The total human host population denoted *N*(*t*), at time *t* > 0 is classified into five compartments of susceptible denoted *S*(*t*), which are the individuals who are at risk of acquiring the disease. Also, people who have been latently infected but are not yet infectious are denoted by *E*(*t*), symptomatic individuals who did not attend medical awareness program are denoted by *I*_{u}(*t*), symptomatic individuals who did attend medical awareness program are denoted by *I*_{v}(*t*), and the recovered individuals are denoted by R(t), so that

$$ N(t)=S(t)+E(t)+{I}_u(t)+{I}_v(t)+R(t). $$

(1)

The susceptible population is increased by the recruitment of individuals at the rate *A*, following an effective contact with infected individuals in the *I*_{u} and *I*_{v} compartment, the force of infection is denoted *λ*(*t*), and described by the quantity

$$ \lambda (t)=\upbeta \left(\frac{{\mathrm{I}}_{\mathrm{u}}}{1+{\upphi}_1{\mathrm{I}}_{\mathrm{u}}} + \frac{\uptheta {\mathrm{I}}_{\mathrm{v}}}{1+{\phi}_2{\mathrm{I}}_{\mathrm{v}}\ }\right). $$

(2)

In (2), *λ*(*t*) is the force of infection that takes into account the high saturation of infected individuals in the human host community, where *β* is the transmission rate of infection and *θ* is the modification parameter that takes into account the relative infectiousness of medical awareness non-compliant individuals to transmit infection at a higher rate than medical awareness compliant individuals. We adopt a nonlinear saturated incidence rate in the two groups of individuals to describe the behavioral change and crowding effect of infected humans where *ϕ*_{1} and *ϕ*_{2} measures the inhibitory effect. But if *ϕ*_{1} and *ϕ*_{2} are zeros, then the incidence function follows a bilinear incidence which is commonly adopted in many models. The population of the susceptible individuals is further decreased by natural death rate *μ*. Thus, the rate of change of the susceptible population is given by

$$ \frac{dS}{\kern0.5em dt}=A-\left(\mu +\lambda (t)\right)S. $$

(3)

The population of exposed individuals is increased by the force of infection *λ*(*t*). The compartment is further on decreased by the development of clinical symptoms, natural death and the disease-induced mortality at the rate ϵ, μ and *d* respectively, so that

$$ \frac{dE}{dt}=\lambda (t)S-\left(\in +\upmu +d\right)E. $$

(4)

The population of the symptomatic individuals who did not attend medical awareness program is increased at the rate *ϵ.* It is decreased by natural recovery rate *γ*_{o}, the rate of emergence of new symptoms *σ*, natural death μ_{1}, and disease-induced mortality rate α_{1}. This is given by

$$ \frac{dI_u}{dt}=\in E-\left({\upgamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right){I}_u. $$

(5)

The population of infected individuals who attended medical awareness program is increased progressively at the rate *σ*. The compartment is decreased by recovery rate γ_{1}, natural death rate μ_{2}, and disease-induced mortality *α*_{2}. It is assumed that the disease-induced mortality rate of individuals who attended medical awareness program is low in comparison with infected individuals who did not attend medical awareness program, such that *α*_{2} < α_{1}. Hence, the rate of change of this population is given by

$$ \frac{d{I}_v}{dt}=\sigma {I}_u-\left({\upmu}_2+{\upgamma}_1+{\alpha}_2\right){I}_v. $$

(6)

Finally, the population of recovered individuals is generated by the recovery of individuals who attend and who did not attend medical awareness program at the rate *γ*_{o} and γ_{1}, while it is decreased by natural death rate *μ*, so that

$$ \frac{dR}{dt}={\upgamma}_o{I}_u+{\upgamma}_1{I}_v-\upmu \mathrm{R}. $$

(7)

Thus, the model for the transmission dynamics of a generalized infectious disease with non-linear incidence of two groups of infected individuals who attend and did not attend medical awareness follows a first order system of ordinary differential equations given by

$$ {\displaystyle \begin{array}{l}\frac{dS}{dt}=A-\left(\mu +\lambda \left(\mathrm{t}\right)\right)S,\\ {}\frac{dE}{dt}=\lambda (t)S-\left(\upepsilon +\mu +d\right)E,\\ {}\begin{array}{l}\ \frac{d{I}_u}{dt}=\epsilon E-\left({\gamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right){I}_u,\kern0.5em \\ {}\frac{d{I}_v}{dt}=\sigma {I}_u-\left({\upmu}_2+{\upgamma}_1+{\alpha}_2\right){I}_v,\\ {}\frac{dR}{dt}={\gamma}_o{I}_u+{\gamma}_1{I}_v-\mu R.\end{array}\end{array}} $$

(8)

Subject to the initial conditions *S*(0) = *S*_{o}, *E*(0) = *E*_{o}, *I*_{u}(0) = *I*_{uo}, *I*_{v}(0) = *I*_{vo}, *R*(0) = *R*_{o}.

### 2.1 Positivity of the model

It is assumed that the initial conditions of the model are non-negative and it is necessary to show that the solution of the model is positive.

**Theorem 1:** Let Ω = {(*S*, *E*, *I*_{u}, *I*_{v}, *R*) ∈ *R*_{+}^{5} : *S*_{o} > 0, *E*_{o} > 0, *I*_{uo} > 0, *I*_{vo} > 0, *R*_{o} > 0}. Then the solutions of *S*, *E*, *I*_{u}, *I*_{v}, *R* are positive for t ≥ 0.

**Proof:** From the model system of differential Eq. (8), considering the first state equation given by

$$ {\displaystyle \begin{array}{l}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{A}-\left(\upmu +\uplambda \right)\mathrm{S},\\ {}\mathrm{so}\ \mathrm{that}\\ {}\begin{array}{l}\frac{\mathrm{dS}\left(\mathrm{t}\right)}{\mathrm{dt}}\ge \left(\upmu +\uplambda \right)\mathrm{S},\\ {}\frac{\mathrm{dS}\left(\mathrm{t}\right)}{\mathrm{S}}\ge \left(\lambda +\mu \right) dt,\\ {}\begin{array}{l}\mathrm{and}\\ {}\int \frac{\mathrm{dS}\left(\mathrm{t}\right)}{\mathrm{S}}\ge \kern0.5em \int \left(\uplambda +\upmu \right)\mathrm{dt}.\end{array}\end{array}\end{array}} $$

(9)

Solving (9) using separation of variable and applying the initial condition *S*(0) = *S*_{o}, yields

$$ S(t)\ge {S}_o{e}^{-\left(\uplambda +\upmu \right)t}\ge 0. $$

(10)

Also, from the second state equation of (8),

$$ \frac{dE}{dt}=\lambda S-\left(\epsilon +\mu +d\right)E. $$

(11)

Simplifying (11) further yields

$$ {\displaystyle \begin{array}{l}\frac{dE}{dt}\ge \kern0.5em \left(\epsilon +\mu +d\right)E\\ {}\mathrm{and}\\ {}\int \frac{dE}{E}\ge \kern0.5em \int \left(\ \epsilon +\mu +d\right) dt.\end{array}} $$

(12)

On solving (12) using separation of variable and applying initial condition *E*(0) = *E*_{o}, yields

$$ E(t)\ge {E}_o{e}^{-\left(\epsilon +\mu +d\right)t}\ge 0. $$

(13)

From the third state equation in (8),

$$ \frac{d{I}_u}{dt}=\epsilon E(t)-\left({\gamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right){I}_u, $$

(14)

Simplifying (14) further become,

$$ {\displaystyle \begin{array}{l}\frac{dI_u}{dt}\ge \left({\upgamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right){I}_u\\ {}\mathrm{and}\\ {}\int \frac{dI_u}{I_u}\ge \int \left({\upgamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right)d(t).\end{array}} $$

(15)

Solving (15) using separation of variable and applying initial condition *I*_{u}(0) = *I*_{uo}, yields

$$ {I}_u(t)\ge {I}_{uo}{e}^{-\left({\gamma}_o+{\upmu}_1+{\upalpha}_1+\sigma \right)t}\ge 0. $$

(16)

In addition, taking the fourth state equation of (8),

$$ \frac{d{I}_v}{dt}=\sigma {I}_u(t)-\left({\upmu}_2+{\upgamma}_1+{\alpha}_2\right){I}_v $$

(17)

where

$$ \int \frac{dI_v}{I_v}\ge -\int \left({\upmu}_2+{\upgamma}_1+{\upalpha}_2\right) dt. $$

(18)

Solving (18) using separation of variable and applying initial condition *I*_{v}(0) = *I*_{vo}, yields

$$ {I}_v(t)\ge {I}_{vo}{e}^{-\left({\upmu}_2+{\upgamma}_1+{\upalpha}_2\right)t}\ge 0. $$

(19)

Finally, taking the fifth state equation of (8),

$$ \frac{dR}{dt}={\gamma}_o{I}_u(t)+{\upgamma}_1{I}_v(t)-\upmu \mathrm{R}. $$

(20)

The simplification of (20) yields

$$ {\displaystyle \begin{array}{l}\frac{dR}{dt}\ge -\upmu \mathrm{R},\\ {}\mathrm{and}\\ {}\begin{array}{l}\frac{dR}{R}\ge -\upmu \mathrm{dt},\\ {}\int \frac{dR}{R}\ge -\int \upmu \mathrm{dt}.\end{array}\end{array}} $$

(21)

Solving (21) using separation of variable and applying initial condition *R*(0) = *R*_{o}, yields

$$ R(t)\ge {R}_o{e}^{-\upmu t}\ge 0. $$

(22)

From (10), (13), (16), (19), and (22), it is clear that at time *t* > 0, the model solutions are positive.

This completes the proof of the theorem.

### 2.2 Invariant region

In this section, the model system is analyzed in an invariant region and shown to be bounded. The addition of the whole model system Eq. (8) yields

$$ \mathrm{N}=\mathrm{S}+\mathrm{E}+{I}_u+{I}_v+R, $$

(23)

such that

$$ \frac{dN}{dt}=\frac{dS}{dt}+\frac{dE}{dt}+\frac{d{I}_u}{dt}+\frac{d{I}_v}{dt}+\frac{dR}{dt} $$

(24)

and

$$ \frac{dN}{dt}=A-\mu N- dE-{\upalpha}_1{I}_u-{\upalpha}_2{I}_v. $$

(25)

In the absence of natural and mortality due to disease, i.e., (*d* = 0, α_{1} = 0, *α*_{2} = 0), (25) becomes

$$ \frac{dN}{dt}=A-\mu N. $$

(26)

Integrating both side of (26) yields

$$ \int \frac{dN}{A-\mu N}\le \int \mathrm{dt} $$

(27)

and

$$ \frac{1}{\upmu}\ \ln\ \left(A-\mu N\right)\le t. $$

(28)

Simplification of (28) become

$$ \mathrm{A}-\mu N\ge {e}^{-\mu t}. $$

(29)

Applying the initial condition, *N*(0) = *N*_{o}, (29) yields *A* = *A* − *μN*_{o}. Substituting *A* = *A* − *μN*_{o} into (29) yields

$$ A-\mu {N}_o\ge \left(A-\mu {N}_o\right){e}^{-\mu t}. $$

(30)

Further simplification and re-arrangement of (30) yields

$$ N\le \left(\frac{A}{\upmu}-\frac{A-\upmu {\mathrm{N}}_{\mathrm{o}}}{\upmu}\right)\ {e}^{-\mu t}. $$

(31)

As *t* → ∞ in (31), the population size \( N\to \frac{A}{\mu } \) implies that \( 0\le N\le \frac{A}{\mu } \). Thus, the feasible solution set of the system equations of the model start and end in the region

$$ \Omega =\left(\left\{S,E,{I}_u,{I}_v,R\right\}\in {R^{+}}^5:N\le \frac{A}{\mu}\right). $$

(32)

Therefore, the basic model (8) is well posed mathematically and epidemiologically reasonable. Hence, it is sufficient to study the dynamics of the model system (8) in Ω.

### 2.3 Equilibria

To find the disease-free equilibrium solutions, the right-hand side of the model system (8) is equated to zero, evaluating it at when there is no disease in the system, i.e., *E* = *I*_{u} = *I*_{v} = 0. Therefore, the disease-free equilibrium solutions are given by

$$ {E}_o=\left(S,\kern0.5em E,{I}_u,{I}_v,R\right)=\left(\frac{A}{\mu },0,0,0,0\right). $$

(33)

The endemic equilibrium is denoted *E*^{∗∗} = (*S*^{∗∗}, *E*^{∗∗}, *I*_{u}^{∗∗}, *I*_{v}^{∗∗}, *R*^{∗∗}) and it occurs when a disease persist in the human host population. Therefore

$$ {E}^{\ast \ast }=\left({S}^{\ast \ast },{E}^{\ast \ast },{I_u}^{\ast \ast },{I_v}^{\ast \ast },{R}^{\ast \ast}\right)=\left({S}^{\ast }=\frac{A}{{\mathrm{m}}_1}, \kern0.5em {E}^{\ast }=\frac{\lambda A}{{\mathrm{m}}_2{\mathrm{m}}_3},\kern0.75em {I}_u^{\ast }=\frac{\lambda \epsilon A}{{\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3},{I}_v^{\ast }=\frac{\lambda \sigma \epsilon A}{{\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4},{R}^{\ast }=\frac{1}{\mu\ }\left[\frac{\lambda \epsilon {\gamma}_oA}{{\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3}\kern0.5em +\frac{\lambda \sigma \epsilon {\upgamma}_1A}{{\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4\ }\right]\right). $$

(34)

Where *m*_{1} = (*λ* + *μ*), *m*_{2} = (*ϵ* + *μ* + *d*), *m*_{3} = (*γ*_{o} + *μ*_{1} + *α*_{1} + *σ*), *m*_{4} = (*μ*_{2} + *γ*_{1} + *α*_{2}).

### 2.4 Basic reproduction number (*R*
_{r})

We want to show how the threshold that governs the spread of a disease, called the basic reproduction number is obtained.

**Theorem 2.**

Define *X*_{s} = {*X* = 0| *X*_{i}, *i* = 1, 2, 3, …}, in order to obtain *R*_{r}, new infections are distinguished from other changes in the populations. Such that, *F*_{i}(*x*) is the rate of new manifestations of clinical symptoms in compartment *i*. Also, let \( {V}_i^{+} \) be the rate at which individuals move out of compartment *i*. Then *x*_{i} = *F*_{i}(*x*) − *V*_{i}(*x*), *i* = 1, 2, 3, …. , and \( {V}_i(x)={V}_i^{-}-{V}_i^{+} \). *F* is a non negative matrix and *V* is a non-singular matrix.

**Proof:** We applied the next generation matrix method to the model equations starting with newly infective classes given by

$$ \left.\begin{array}{l}\frac{dE}{dt}=\lambda (t)S-{\mathrm{m}}_2E,\\ {}\frac{dI_u}{dt}=\varepsilon E-{\mathrm{m}}_3{I}_u,\\ {}\frac{dI_v}{dt}=\sigma {I}_{\mathrm{u}}-{\mathrm{m}}_4{I}_v.\end{array}\right\} $$

(35)

The rate of new clinical symptoms is given by

$$ \mathrm{F}=\left(\genfrac{}{}{0pt}{}{\upbeta \mathrm{S}\left(\frac{{\mathrm{I}}_{\mathrm{u}}}{1+{\upphi}_1{\mathrm{I}}_{\mathrm{u}}} + \frac{\uptheta {\mathrm{I}}_{\mathrm{v}}}{1+{\phi}_2{\mathrm{I}}_{\mathrm{v}}\ }\right)}{\begin{array}{c}0\\ {}0\end{array}}\right) $$

(36)

And the rate of transfer terms of individuals is given by

$$ \mathrm{V}=\left(\genfrac{}{}{0pt}{}{\begin{array}{c}{\mathrm{m}}_2E\\ {}\epsilon E-{\mathrm{m}}_3{I}_u\end{array}}{\sigma {I}_{\mathrm{u}}-{\mathrm{m}}_4{I}_v}\right). $$

(37)

The Jacobian matrices of *F* and *V* evaluated at disease-free equilibrium solution (33) are given by

$$ \mathrm{F}=\left(\begin{array}{c}0\kern3em 0\kern4em \frac{\upbeta \mathrm{A}}{\mu\ }\\ {}0\kern3.5em 0\kern4.5em 0\\ {}0\kern3.25em 0\kern4.5em 0\ \end{array}\right) $$

(38)

and

$$ V=\left(\begin{array}{c}{\mathrm{m}}_2\kern4.25em 0\kern5.25em 0\\ {}\upepsilon \kern4.75em {\mathrm{m}}_3\kern5em 0\\ {}0\kern5em \sigma \kern5em {\mathrm{m}}_4\end{array}\right). $$

(39)

The inverse of *V* is given by

$$ {V}^{-1}=\left(\begin{array}{ccc}\frac{1}{{\mathrm{m}}_2}& 0& 0\\ {}\frac{\upepsilon}{{\mathrm{m}}_2{\mathrm{m}}_3}&\ \frac{1}{{\mathrm{m}}_3}& 0\\ {}\frac{\upepsilon \sigma }{{\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4}& \frac{\sigma }{{\mathrm{m}}_4}&\ \frac{1}{{\mathrm{m}}_4}\end{array}\right) $$

(40)

and

$$ F{V}^{-1}=\left(\begin{array}{ccc}\frac{\lambda A\upepsilon \sigma }{\mu {\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4}& \frac{\lambda A\sigma}{\mu {\mathrm{m}}_3{\mathrm{m}}_4}& \frac{\lambda A}{\mu {\mathrm{m}}_4}\\ {}0& 0& 0\\ {}0& 0& 0\end{array}\right). $$

(41)

The eigenvalues of *FV*^{−1} in (41) are given by

$$ {\displaystyle \begin{array}{l}{\uplambda}_1={\uplambda}_2=0\\ {}\mathrm{and}\\ {}{\uplambda}_3=\frac{\lambda A\in \sigma }{\mu {\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4}.\end{array}} $$

(42)

The dominant eigenvalue in (42) is *λ*_{3}. Therefore, the basic reproduction number *R*_{r} is given by

$$ {R}_r=\frac{\lambda A\upepsilon \sigma }{\mu {\mathrm{m}}_2{\mathrm{m}}_3{\mathrm{m}}_4}. $$

(43)

The threshold in (43) measures the rate at which new cases of infection arises, when a typical infected individual is introduced into a susceptible population of humans during their course of infection.