model for transmission of monkey-pox disease in humans. Therefore, there is non-numerous work on mathematical modelling of monkey pox transmission. This Paper extends the work of [7] by incorporating Vaccination class for migrants, Exposed classes for both human and non-human population.

2. Model formulation

We formulate a model for the spread of Monkey-pox in human and primates (Monkey) population with the total population size at time t given by $N_h\left(t\right)$ and $N_p\left(t\right)$. The populations are further compartmentalized into epidemiological classes as shown in the model flow diagram in Fig. 1. The total human population is divided into five subgroups that is the Susceptible, $S_h\left(t\right)$, the Vaccinated, $V_h\left(t\right)$, the Exposed, $E_h\left(t\right)$, the Infected, $I_h\left(t\right)$, and the Recovery with permanent immunity, $R_h\left(t\right)$. The total primates (Monkey) population model divides into the Susceptible, $S_p\left(t\right)$, the Infected primates, $I_p\left(t\right)$, the Exposed, $E_p\left(t\right)$ and the recovery with permanent immunity, $R_p\left(t\right)$. As indicated in the compartmental diagram in Fig. 1, people enter Susceptible class through birth and immigration, (${\mathrm{\Lambda }}_h$), where a proportion of vaccinated human immigrants ($f$) enter to the vaccinated class and proportion of unvaccinated immigrants ($1-f$) enter to the susceptible class. We do not consider the immigration of infection person, because we assume that people who are coming from monkey-pox endemic zones have to be vaccinated. The susceptible individuals vaccinated at the rate $\gamma$ and loss the vaccination at the rate $\omega$. Susceptible human get contact from primate at rate ${\sigma }_{p_1}$, $S_h$ are exposed to monkey-pox infection at the rate ${\sigma }_h$ and infected at the rate ${\beta }_h$, with natural death ${\mu }_h$ and die due to the infection at the rate ${\delta }_h$ and recovery with permanent immunity at a rate ${\rho }_h$. The Susceptible primates, $S_p$ class is generated from the daily recruitment of individuals through birth and immigration at the ${\mathrm{\Lambda }}_p$ and natural death rate ${\mu }_p$. They become exposed to monkey-pox virus at the rate ${\sigma }_{p_2}$ and leave the class to infected class at the rate ${\beta }_p$. Individuals primate infected die due to the infection at the rate ${\delta }_p$ and recovery with the permanent immunity at the rate ${\rho }_p$.

Fig. 1Schematic diagram of transmission of monkey-pox disease

Base on the above assumptions and schematic diagram of Fig. 1, the model for transmission dynamics of Monkey-pox infection in human and primates (Monkey) is described by a system of Ordinary Differential Equations (ODEs) given below:

where:

The susceptible human host get infected from both the infected primate ($I_p$) and infected human ($I_h$) which is called force of infection from primates to human as well as human to human [7]. The term ${\sigma }_{p1}$ is the product of the effective contact rate and probability of the human being get infected from the infected primates ($I_p$) and ${\sigma }_h$ is the product of the effective contact and probability of the human get infected from the infected persons ($I_h$). Similarly, the $S_p$ get infected from infected primate, where ${\sigma }_{p2}$ is the product of the effective contact rate and probability of the primates (monkey) get infected per contact with an infected primate ($I_p$) [7]. The modification parameter ${\epsilon }_h$account for the assumption that exposed human transmits at a rate lower than symptomatic humans. The modification parameter ${\epsilon }_p$ account for the assumption that exposed non-human transmits at a rate lower than symptomatic non-humans. It is assumed here that mortality of monkeys due to being hunted by humans is negligible and can be safely ignored.

3. Analysis of the model equations

Theorem 1: (Invariant Region) the following biological feasible region of the model Eqs. (1)-(9) $\mathrm{\Omega }={\{S_h,E_h,I_h,V_h,R}_h$, ${S_p,E_p,I_p,R}_p)\in {\mathfrak{R}}_{+}^9$: ${{\{S}_h\left(t\right)+E_h\left(t\right)+I_h\left(t\right)+V_h\left(t\right)+R}_h\left(t\right)\le {\mathrm{\Lambda }}_h/{\mu }_h;Spt+Ept+Ipt+Rp\left(t\right)\le {\mathrm{\Lambda }}_p/{\mu }_p\}\mathrm{}\mathrm{}$is positively invariant and attracting.

Proof; the addition of all the equations model in Eqs. (1)-(9) give:

So that:

It follows from [8], the Gronwall inequality, that:

In particular, $N_h\left(t\right)\le {\mathrm{\Lambda }}_h/{\mu }_h$ if $N_h\left(0\right)\le {\mathrm{\Lambda }}_h/{\mu }_h$ and $N_p\left(t\right)\le {\mathrm{\Lambda }}_p/{\mu }_p$ if $N_p\left(0\right)\le {\mathrm{\Lambda }}_p/{\mu }_p$. Thus, $\Omega$ is positively invariant. Hence, it is sufficient to consider the model dynamics Eqs. (1)-(9) in $\Omega$, in this region, the model equations can be considered as been epidemiologically and mathematically well posed.

Theorem 2: (Positivity of the Solution for the Model). Let $t_0>0$ and the initial conditions satisfied $S_h\left(0\right)>0$, $E_h\left(0\right)>0$, $I_h\left(0\right)>0$, $V_h\left(0\right)>0$, $R_h\left(0\right)>0$, $S_p\left(0\right)>0$, $E_p\left(0\right)>$ 0, $I_p\left(0\right)>$0, $R_p\left(0\right)>0$, then the solutions $S_h\left(t\right)$, $E_h\left(t\right)$, $I_h\left(t\right)$, $V_h\left(t\right)$, $R_h\left(t\right)$, $S_p\left(t\right)$, $E_p\left(t\right)$, $I_p\left(t\right)$, $R_p\left(t\right)$, of the model Eqs. (1)-(9) are positive for all $t\ge 0$.

Proof:

Proving that for all $t\in [0,t_0]$, $S_h\left(t\right)$, $E_h\left(t\right)$, $I_h\left(t\right)$, $V_h\left(t\right)$, $R_h\left(t\right)$, $S_p\left(t\right)$, $E_p\left(t\right)$, $I_p\left(t\right)$, $R_p\left(t\right)$ will be positive in ${\mathfrak{R}}_{+}^9$, Since all the parameters used in the system are positive. Thus, it is clear from Eq. (1) that:

So that:

The similar approach can be used to show that $E_h\left(t\right)>0$, $I_h\left(t\right)>0$, $V_h\left(t\right)>0$, $R_h\left(t\right)>0$, $S_p\left(t\right)>0$, $E_p\left(t\right)>$ 0, $I_p\left(t\right)>$0, $R_p\left(t\right)>0$. Thus, for all $t\in [0,t_0]$, $S_h\left(t\right)$, $E_h\left(t\right)$, $I_h\left(t\right)$, $V_h\left(t\right)$, $R_h\left(t\right)$, $S_p\left(t\right)$, $E_p\left(t\right)$, $I_p\left(t\right)$, $R_p\left(t\right)$ will be positive and remain in ${\mathfrak{R}}_{+}^9$.

4. Existence of the equilibrium

At equilibrium state, we let:

From Eq. (3), we have:

Substitute Eq. (16) into Eq. (2), we have:

Equation (17) gives:

From Eq. (8), we have:

Substitute Eq. (19) into Eqs. (7), we have:

Equation Eq. (20) gives:

Substitute $I_h=0$ in Eq. (9) into Eq. (16) and Eq. (5), we have:

Making $V_h$ subject of expression in Eq. (4) and Eq. (1), and equating them, we have:

Implies:

Since $I_h=0$, implies ${\alpha }_h=0$, then we have:

Reduced to:

If there is no vaccination, then $S_h^0=\frac{{\mathrm{\Pi }}_h{\mu }_h}{{{\mu }_h}^2}=\frac{{\mathrm{\Pi }}_h}{{\mu }_h}$ as $\omega =\gamma =f=0$.

Substitute Eq. (25) into Eq. (4), we have:

Substitute $I_p=0$ in Eq. (21) into Eqs. (19) and (9), we have:

From Eq. (6), we have:

The disease free equilibrium (DFE) state is given as:

5. Effective basic reproduction number (${\mathit{R}}_{\mathit{e}}$)

We applying next generation matrix operator to compute the effective basic reproduction number [9]. The largest Eigenvalue or spectral radius of $F{V}^{-1}$ is the effective basic reproduction number of the model:

where $F_i$ is the rate of appearance of new infection in compartment $i$, $V_i$ is the transfer of infection from one compartment $i$ to another and $E^0$is the Disease-Free Equilibrium.

Using this technique, we have spectral radius ($\rho$) of the next generation matrix, $F{V}^{-1}$ that is, $R_0=\rho \left(F{V}^{-1}\right)$. Both $F$ and $V$ are obtained from the Jacobian matrix of the linearization of system Eqs. (1)-(9) disease free equilibrium. Therefore, the vector $F$ and $V$ representing inflow and outflow from compartments $E_h$, $E_p$, $I_h$ and $I_P$ are given by:

where, $A_1={\beta }_h+{\mu }_h$; $A_2={\rho }_h+{\mu }_h+{\delta }_h$; $A_3={\beta }_p+{\mu }_p$; $A_4={\rho }_p+{\mu }_p+{\delta }_p$ and:

From Eqs. (34), we have:

At DFE, and since $N_h\left(t\right)\le {\mathrm{\Pi }}_h/{\mu }_h$ and $N_p\left(t\right)\le {\mathrm{\Pi }}_p/{\mu }_p$ we have:

From Eq. (36), we have:

Multiplying Eq. (37) and (35) together, we have:

Equation (39) implies:

The characteristics equation of Eq. (40), gives $\left|{FV}^{-1}-\lambda I\right|=0$:

where:

Determinant of Eq. (41) gives:

To solve Eq. (43) with completing the square method, we have:

${\lambda }_1$ is the spectral radius of $\rho \left(F{V}^{-1}\right)$ and the reproduction number is the largest eigenvalue from Eq. (44):

Implies:

${\lambda }_2$ is the spectral radius of $\rho \left(F{V}^{-1}\right)$:

Then, we obtained effective basic reproduction number as Eqs. (48) and (45). There are two host populations and it was shown from the model flow diagram in Fig. 1 that the monkey transmits the infection to human host and human to human. Hence, the effective basic reproduction number can be represented as: