Final epidemic size and optimal control of socio-economic multi-group influenza model

Abstract

Flu, a common respiratory disease is caused mainly by the influenza virus. The Avian influenza (H5N1) outbreaks, as well as the 2009 H1N1 pandemic, have heightened global concerns about the emergence of a lethal influenza virus capable of causing a catastrophic pandemic. During the early stages of an epidemic a favourable change in the behaviour of people can be of utmost importance. An economic status-based (higher and lower economic class) structured model is formulated to examine the behavioural effect in controlling influenza. Following that, we have introduced controls into the model to analyse the efficacy of antiviral treatment in restraining infections in both economic classes and examined an optimal control problem. We have obtained the reproduction number $R_0$ along with the final epidemic size for both the strata and the relation between reproduction number and epidemic size. Through numerical simulation and global sensitivity analysis, we have shown the importance of the parameters ${\varphi }_i,{\varphi }_s,{\eta }_2,\beta$ and $\theta$ on reproduction number. Our result shows that by increasing ${\varphi }_1$, ${\eta }_2$ and by decreasing $\beta$, $\theta$ and ${\varphi }_s$, we can reduce the infection in both the economic group. As a result of our analysis, we have found that the reduction of infections and their level of adversity is directly influenced by positive behavioural patterns or changes as without control susceptible population is increased by $23\%$, the infective population is decreased by $48.54\%$ and the recovered population is increased by $23.23\%$ in the higher economic group who opted changed behaviour as compared to the lower the economic group (people living with normal behaviour). Thus normal behaviour contributes to the spread and growth of viruses and adds to the hassle. We also examined how antiviral drug control impacts both economic strata and found that in the higher economic strata, the susceptible population increased by $53.84\%$, the infective population decreased by $33.6\%$ and the recovered population improved by $62.29\%$ as compared to the lower economic group, the susceptible population has increased by $19.04\%$, the infective population is decreased by $17.29\%$ and the recovered population is improved by $47.82\%$. Our results enlighten the role that how different behaviour in separate socio-economic class plays an important role in changing the dynamics of the system and also affects the basic reproduction number. The results of our study suggest that it is important to adopt a modified behaviour like social distancing, wearing masks accompanying the time-dependent controls in the form of an antiviral drug’s effectiveness to combat infections and increasing the proportion of the susceptible population.

Introduction

Historical background

The influenza virus was originally discovered in a laboratory in 1932 and was the most studied of virus [1]. The studies by several researchers, epidemiologists, clinicians and the pharmaceutical business were published in a large body of literature [1–3]. Influenza has symptoms which involve three-day fever, muscle discomfort and prostration that are out of proportion to the severity of the other symptoms [1]. Till now, there have been three possible influenza pandemics, according to the historical record [4]. The main reason for the pandemic is the mutation of the virus rapidly. Every year, influenza epidemics kill approximately 250,000–500,000 people all over the world [5] and hence call for effective pharmaceutical and non-pharmaceutical interventions play a major role in it. Coughing, sneezing and hands contacting eyes, mouth, or nose are all common ways for the flu to transmit from person to person [2].

Interventions to prevent influenza

Several non-pharmaceutical and pharmaceutical interventions are used to protect against this virus. Vaccines fall under pharmaceutical intervention but as the influenza viruses are constantly evolving, new vaccinations are necessary on a regular basis to keep up with rapid changes in their antigenic features [2, 3]. Antigenic diversity in the influenza virus is a critical feature in predicting epidemics and in vaccine development, as the prevalent viral strain(s) must be integrated into current vaccines for every given year [1]. Influenza vaccines have a protective value of 70 to 90% [6] along with non-pharmaceutical interventions like social distancing, travel restrictions, work from home/work leave, limited exposure to human interactions i.e. community activities, gatherings, etc also. All of these aforementioned non-pharmaceutical factors are greatly affected with socio-economic conditions of an individual. People with low economic conditions tends to live in crowded places with less hygienic surroundings and therefore are likely to catch/spread infection easily than the population who live in good economic conditions and dwell in decent surroundings [2]. Travelling through local transport in the case of individuals with low income can trigger widespread infections as compared to people with good economic status who are able to afford their own vehicles [2]. Also, human behaviour plays a critical role in the development of new infectious illnesses [7, 8], especially when economic disparity is present. Our work captures this aspect at heart and digs at how an individual behaviour varies during an infection as a result of their economic level and how this affects the spread of new infectious diseases, influenza in our case.

Literature on mathematical models

Several mathematical models have been suggested for the transmission dynamics of influenza [9–14]. The authors of [15] investigated a two-dimensional SIS model with vaccination in a two-group population, taking into account disease transmission inside each population as well as between them. Multi-group approach is a sparse event in this domain and has been attempted considerably in this piece of research. Authors in [2] have dissected an SVIR model, in order to study the spread of influenza, assuming two strata of the population on the basis of income. The model has been subjected to linear and non-linear stability assessments. In another work [3], authors used non-linear least square fitting to estimate model parameters for the two cities of Canada namely Montreal and Winnipeg, utilizing data from the 1984 fall wave epidemic. They then used a two-pattern model to investigate the role of heterogeneity. A simple differential equation model developed by researchers in [8] allows people to change their behaviour to reduce their risk of infection. They have also described a large-scale agent-based model that will be used to analyse the impact of isolation scenarios like school closures and fear-based home isolation during a pandemic, in addition to the prior model. Changes in behaviour can be beneficial in decreasing the transmission of disease, according to both models. Hence, we have investigated an SIR model with population being divided into two groups on the basis of their economic status, high or low, as the case may be. To better forecast the transmission dynamics of an epidemic, our model reflects realistic individual-level mixing patterns and coordinated reactive changes in human behaviour. The optimal control assumption is an excellent systematic tool that makes decisions under uncertain dynamical situations. It is also been connected to our model, and this is a great approach for determining how well a disorder can be controlled [16–19]. In the paper [20], optimal control (vaccine and antiviral) in the context of influenza among strata differentiated on the basis of geographical regions were studied. However, the angle related to behavioural change along with the efficacy of antiviral on strata divided on the basis of economic status needs to be addressed. The paper is an attempt to address a few research questions that how behavioural change can help in reducing the infection? What is the final epidemic size of strata from different economic class and how they are related to the basic reproduction number. Further, what happens to the dynamics of the system if we involve time-dependent control and which are the sensitive parameters to be controlled to reduce the infection.

Structure of the paper

We described the development of the mathematical model in Sect. 2 followed by the theoretical analysis in Sect. 3 which includes positivity of the system, basic reproduction number, final epidemic size with its relation to $R_0$. Section 4 deals with the optimal control problem involving time-dependent control variables (efficacy of antiviral drug in higher and lower economic group). Numerical results are discussed in Sect. 5 which involves the contour plots of $R_0$ with different parameters, comparison of the state variables with and without control and global sensitivity analysis using PRCC. Following this, in Sect. 6, we provide a comprehensive summary of our results.

Model development and analysis

The model classifies the population into two subgroups: those who do not change their behaviour or who exhibit normal behaviour (${\lambda }_n$) and those who modify their behaviour in response to an outbreak (${\lambda }_m$). People switch among the two groups depending on the behaviour they adopt (reducing susceptibility or infectiousness). There are three types of individuals in each of the activity groups: susceptible, $S_1$, and $S_2$, infectious, $I_1$, and $I_2$, and recovered population $R_1$ and $R_2$. We use a system with no demographic change to or from the inhabitants.

The following are the model’s assumptions:

• Total population is separated into two categories: (i) The first group consists of persons from economically higher strata, a group who changes their behaviour in response to an outbreak. (ii) The second group consists of individuals from the economically lower status, a group who does not adjust their behaviour or behaves normally in response to an outbreak.

• All susceptible individuals are at risk of infection and can only leave the group through infection.

• The population is protected by behavioural changes such as social distancing, masks and so on.

• The system assumes no migration into or out of the population and does not include births or deaths.

Hence, the model is defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(\mathbf{Higher}\mathbf{Economic}\mathbf{Class})\left\{\begin{array}{c}\frac{\text{d}{S}_1}{\text{d}t}=-{\lambda }_m{S}_1,\hfill \\ \frac{\text{d}{I}_1}{\text{d}t}={\lambda }_m{S}_1-{\eta }_1{I}_1,\hfill \\ \frac{\text{d}{R}_1}{\text{d}t}={\eta }_1{I}_1,\hfill \end{array}\right)\hfill \\ (\mathbf{Lower}\mathbf{Economic}\mathbf{Class})\left\{\begin{array}{c}\frac{\text{d}{S}_2}{\text{d}t}=-{\lambda }_n{S}_2,\hfill \\ \frac{\text{d}{I}_2}{\text{d}t}={\lambda }_n{S}_2-{\eta }_2{I}_2,\hfill \\ \frac{\text{d}{R}_2}{\text{d}t}={\eta }_2{I}_2,\hfill \end{array}\right)\hfill \end{array}\right)\end{array}$$ 1

with initial conditions

$$\begin{array}{c}\hfill S_i(0)=S_{i0},I_i(0)=I_{i0}\&R_i(0)=R_{i0}\text{for}i=1,2.\end{array}$$ 2

Here, ${\lambda }_n=\beta [\theta \frac{I_2}{N_2}+(1-{\varphi }_i)\theta \frac{I_1}{N_1}],{\lambda }_m=\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1}{N_1}+(1-{\varphi }_s)\frac{\theta I_2}{N_2}]$

We are assuming ${\lambda }_n$(normal behaviour) and ${\lambda }_m$ (modified behaviour) as the force of infection. ${\varphi }_i$ is implemented into $\frac{\theta I_1}{N_1}$ contagious fragments in the forces of infection seeing as individual people in the $I_1$ class have adopted a changed behaviour, and ${\varphi }_s$ is implemented into the contagious fractions in ${\lambda }_m$ seeing as individual people in the susceptible class ($S_2$) have also introduced a comprehensive behaviour (Table 1).

Table 1.

Variable/parameter description

Variable/parameter	Description
$S_1$ or $S_2$	Susceptible individuals
$I_1$ or $I_2$	Infected individuals
$R_1$ or $R_2$	Recovered individuals
${\lambda }_m$ or ${\lambda }_n$	Force of infection for Higher & Lower economic group
${\eta }_1$	Removal rate from first group
${\eta }_2$	Removal rate from second group
$\beta$	Contact rate
$\theta$ & $1-{\varphi }_j$, $j=s$ or i	Effectiveness of behaviour in reducing either susceptibility or infectivity

Theoretical analysis

Positivity of the system

This section will go over the system’s (1) positive invariance for all $t>0$.

Theorem 1

The system (1) is positively invariant in ${\mathbb{R}}_{+}^6$ $\forall t>0$.

The proof is mentioned in the Appendix (Fig. 1).

Fig. 1.

The figure depicts a transmission flowchart of the state’s specific flow rates of the system (1). Each steady pointer of green colour in this graph represents a flow rate of individuals, whereas the dotted arrows of red colour depict the impact either of contagious of higher economic group 1 or infective of lower economic group 2

Basic reproduction number

The basic reproduction number $R_0$ is defined as

$$\begin{array}{c}\hfill R_0=\beta \theta \left[\frac{(1-{\varphi }_s)(1-{\varphi }_i)S_{10}}{{\eta }_1{N}_1},+,\frac{S_{20}}{{\eta }_2{N}_2}\right]=R_{01}+R_{02}(\text{Say}).\end{array}$$ 3

Biological interpretation of $R_0$ The basic reproduction number $R_0$ is the sum of two basic reproduction numbers $R_{01}$ and $R_{02}$, where $R_{01}$ represents the average number of secondary infections produced by an infectious individual from higher economic class into the entire susceptible population in that strata in its average infectious period $\frac{1}{{\eta }_1{N}_1}$ with the transmission rate $\beta \theta (1-{\varphi }_s)(1-{\varphi }_i)S_{10}$. $R_{02}$ represents the average number of secondary infections produced by an infectious individual from a higher economic class into the entire susceptible population in that strata in its average infectious period $\frac{1}{{\eta }_2{N}_2}$ with the transmission rate $\beta \theta S_{20}$.

Final epidemic size

Theorem 2

Let $S(0)=S_0\ge 0$ and $I(0)=I_0>0$, then the final size of an epidemic model (1) is given by

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}S(t)={\mathfrak{L}}_1,\underset{t\to +\infty }{lim}I(t)=0,\underset{t\to +\infty }{lim}R(t)={(N_1,N_2)}^T-{\mathfrak{L}}_1.\end{array}$$

For the proof of the above theorem, one may refer to Appendix.

Basic reproduction number: relation with final epidemic size

In this subsection, we will address the general relationship between the epidemic’s final size and $R_0$ as defined in (16). From system (1) as

$$\begin{array}{c}\hfill \text{ln}(S(t))-\text{ln}(S_0)=C{D}^{-1}(S(t)+I(t)-S_0-I_0)\forall t\ge 0.\end{array}$$ 4

At $t\to +\infty$, Eq. (4) gives

$$\begin{array}{c}\hfill \text{ln}(S(+\infty ))-ln(S_0)=C{D}^{-1}(S(+\infty )-S_0-I_0),I(+\infty )=0.\end{array}$$ 5

As $A=\text{diag}(S_0)C{D}^{-1}$, we get

$$\begin{array}{c}\hfill \text{diag}(S_0)[ln(S(+\infty )-ln(S_0)]=A(S(+\infty )-S_0-I_0).\end{array}$$

Since A is an irreducible matrix and $R_0=r(A)$, a left eigenvector can be found as $P={(P_1,P_2)}^T>0$ that will give $PA=R_{0}A$ provided that

$$\begin{array}{c}\hfill P\text{diag}(S_0)[\text{ln}(S(+\infty ))-\text{ln}(S_0)]=R_{0}A(S(+\infty )-S_0-I_0),\end{array}$$ 6

which gives the general relationship between $R_0$ and final epidemic size and for one-dimensional SIR model it is, $\text{ln}(\frac{S(+\infty )}{S_0})=R_0(\frac{S(+\infty )}{S_0}-1)-\frac{R_0}{S_0}{I}_0$.

Optimal control

The optimized control issue will be the focus of this section. Our primary goal is to keep infection and infected populations under control. During an influenza outbreak of disease, the goal is to limit the number of medically unhealthy population line up over a finite period of time duration [0, T] at the lowest possible cost of effort. We used two controls $T_a$ for the upper high economic class and, $T_b$ for the lower economic class, both groups yielding the controlled equations as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{\text{d}{S}_1}{\text{d}t}=-(1-T_a){\lambda }_m{S}_1,\hfill \\ \frac{\text{d}{I}_1}{\text{d}t}=(1-T_a){\lambda }_m{S}_1-{\eta }_1{I}_1,\hfill \\ \frac{\text{d}{R}_1}{\text{d}t}={\eta }_1{I}_1,\hfill \\ \frac{\text{d}{S}_2}{\text{d}t}=-(1-T_b){\lambda }_n{S}_2,\hfill \\ \frac{\text{d}{I}_2}{\text{d}t}=(1-T_b){\lambda }_n{S}_2-{\eta }_2{I}_2,\hfill \\ \frac{\text{d}{R}_2}{\text{d}t}={\eta }_2{I}_2,\hfill \end{array}\right)\end{array}$$ 7

$T_a$ = Efficacy of an antiviral drug in preventing new infections in a higher social class,

$T_b$ = Efficacy of an antiviral drug in preventing new infections in a lower middle class,

$(1-T_a)$ represents the rate of infection all through antiviral therapy in a higher class,

$(1-T_b)$ represents the rate of infection all through antiviral therapy in a lower class. The goal of optimization is to maximize the following objective function:

$$\begin{array}{c}\hfill J(T_a,T_b)={\int }_0^{t_f}{S}_1+R_1+S_2+R_2-\left[\frac{C_1}{2},T_a^2,+,\frac{C_2}{2},T_b^2\right]\text{d}t,\end{array}$$ 8

where $t_f$ is the treatment time period and introduced treatments’ benefits and costs are denoted by $C_1$ and $C_2$. $T_a(t)$ and $T_b(t)$ are assumed to be bounded and Lebesgue integrable.

$$\begin{array}{c}\hfill J(T_a^{\ast },T_b^{\ast })=max(J(T_a,T_b):(T_a,T_b)\in T).\end{array}$$ 9

The controls set is defined below as

$$\begin{array}{c}\hfill T=\{T_a(t),T_b(t):\mathrm{is}\mathrm{measurable},0\le T_i\le 1,t\in [0,t_f],i=a,b\}\end{array}$$

We are now moving on to the Existence Theorem. We use results to demonstrate the existence of a control system from Lukes [21, 22].

Theorem 3

There exist a function used for control pair as $(T_a^{\ast },T_b^{\ast })$ such that

$$\begin{array}{c}\hfill J(T_a^{\ast },T_b^{\ast })=max(J(T_a,T_b)),(T_a,T_b)\in T).\end{array}$$ 10

Proof

For the proof following properties are needed [23]:

1. Sets of control variables and state variables corresponding to them are not vacant.

2. T is closed as well as convex.

3. The state function and the control function in the RHS are linear functions.

4. Integrand of objective function is concave on T.

5. There exist constants $p_1,p_2>0$ and $q>1$ such that the integrand $I(S_1,R_1,S_2,R_2,T_a,T_b)$ of the objective function satisfies

   $$\begin{array}{c}\hfill I(S_1,R_1,S_2,R_2,T_a,T_b)\le p_2-p_1(|T_a|+|T_b{|)}^{\frac{q}{2}}.\end{array}$$ 11

To demonstrate these requirements, we used Lukes’ findings [21] to demonstrate the existence of system solutions (7), that also generate condition (1). The monitoring set is convex and closed by definition, giving rise in condition (2). Because our framework is bilinear in $T_a$ and $T_b$, the RHS is also bilinear. (7), satisfies condition (3) by virtue of the boundedness of the solutions. Because the integrand of our objective function is concave, we also have the situation which is now required.

$$\begin{array}{c}\hfill I(S_1,R_1,S_2,R_2,T_a,T_b)\le p_2-p_1(|T_a|+|T_b{|)}^{\frac{q}{2}}.\end{array}$$ 12

Because $C_1>0$ and $C_2>0$, the upper bound on $S_1,R_1$ and $S_2,R_2$ determines $q_2$. We conclude that an optimal control couple $(T_a^{\ast },T_b^{\ast })\in T$ exists such that $J(T_a^{\ast },T_b^{\ast })=max(J(T_a,T_b)),(T_a,T_b)\in T$. $\square$

Pontryagin’s maximum principle [24] demonstrates the conditions required for an optimization problem.

The Hamiltonian is specifically defined as: The maximum principle proposed by Pontryagin [24] establishes the necessary conditions for an optimal control problem. The Hamiltonian is defined as

$$\begin{array}{cc}\hfill & H(t,S_1,R_1,S_2,R_2,T_a,T_b,\chi )\hfill \\ \hfill & =\left[\frac{C_1}{2},T_a^2,+,\frac{C_2}{2},T_b^2\right]-S_1-R_1-S_2-R_2+\sum _{i=1}^6{\chi }_i{f}_i,\hfill \end{array}$$ 13

where $f_1=-(1-T_a){\lambda }_m{S}_1$, $f_2=(1-T_a){\lambda }_m{S}_1-{\eta }_1{I}_1$, $f_3={\eta }_1{I}_1$, $f_4=-(1-T_b){\lambda }_n{S}_2$, $f_5=(1-T_b){\lambda }_n{S}_2-{\eta }_2{I}_2$ $f_6={\eta }_2{I}_2$

Pontryagin’s maximum principle is used for further solution, yielding the following theorem.

Theorem 4

With the optimized conditions $T=(T_a,T_b)$, solutions $A=(S_1^{\ast },R_1^{\ast },S_2^{\ast },R_2^{\ast })$ for correlating state system (7), there exists adjoint variable ${\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4,{\chi }_5,{\chi }_6$, which satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\chi }_1^{\prime }=1+\beta [\frac{(1-{\varphi }_s)(1-{\varphi }_i)I_1}{N_1}+\frac{(1-{\varphi }_s)I_2}{N_2}]\theta (1-T_a)({\chi }_1^{\prime }-{\chi }_2^{\prime }),\hfill \\ {\chi }_2^{\prime }=\beta [\frac{(1-{\varphi }_s)(1-{\varphi }_i)}{N_1}{S}_1(1-T_a)({\chi }_1-{\chi }_2)\hfill \\ +\frac{(1-{\varphi }_i)}{N_2}{S}_2(1-T_b)({\chi }_4^{\prime }-{\chi }_5^{\prime })]\theta +{\eta }_1({\chi }_2^{\prime }-{\chi }_3^{\prime }),\hfill \\ {\chi }_3^{\prime }=1,\hfill \\ {\chi }_4^{\prime }=1+\beta [\frac{(1-{\varphi }_i)I_1}{N_1}+\frac{I_2}{N_2}]\theta (1-T_b)({\chi }_4-{\chi }_5),\hfill \\ {\chi }_5^{\prime }=\beta [\frac{(1-{\varphi }_s)}{N_2}{S}_1(1-T_a)({\chi }_1^{\prime }-{\chi }_2^{\prime },\hfill \\ +\frac{1}{N_2}{S}_2(1-T_b)({\chi }_4^{\prime }-{\chi }_5^{\prime })]\theta +{\eta }_2({\chi }_5^{\prime }-{\chi }_6^{\prime }),\hfill \\ {\chi }_6^{\prime }=1.\hfill \end{array}\right)\end{array}$$

Conditions of transversality ${\chi }_i=0,i=1,\dots ,6.$ Furthermore, the best optimality control is given as

$T_a^{\ast }=min(1,max(0,\frac{1}{C_1}(({\chi }_2-{\chi }_1)\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1^{\ast }}{N_1}+(1-{\varphi }_s)\frac{\theta I_2^{\ast }}{N_2}]S_1^{\ast })))$, $T_b^{\ast }=min(1,max(0,\frac{1}{C_2}(({\chi }_5-{\chi }_4)\beta [\frac{(1-{\varphi }_i)\theta I_1^{\ast }}{N_1}+\frac{\theta I_2^{\ast }}{N_2}]S_2^{\ast }))).$

Proof

The above equations can be obtained by using Pontryagin’s Maximum Principal.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\chi }_1^{{}^{\prime }}=-\frac{\partial H}{\partial S_1}(t),{\chi }_1(t_f)=0,\hfill \\ {\chi }_2^{{}^{\prime }}=-\frac{\partial H}{\partial I_1}(t),{\chi }_2(t_f)=0,\hfill \\ {\chi }_3^{{}^{\prime }}=-\frac{\partial H}{\partial R_1}(t),{\chi }_3(t_f)=0,\hfill \\ {\chi }_4^{{}^{\prime }}=-\frac{\partial H}{\partial S_2}(t),{\chi }_1(t_f)=0,\hfill \\ {\chi }_5^{{}^{\prime }}=-\frac{\partial H}{\partial I_2}(t),{\chi }_1(t_f)=0,\hfill \\ {\chi }_6^{{}^{\prime }}=-\frac{\partial H}{\partial R_2}(t),{\chi }_1(t_f)=0.\hfill \end{array}\right)\end{array}$$ 14

$\square$

The results of both controls, which include efficacy of an antiviral drug in preventing new infections in a higher economic class & efficacy of an antiviral drug in preventing new infections in a lower economic class, are represented graphically (Figs. 2, 3, 4, 5, 6, 7).

Fig. 2.

Effect of $\theta$ on $S_1$

Fig. 3.

Effect of $\theta$ on $S_2$

Fig. 4.

Effect of $\theta$ on $I_1$

Fig. 5.

Effect of $\theta$ on $I_2$

Fig. 6.

Effect of $\theta$ on $R_1$

Fig. 7.

Effect of $\theta$ on $R_2$

Numerical section and sensitivity analysis

In this section, we compared the models with and without control using the numerical values in Table 2. We have obtained the sensitive parameters which are responsible for change in the reproduction number followed by probability sensitivity analysis by obtaining partial rank correlation coefficient using latin hypercube sampling. This method will help us to determine the crucial parameters which can be controlled to reduce the final epidemic size. PRCC (partial rank correlation coefficient) [25, 26] is a methodology for analysing the volatility in any model. For our basic reproduction number, we will use PRCC to determine and assess how such volatility of the parameters (input) impacts infection transmission. 1000 calculations are performed on each input parameter. Each input parameter is calculated 1000 times. The PRCC for $R_0$ is discussed to gain a deeper understanding of the parameters, and it is believed that if PRCC is close to $+1$ or $-1$, the variables and parameters are highly correlated, with PRCC $>0.4$ indicating significant correlation, $0.1-0.4$ indicating moderate correlation and $0-0.1$ indicating no correlation [27]. This analysis is carried out at the $5\%$ level of significance. We assumed normal distribution for the parameters $\beta ,{\varphi }_i,{\varphi }_s,{\eta }_1$ and ${\eta }_2$, and uniform distribution for $N_1$ and $N_2$. Tables with the parameters’ mean, standard deviation and range are shown in Table 3.

Table 2.

Parameters values

Parameter	Value	Reference
${\eta }_1$	0.001	Assumed
${\eta }_2$	0.001	Assumed
$\beta$	0.4	[8]
$\theta$	0.1–0.9	Assumed
${\varphi }_i$	0.5	[8]
${\eta }_j$	0.5	[8]
$N_1$	1000	[8]
$N_2$	1000	[8]

Table 3.

For normal distribution

Parameters	Distribution
$\beta$	Normal (0.4,0.001)
${\varphi }_i$	Normal (0.5,0.01)
${\varphi }_s$	Normal (0.5,0.01)
$\theta$	Normal (0.45,0.01)
${\eta }_1$	Normal (0.001,0.000001)
${\eta }_2$	Normal (0.001,0.000001)
$N_1$	Uniform (1–1000 )
$N_2$	Uniform (1–1000 )

• Effect of $\theta$ on individuals We have shown the effect of $\theta$ (effectiveness of behaviour in reducing susceptibility or infectivity) on susceptible, infected and recovered populations of both the strata: higher economic as well as lower economic class. From Figs. 2 and 7, we have observed that for both the economic groups, $\theta$ shows a direct effect on the susceptible and recovered class and an adverse effect on infective as on increasing $\theta$, susceptible and recovered individuals are increasing while infective is decreasing which means a proper behaviour can have promising results in the context of reducing the infective.

• Effect of ${\varphi }_s$ and ${\varphi }_i$ on $R_0$: From Fig. 8, we have concluded that when $0.2<{\varphi }_s<0.9$ and $0<{\varphi }_i<0.2$ or $0.2<{\varphi }_s<0.4$ and $0.8<{\varphi }_i<0.9$ holds, $R_0$ is small which means that it is mostly affected when ${\varphi }_i$ is large and ${\varphi }_s$ is small or ${\varphi }_s$ is large and ${\varphi }_i$ is small. However, when $0.2<{\varphi }_s<0.35$ and $0.2<{\varphi }_i<0.3$, $R_0$ is maximum. So, $R_0$ can be reduced by either increasing ${\varphi }_i$ and decreasing ${\varphi }_s$ or vice versa.

• Effect of ${\varphi }_i$, ${\varphi }_s$ and $\theta$ on $R_0$: Figs. 9 and 10 depict that $R_0$ will increase as ${\varphi }_i$ decreases and ${\varphi }_s$ increases. The terms ${\varphi }_i$ and ${\varphi }_s$ refer to the effectiveness of behaviour in lowering infection rate and susceptibility. Since, force of infection ${\lambda }_n$ for the lower group depends on $(1-{\varphi }_i)$ and ${\lambda }_m$ for the higher group depends on $(1-{\varphi }_s)$ and $(1-{\varphi }_i)$, so by enhancing ${\varphi }_s$ and reducing ${\varphi }_i$ infection will rise. $R_0$ has a direct response with $\theta$ and ${\varphi }_s$ and indirect response with ${\varphi }_i$ as on increasing $\theta$ as well as ${\varphi }_s$ and by decreasing ${\varphi }_s$, $R_0$ increases or vice versa.

• Effect of control on individuals Figures 11 and 12 depict the strata opting for changed behavioural pattern have a higher susceptible population in case of with & without control situations as compared to the lower economic group. Furthermore, it can be seen even if the lower economic group does not opt for changed behaviour, it can have a higher susceptible population just by using control measures. However, we cannot subside the significance of changed behavioural patterns in increasing the susceptible population percentage, therefore changed behaviour patterns combined with appropriate control measures will surely be useful in combating infections and increasing the proportions of susceptible.

• From Figs. 13, 14, 15 and 16, we observed that after applying control, the population bearing modified behaviour will recover early and infected and infectiousness of the susceptible population will reduce more in comparison to the population following normal behaviour. In other words, modified behaviour consisting of elements like taking effective precautions, adhering to social distancing norms and wearing masks etc. is an advisable course of action if we wish to combat & face infectious situations. On the other hand, normal behavioural patterns will add to the trouble and will be supportive of the growth & spread of viruses. By applying control, our susceptible population for the higher economic group is increased by $53.84\%$, infective are decreased by $33.6\%$ and the recovered population is improved by $62.29\%$ while in the lower economic group, the susceptible population is increased by $19.04\%$, infectives are decreased by $17.29\%$ and the recovered population is improved by $47.82\%$.

• Sensitive parameter analysis of $R_0$ Using our input parameters, we obtain PRCC values as shown in Fig. 17. Values of $\beta$ and $\theta$ are close to 1 indicating a highly positive correlation with $R_0$ as on increasing $\beta$(contact rate) and $\theta$ (Effectiveness of behaviour in reducing either susceptibility or infectivity) will increase the transmission rate. In order to suppress the infection, $\beta$ and $\theta$ must be decreased. The value of ${\eta }_2$ (recovery rate of the lower economic group) is close to $-1$ indicating a highly negative correlation with $R_0$ as on increasing $\beta$(contact rate) and $\theta$ (Effectiveness of behaviour in reducing either susceptibility or infectivity) will decrease transmission rate. In order to suppress the infection, the recovery rate of the lower-income group must be increased. We know that the lower economic strata is associated with higher economic strata as per our model which implies that despite practising modified behaviour by higher economic strata recovered population cannot increase till the recovery rate of lower strata also gets improved.

Fig. 8.

Effect of ${\varphi }_i$ and ${\varphi }_s$ on $R_0$

Fig. 9.

Effect of ${\varphi }_i$ and $\theta$ on $R_0$

Fig. 10.

Effect of ${\varphi }_s$ and $\theta$ on $R_0$

Fig. 11.

With and without control graph of $S_1$

Fig. 12.

With & without control graph of $S_2$

Fig. 13.

With & without control graph of $I_1$

Fig. 14.

With & without control graph of $I_2$

Fig. 15.

With & without control graph of $R_1$

Fig. 16.

With & without control graph of $R_2$

Fig. 17.

PRCC: Sensitivity index of $R_0$

Conclusion

In this paper, we have conducted a comparative analysis of the behaviour effect for model (1) with higher and lower economic strata. In order to better predict the transmission dynamics of an epidemic, the model captures realistic individual-level mixing patterns and coordinated reactive changes in human behaviour. The model confirms that behavioural changes can be effective in reducing disease spread which can be also visualized by the contour plots between the parameters and $R_0$. We have also shown that the disease persists for $R_0>1$. Furthermore, based on the final epidemic size, it is concluded that infection load is lower in higher economic groups than in lower economic groups. The reason is due to the modified behaviour followed by people from higher economic class. Further, we have incorporated time-dependent controls in the form of efficacy of an antiviral drug in both the classes to analyse the comparison between both the classes and we obtained that the recovered population of the higher economic class is $14.47\%$ higher in comparison to the lower economic class after applying control. The findings reveal that modified behaviour along with higher efficacy of the antiviral drug can reduce the infection specially in the scenarios when the virus has the ability to mutate. Finally, our numerical analysis using PRCC also gave us important information regarding the sensitive parameters ${\varphi }_i$, ${\varphi }_s$, $\beta$, ${\eta }_2$ and $\theta$ influencing $R_0$.

As a result, medical procedures should be made available to all populations in an impartial manner. Also, timely screening of symptomatic patients, as well as earlier disclosure, is required to encourage residents to act rationally, such as through self-quarantine, mask-wearing and so on. Controlling human behaviour can have a significant impact on disease propagation, forecasting and on the resources required to contain an outbreak. Modelling studies like the ones proposed here could be useful in assessing the implications of differences in human behaviour for future global epidemic regulations. Such studies and their findings could also be helpful in making appropriate policies related to social and behavioural norms, medical procedures and general measures to fight with such diseases and infections.

Appendix

Proof of positivity of the system

Proof

Let $C(0)=C_0\in {\mathbb{R}}_{+}^6$, where $C=(S_1,I_1,R_1,S_2,I_2,R_2)\in {\mathbb{R}}_{+}^6$ than the system (1) can be written in matrix form as $\dot{C}=D(C)$ given as

$$\begin{array}{c}\hfill D(C)=\left[\begin{array}{c}-{\lambda }_m{S}_1\\ {\lambda }_m{S}_1-{\eta }_1{I}_1\\ {\eta }_1{I}_1\\ -{\lambda }_n{S}_2\\ {\lambda }_n{S}_2-{\eta }_2{I}_2\\ {\eta }_2{I}_2.\end{array}\right]\end{array}$$ 15

Let $D:\mathbb{R}{}_{+}^6\to {\mathbb{R}}^6$ be locally Lipschitz and using the standard theory of the ordinary differential equations (ODE) system, the unique solution for any initial condition $D(0)=D_0=(S_1{}_0,I_1{}_0,R_1{}_0,S_2{}_0,I_2{}_0,R_2{}_0)\in \mathbb{R}{}_{+}^6$ of the system (1) is obtained. In addition, system (1) can be rewritten as $\frac{\text{d}{S}_1}{\text{d}t}=S_1{\psi }_1(S_1,I_1,I_2),\frac{d{I}_1}{\text{d}t}=I_1{\psi }_2(S_1,I_1,I_2),\frac{d{R}_1}{\text{d}t}=R_1{\psi }_3(I_1),\frac{\text{d}{S}_2}{\text{d}t}=S_2{\psi }_4(S_2,I_1,I_2),\frac{d{I}_2}{\text{d}t}=I_2{\psi }_5(S_2,I_1,I_2),\frac{d{R}_2}{\text{d}t}=R_2{\psi }_6(I_2),$ where

$$\begin{array}{cc}\hfill {\psi }_1=& -\beta \left[(1-{\varphi }_s),(1-{\varphi }_i),\frac{\theta I_1}{N_1},+,(1-{\varphi }_s),\frac{\theta I_2}{N_2}\right],\hfill \\ \hfill {\psi }_2=& \beta \left[(1-{\varphi }_s),(1-{\varphi }_i),\frac{\theta }{N_1},+,(1-{\varphi }_s),\frac{\theta I_2}{N_2{I}_1}\right]S_1-{\eta }_1,\hfill \\ \hfill {\psi }_3=& {\eta }_1\frac{I_1}{R_1},\hfill \\ \hfill {\psi }_4=& -\beta \left[\theta ,\frac{I_2}{N_2},+,(1-{\varphi }_i),\theta ,\frac{I_1}{N_1}\right],\hfill \\ \hfill {\psi }_5=& \beta \left[\frac{\theta }{N_2},+,(1-{\varphi }_i),\theta ,\frac{I_1}{N_1{I}_2}\right]S_2-{\eta }_2,\hfill \\ \hfill {\psi }_6=& {\eta }_2\frac{I_2}{R_2}.\hfill \\ \hfill \end{array}$$

From the first equation of the system (1),

$\frac{1}{S_1}\frac{\text{d}{S}_1}{\text{d}t}=-\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1}{N_1}+(1-{\varphi }_s)\frac{\theta I_2}{N_2}]\Rightarrow \frac{d(\text{ln}{S}_1)}{\text{d}t}=-\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1}{N_1}+(1-{\varphi }_s)\frac{\theta I_2}{N_2}]$ Integrating from 0 to t, we get,

$\text{ln}{S}_1(t)-\text{ln}{S}_1(0)=({\int }_0^t-\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1}{N_1}+(1-{\varphi }_s)\frac{\theta I_2}{N_2}]\text{d}t)$

$S_1(t)=S_1(0)e^{({\int }_0^t-\beta [(1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta I_1}{N_1}+(1-{\varphi }_s)\frac{\theta I_2}{N_2}]\text{d}t)}>0$

Similarly, by the theorem in [28, 29], $D_i{(C)|}_{C_i=0}=0$ for $(i=1,2,3,4,5,6)$, where $C(0)\in {\mathbb{R}}_{+}^6$ as a result $C_i=0$.

thus, the system (1) is positively invariant in ${\mathbb{R}}_{+}^6$ $\forall$ $t>0$. $\square$

Proof of basic reproduction number. The basic reproduction number $R_0$ is defined as the spectral radius of A, where $A=\text{diag}(S_0)C{D}^{-1}$, $\text{diag}(S_0)=\left[\begin{array}{cc}{S}_{10}& 0\\ 0& S_{20}\\ \end{array}\right]$ where $S_1(0)$ and $S_2(0)$ are the initial sizes of the susceptible people of both groups. C represents the rate of appearance of new infections in a compartment and D represents the rate of transfer of individuals into other compartments. Using the next-generation method [30–32], we obtain the below matrices for our model, $C=\left[\begin{array}{cc}\beta (1-{\varphi }_s)(1-{\varphi }_i)\frac{\theta }{N_1}& \beta (1-{\varphi }_s)\frac{\theta }{N_2}\\ \beta \frac{(1-{\varphi }_i)\theta }{N_1}& \beta \frac{\theta }{N_2}\\ \end{array}\right],D=\left[\begin{array}{cc}{\eta }_1& 0\\ 0& {\eta }_2\\ \end{array}\right],D^{-1}=\left[\begin{array}{cc}\frac{1}{{\eta }_1}& 0\\ 0& \frac{1}{{\eta }_2}\\ \end{array}\right]$ $A=C{D}^{-1}=\left[\begin{array}{cc}\frac{\beta (1-{\varphi }_s)(1-{\varphi }_i)\theta S_{10}}{{\eta }_1{N}_1}& \frac{\beta (1-{\varphi }_s)\theta S_{10}}{{\eta }_2{N}_2}\\ \frac{\beta (1-{\varphi }_i)\theta S_{20}}{{\eta }_1{N}_1}& \frac{\beta \theta S_{20}}{{\eta }_2{N}_2}\\ \end{array}\right].$ The basic reproduction number is $R_0=r(A)$, where r is the spectral radius of the matrix A.

$$\begin{array}{c}\hfill R_0=\beta \theta \left[\frac{(1-{\varphi }_s)(1-{\varphi }_i)S_{10}}{{\eta }_1{N}_1},+,\frac{S_{20}}{{\eta }_2{N}_2}\right].\end{array}$$ 16

Further, the Perron–Frobenius theorem can be used to find left eigenvector $P={(P_1,P_2)}^T$ and right eigenvector $Q={(Q_1,Q_2)}^T$ for the matrix A, which is non-negative and irreducible, such that $P\ge 0$ and $Q\ge 0$ with $P\text{diag}(S_0)C{D}^{-1}=R_{0}P$ and $\text{diag}(S_0)C{D}^{-1}Q=R_{0}Q.$

From second and fifth equation of the system (1) we obtain

$\frac{\text{d}I}{\text{d}t}=\text{diag}(S(t))CI(t)-DI(t)=[\text{diag}(S(t))C{D}^{-1}-I]DI(t),t\ge 0$ Here $S=\left[\begin{array}{c}{S}_1(t)\\ S_2(t)\\ \end{array}\right],I=\left[\begin{array}{c}{I}_1(t)\\ I_2(t)\\ \end{array}\right]$

As a consequence, the following lemma holds true.

Lemma 1

Suppose that DI(0) is directly proportional to Q, the eigenvector connected with the matrix diag$(S(0))C{D}^{-1}$ dominant’s eigenvalue (i.e. $R_0$). Therefore, when $t=0$, $\frac{\text{d}I(0)}{\text{d}t}=(R_0-1)DI(0)$.

Furthermore, along with DI(0) is directly proportionate to Q if $R_0>1$, both the infective population $I_1(t)$ and $I_2(t)$ enhance locally across $t=0$ and if $R_0<1$, decrease locally across $t=0$.

Furthermore for any initial distribution I(0) we have

$\frac{P\text{d}I(0)}{\text{d}t}=(R_0-1)PDI(0)$ = $\frac{P\text{d}I(0)}{\text{d}t}=(R_0-1)(P_1{\eta }_1{I}_1(0)+P_2{\eta }_2{I}_2(0))$. It is obvious to see that when $R_0>1$ we have at least one component increasing locally around $t=0$. Indeed when $R_0>1$ we may obtain complex dynamics.

Derivation of final epidemic size

From first equation of system (1), we have

$$\begin{array}{c}\hfill \frac{\text{d}{S}_1}{\text{d}t}=-\beta \left[(1-{\varphi }_s),\frac{\theta I_2}{N_2},+,(1-{\varphi }_s),(1-{\varphi }_i),\frac{\theta I_1}{N_1}\right]S_1.\end{array}$$

Then, on integration, we get

$$\begin{array}{cc}\hfill & \text{ln}{S}_1(t)-\text{ln}{S}_1(0)=-\frac{\beta (1-{\varphi }_s)\theta }{N_2}{\int }_0^t{I}_2(s)\text{d}s\hfill \\ \hfill & -\frac{\beta (1-{\varphi }_s)(1-{\varphi }_i)\theta }{N_1}{\int }_0^t{I}_1(s)\text{d}s\hfill \end{array}$$ 17

and by adding the first two equations of the system (1), we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\text{d}(S_1+I_1)}{\text{d}t}}=-{\eta }_1{I}_1\hfill \\ \hfill & \Rightarrow (S_1+I_1)(t)-(S_1+I_1)(0)=-{\eta }_1{\int }_0^t{I}_1(s)\text{d}s.\hfill \end{array}$$

Now, putting the value ${\eta }_1{\int }_0^t{I}_1(s)\text{d}s$ into (17), we get

$$\begin{array}{cc}\hfill \text{ln}{S}_1(t)-\text{ln}{S}_1(0)=& -\frac{\beta (1-{\varphi }_s)\theta }{N_2}{\int }_0^t{I}_2(s)\text{d}s\hfill \\ \hfill & +\frac{\beta (1-{\varphi }_s)(1-{\varphi }_i)\theta }{{\eta }_1{N}_1}\left((S_1+I_1),(t),-,(S_1+I_1),(0)\right).\hfill \end{array}$$ 18

Similarly, from fourth equation of system (1), we have

$$\begin{array}{c}\hfill \text{ln}{S}_2(t)-\text{ln}{S}_2(0)=-\frac{\beta \theta }{N_1}{\int }_0^t{I}_1(s)\text{d}s-\beta \frac{(1-{\varphi }_i)\theta }{N_2}{\int }_0^t{I}_2(s)\text{d}s\end{array}$$ 19

and

$$\begin{array}{c}\hfill (S_2+I_2)(t)-(S_2+I_2)(0)=-{\eta }_2{\int }_0^t{I}_2(s)\text{d}s.\end{array}$$ 20

Again, putting the value of ${\eta }_2{\int }_0^t{I}_2(s)\text{d}s$ in (18), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{ln}{S}_1(t)-\text{ln}{S}_1(0)& =\frac{\beta (1-{\varphi }_s)\theta }{{\eta }_2{N}_2}\left[(S_2+I_2),(t),-,(S_2+I_2),(0)\right]\hfill \\ \hfill & +\frac{\beta (1-{\varphi }_s)(1-{\varphi }_i)\theta }{{\eta }_1{N}_1}\left[(S_1+I_1),(t),-,(S_1+I_1),(0)\right],\hfill \end{array}\end{array}$$ 21

which can be rewritten as

$$\begin{array}{cc}\hfill & \text{ln}{S}_1(t)-\text{ln}{S}_1(0)\hfill \\ \hfill & =M[(S_1+I_1)(t)-(S_1+I_1)(0)]+N[(S_2+I_2)(t)-(S_2+I_2)(0)],\hfill \end{array}$$ 22

where

$$\begin{array}{c}\hfill M=\frac{\beta (1-{\varphi }_s)(1-{\varphi }_i)\theta }{{\eta }_1{N}_1}\text{and}N=\frac{\beta (1-{\varphi }_s)\theta }{{\eta }_2{N}_2}.\end{array}$$

In the similar manner, by putting the value of ${\eta }_1{\int }_0^t{I}_1(s)\text{d}s$ and ${\eta }_2{\int }_0^t{I}_2(s)\text{d}s$ in Eq. (19), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{ln}{S}_2(t)-\text{ln}{S}_2(0)& =A[(S_1+I_1)(t)-(S_1+I_1)(0)]\hfill \\ \hfill & +B[(S_2+I_2)(t)-(S_2+I_2)(0)],\hfill \end{array}\end{array}$$ 23

where

$$\begin{array}{c}\hfill A=\frac{\beta \theta }{{\eta }_1{N}_1}\text{and}B=\frac{\beta (1-{\varphi }_i)\theta }{{\eta }_2{N}_2}.\end{array}$$

On taking limit $t\to +\infty$ in Eq. (22), we get

$$\begin{array}{cc}\hfill & M(S_1+I_1)(+\infty )+N(S_2+I_2)(+\infty )-\text{ln}{S}_1(+\infty )\hfill \\ \hfill & =M(S_1+I_1)(0)+N(S_2+I_2)(0)-\text{ln}{S}_1(0).\hfill \end{array}$$

$\Rightarrow \text{ln}(\frac{S_1(+\infty )}{S_1(0)})=M[(S_1+I_1)(+\infty )-V]+N[(S_2+I_2)(+\infty )-U]$,

where, $V=(S_1+I_1)(0)$ and $U=(S_2+I_2)(0)$.

Hence, we get $S_1(+\infty )=S_1(0)\text{exp}[M((S_1+I_1)(+\infty )-V)+N((S_2+I_2)(+\infty )-U)]$.

Similarly, $S_2(+\infty )=S_2(0)\text{exp}[A((S_1+I_1)(+\infty )-V)+B((S_2+I_2)(+\infty )-U)]$.

The fixed problem thus defined in $0\le S(+\infty )\le S_1(0),0\le S(+\infty )\le S_2(0)$ is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1(+\infty )& =S_1(0)\text{exp}\left[(M(S_1)(+\infty )-V),+,N,((S_2)(+\infty )-U)\right]\hfill \\ \hfill S_2(+\infty )& =S_2(0)\text{exp}\left[(A(S_1)(+\infty )-V),+,B,((S_2)(+\infty )-U)\right].\hfill \end{array}\end{array}$$ 24

Remark 1

[32] The following notations will be used to prove further results:

Let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ be two vectors in ${\mathbb{R}}^2$, then

1. $x\le y⟺x_i\le y_i,\text{for all}i=1,2$.

2. $x<y⟺x\le y$ and $x_i<y_i,\text{for some}i=1,2$.

3. $x<<y⟺x_i<y_i,\text{for all}i=1,2$.

Now, we define a mapping by system (24), as $\mathfrak{T}:{\mathbb{R}}^2\to {\mathbb{R}}^2$, which is given by

$$\begin{array}{c}\hfill \mathfrak{T}(x_1,x_2)={({\mathfrak{T}}_1(x_1,x_2){\mathfrak{T}}_{\mathfrak{2}}(x_1,x_2))}^T\end{array}$$ 25

with

$$\begin{array}{c}\hfill {\mathfrak{T}}_1(x_1,x_2)=S_1(0)\text{exp}\left(M,[x_1-V],+,N,[x_2-U]\right)\end{array}$$

and

$$\begin{array}{c}\hfill {\mathfrak{T}}_2(x_1,x_2)=S_2(0)\text{exp}\left(A,[x_1-V],+,B,[x_2-U]\right).\end{array}$$

In view of the above definition, we have the following observations regarding the properties of $\mathfrak{T}$:

Observations: (I) $\mathfrak{T}$ is monotonic increasing function, i.e.

$$\begin{array}{c}\hfill x\le y\Rightarrow \mathfrak{T}(x)\le \mathfrak{T}(y).\end{array}$$

(II) Since $M,N,A,B$ are all strictly positive constants, then

$$\begin{array}{c}\hfill x<<y\Rightarrow \mathfrak{T}(x)<<\mathfrak{T}(y).\end{array}$$

(III) $0<<\mathfrak{T}(0)<\mathfrak{T}(S(0))<S(0)$, where $S(t)={(S_1(t)S_2(t))}^T$.

(IV) By the induction on $n\ge 1$, we can deduce from observations (III) that

$$\begin{array}{c}\hfill 0<<\mathfrak{T}(0)\cdots <<{\mathfrak{T}}^n(0)<<{\mathfrak{T}}^{n+1}(0)\le {\mathfrak{T}}^{n+1}(S(0))<\cdots <{\mathfrak{T}}^n(S(0))<S(0).\\ \hfill \end{array}$$ 26

(V) On taking limit $n\to +\infty$ in (26), we get

$$\begin{array}{c}\hfill 0<<\underset{n\to +\infty }{lim}{\mathfrak{T}}^n(0)={\mathfrak{L}}_1\le {\mathfrak{L}}_2=\underset{n\to +\infty }{lim}{\mathfrak{T}}^n(S(0))<S(0).\end{array}$$

(VI) By the continuity of $\mathfrak{T}$, we have

$$\begin{array}{c}\hfill \mathfrak{T}({\mathfrak{L}}_1)={\mathfrak{L}}_1\text{and}\mathfrak{T}({\mathfrak{L}}_2)={\mathfrak{L}}_2.\end{array}$$

Now, in view of above observations, we give the following lemmas which can be proved on similar arguments of lemmas (4–8) given in [32].

Lemma 2

The interval $[{\mathfrak{L}}_1,{\mathfrak{L}}_2]$ contains all the fixed points of $\mathfrak{T}$ which lie in [0, S(0)].

Lemma 3

If ${\mathfrak{L}}_1<{\mathfrak{L}}_2$, then ${\mathfrak{L}}_1<<{\mathfrak{L}}_2$.

Lemma 4

For each $\nu >1,x>>0$, we have

$$\begin{array}{c}\hfill \mathfrak{T}(\nu x+{\mathfrak{L}}_1)-\mathfrak{T}({\mathfrak{L}}_1)>>\nu (\mathfrak{T}(x+{\mathfrak{L}}_1)-\mathfrak{T}({\mathfrak{L}}_1))\end{array}$$

with the differential of operator $\mathfrak{T}$, $\mathfrak{D}(\mathfrak{T}(x))$ is given by

$$\begin{array}{c}\hfill \mathfrak{D}(\mathfrak{T}(x))=\left[\begin{array}{cc}M{\mathfrak{T}}_1(x_1,x_2)& N{\mathfrak{T}}_1(x_1,x_2)\\ A{\mathfrak{T}}_2(x_1,x_2)& B{\mathfrak{T}}_2(x_1,x_2).\end{array}\right]\end{array}$$

Lemma 5

The operator $\mathfrak{T}$ has two equilibrium points at most in the interval [0, S(0)]. Moreover,

1. If ${\mathfrak{L}}_1={\mathfrak{L}}_2$, then $\mathfrak{T}$ has only one equilibrium point in the interval [0, S(0)].

2. If ${\mathfrak{L}}_1<<{\mathfrak{L}}_2$, then $\mathfrak{T}$ has exactly two equilibrium points ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$ in the interval [0, S(0)].

Lemma 6

The spectral radius of the matrices $\mathfrak{D}(\mathfrak{T}({\mathfrak{L}}_1))$ and $\mathfrak{D}(\mathfrak{T}({\mathfrak{L}}_2))$ satisfy the following inequality:

$$\begin{array}{c}\hfill r(\mathfrak{D}(\mathfrak{T}({\mathfrak{L}}_1)))<1<r(\mathfrak{D}(\mathfrak{T}({\mathfrak{L}}_2))).\end{array}$$

Now, by using of above lemmas and Theorem 9 in [32], it can be easily shown that the final epidemic size of model (1) under the assumptions that $S(0)=S_0\ge 0$ and $I(0)=I_0>0$, is given by

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}S(t)={\mathfrak{L}}_1,\underset{t\to +\infty }{lim}I(t)=0,\underset{t\to +\infty }{lim}R(t)={(N_1,N_2)}^T-{\mathfrak{L}}_1.\end{array}$$

Author contributions

SC and OPM conceived of the presented idea. MB developed the theory and performed the computations. SG and SC verified the analytical results. SC and OPM supervised the findings of this work. All authors read and approved the final manuscript.

Funding

No funding is available.

Data availibility

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Ethical approval

Not applicable

Consent to participate

Not applicable