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 Iu(t), symptomatic individuals who did attend medical awareness program are denoted by Iv(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 Iu and Iv 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 ϕ1 and ϕ2 measures the inhibitory effect. But if ϕ1 and ϕ2 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 γo, the rate of emergence of new symptoms σ, natural death μ1, and disease-induced mortality rate α1. 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 γ1, natural death rate μ2, and disease-induced mortality α2. 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 α2 < α1. 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 γo and γ1, 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) = So, E(0) = Eo, Iu(0) = Iuo, Iv(0) = Ivo, R(0) = Ro.
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, Iu, Iv, R) ∈ R+5 : So > 0, Eo > 0, Iuo > 0, Ivo > 0, Ro > 0}. Then the solutions of S, E, Iu, Iv, 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) = So, 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) = Eo, 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 Iu(0) = Iuo, 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 Iv(0) = Ivo, 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) = Ro, 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, α1 = 0, α2 = 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) = No, (29) yields A = A − μNo. Substituting A = A − μNo 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 = Iu = Iv = 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∗∗, Iu∗∗, Iv∗∗, 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 m1 = (λ + μ), m2 = (ϵ + μ + d), m3 = (γo + μ1 + α1 + σ), m4 = (μ2 + γ1 + α2).
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 Xs = {X = 0| Xi, i = 1, 2, 3, …}, in order to obtain Rr, new infections are distinguished from other changes in the populations. Such that, Fi(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 xi = Fi(x) − Vi(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 λ3. Therefore, the basic reproduction number Rr 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.