Dynamics of the optimality control of transmission of infectious disease: a sensitivity analysis

Abstract

Over the course of history global population has witnessed deterioration of unprecedented scale caused by infectious transmission. The necessity to mitigate the infectious flow requires the launch of a well-directed and inclusive set of efforts. Motivated by the urge for continuous improvement in existing schemes, this article aims at the encapsulation of the dynamics of the spread of infectious diseases. The objectives are served by the launch of the infectious disease model. Moreover, an optimal control strategy is introduced to ensure the incorporation of the most feasible health interventions to reduce the number of infected individuals. The outcomes of the research are facilitated by stratifying the population into five compartments that are susceptible class, acute infected class, chronic infected class, recovered class, and vaccinated class. The optimal control strategy is formulated by incorporating specific control variables namely, awareness about medication, isolation, ventilation, vaccination rates, and quarantine level. The developed model is validated by proving the pivotal delicacies such as positivity, invariant region, reproduction number, stability, and sensitivity analysis. The legitimacy of the proposed model is delineated through the detailed sensitivity analysis along with the documentation of local and global features in a comprehensive manner. The maximum sensitivity index parameters are disease transmission and people moved from acute stages into chronic stages whose value is (0.439, 1) increase in parameter by 10 percent would increase the threshold quantity by (4.39, 1). Under the condition of a stable system, we witnessed an inverse relationship between susceptible class and time. Moreover, to assist the gain of the fundamental aim of this research, we take the control variables as time-dependent and obtain the optimal control strategy to minimize infected populations and to maximize the recovered population, simultaneously. The objectives are attained by the employment of the Pontryagin maximum principle. Furthermore, the efficacy of the usual health interventions such as quarantine, face mask usage, and hand sanitation are also noticed. The effectiveness of the suggested control plan is explained by using numerical evaluation. The advantages of the new strategy are highlighted in the article.

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Introduction

The capability of infectious diseases in disrupting daily life is well documented in the existing literature of multi-disciplinary nature. In recent years, more aggressive eruptions of the viral flow are witnessed¹. For example, the veracity of the more recent viral explosion of COVID-19 is unprecedented in modern history. To date, more than 6.8 million deaths have been reported globally due to the aforementioned virus². More disturbingly, the transmission of the SARS-CoV-2 virus was accompanied by existing pathogens such as Ebola, SARS, and ZIKA. Recently, the World Health Organization (WHO) announced the onset of the newest transmissible infection called Monkeypox (MPX), named after the host of the virus³. More alarmingly, many researchers have noticed and documented the novel biological alterations assisting the flow of viral transmission. For example, the case of COVID-19⁴ noted consistent alterations in the gene expression of many different immune cell types. Similarly,⁵ documented the presence of EBOLA virus in epidermal and seminal vesicular tubular epithelial cells. Furthermore,^6,7,8 has reported one case in Spain where an individual was detected with the co-presence of MPX, COVID-19, and HIV viruses. The scope and the coverage of research activities are mammoth while researchers from every corner of the globe are responding to the wake of infectious transmissions. A careful and tedious review of the available literature reveals that the ongoing research efforts regarding infectious diseases most commonly target three fronts such as, (i) - exploration of the clinical nature of the virus⁹, (ii) - mathematical encapsulation of the flow dynamics of the infection¹⁰, and (iii) - study of the socio-economic impacts of the upshots of viral flow¹¹. This research is aligned with the second stream of the ongoing multi-disciplinary efforts aiming at understanding the flow dynamics of infectious transmissions.

A considerable amount of time and energy is being dedicated to understanding the mathematical nature of infectious disease transmission. For example, Biswas, launched a Susceptible Exposed Infected Recovered (SEIR) model to explore the control of infectious disease when health interventions are employed¹². Similarly, Wangari (2015) studied the utility of the SEIR model to establish the conditions for global stability of the model when interest lies in the explanation of primary infection mechanisms¹³. Moreover, in adjacent past the consideration of optimal control strategies is gaining momentum to attain better management of the viral flow. The usage of optimal control is not restricted towards medical applications but covers numerous phenomena including, policy formulation¹⁴, emergency planning¹⁵, engineering hazard assessment¹⁶, and the development of more effective control program¹⁷. Fundamentally, an optimal control strategy is defined to assist the determining optimal protocols while dealing with complex multi-frontier problems. The resultant optimization problem is then persuaded by specifying control variables and profiling dynamics of the system under a set of given parameters, while maintaining the notion of parsimony¹⁸ and¹⁹. Many researchers have appreciated the utility of these methods to gain better control over the ongoing stochastic processes. For example, Zhilan Feng advocated the optimal control strategy in order to gain timely identification of the emergence of infectious diseases²⁰. Furthermore, Kar and Soovoojeet (2016)²¹ conducted a theoretical investigation into the mathematical modeling of infectious disease with the application of optimal control²¹. Moreover, Liddo provided an elaborative account of the applicability of optimal control in demonstrating the treatment of infectious diseases²². Additionally²³, Lennon-O- Naraigh and Aine Byrne studied about the Piece wise-constant optimal control strategies for controlling the outbreak of COVID-19 in the Irish population.

By using optimal control, Verma²⁴ proposed and analyzed a non-linear smoking model to prevent interaction between those who are smokers and those who are smoking quitters. A mathematical model was proposed by Verma et al.²⁵ to investigate the dynamics of coronavirus with lockdown effects as an epidemiological model. Moreover, the introduction of the optimal control strategy by²⁶ for COVID-19, Dengue, and HIV through a mathematical model represents interactions between these infectious diseases. Further,²⁷ aimed at the stability analysis of a co-circulation model for COVID-19, Dengue, and Zika, incorporating nonlinear incidence rates and vaccination strategies to enhance our understanding of the dynamic interactions among these infectious diseases. Many researchers have employed the optimal approach to model and predict the efficiency of health surveillance synergies. One may consult the account of²⁸ aimed at SARS-COV-2 and HBV co-dynamics modeling, and²⁹ explores the modeling of backward bifurcation and optimal control in a co-infection model for SARS-CoV-2 and ZIKV. Researchers have examined pioneer work in different areas of mathematical modeling, economics problems such as^30,31,32,33.

Stirred by the significance of the above-documented issue, this research contributes to the existing literature on the applicability of the optimal control theory by introducing a new mathematical formation capable of explaining the dynamics of the infectious disease flow in a community. While simultaneously creating a trade between the number of infectious individuals and the cost of the considered health interventions. The whole population is classified into five mutually inclusive classes namely, susceptible class, acute infected class, chronic infected class, recovered class, and vaccinated class. Firstly, we will thoroughly examine the developed model to ensure its validity by proving pivotal delicacies such as positivity, invariant region, equilibrium points, reproduction number, and stability analysis. Along with showing complete stability analyses at equilibrium points to evaluate both local and global stability. The rationale of the optimal control scheme remains to assert plausible control law facilitating the optimization of certain performance indices. Further, the research environment is enriched by the addition of relevant control parameters such as medication, isolation, ventilation, vaccination, and quarantine levels. The existence of optimal control is proved with Pontryagin's maximum principle (Pontryagin et al. 1962), and examine the optimal control problem to identify the required conditions for optimality. The parametric setting incorporates various scenarios.

This article is primarily alienated into six sections. Section 2 presents the qualitative analysis of the model, and Section 3 is dedicated to the sensitivity analysis of the infectious disease model, whereas, Section 4 reports the stability features of the considered model. Section 5 outlines the incorporation of optimal control with the focus of launching more feasible health interventions. Lastly, Section 6 concludes all the major findings of this study.

Model structure

Mathematical models of infectious diseases are crucial for comprehending the patterns of disease spread and devising effective disease control strategies. Consequently, when constructing such models, emphasis should be placed on delineating the epidemiological characteristics of the disease and identifying pivotal, modifiable parameters for disease control. Numerous epidemiological models, rooted in diverse disease transmission mechanisms, have been established and documented in the literature, see³⁴. These models have played a pivotal role in shaping and crafting control strategies for various diseases.

Let us consider that, for any time $t>0$, the vulnerable populations under the study, say N(t), is divided into five compartments such as Susceptible class ${\mathbb {S}}(t)$, Acute infected class $\mathbb { I}_1(t)$, Chronic infected class ${\mathbb {I}}_2(t)$, Recovered class ${\mathbb {R}}(t)$, and Vaccinated class ${\mathbb {V}}(t)$. The humans specified by ${\mathbb {S}}(t)$ are those who are at risk of contracting a specific infectious disease. Thus, the susceptible who are currently infected are moved into two classes namely Acute infected class ${\mathbb {I}}_1(t)$ and Chronic infected class ${\mathbb {I}}_2(t)$ means that if people have the time period of disease are less than six month they moved into acute infected class, if people have time period of disease beyond the six month they moved into chronic infected class, and those who are vaccinated at the rate $\sigma$ move to ${\mathbb {V}}(t)$. The vaccinated individuals may either move to Recovered class ${\mathbb {R}}(t)$ at the rate $\phi$ after receiving an infection as a result of having contact with an infectious person. Moreover, some parameters are also used in the model formulation, such as the u rate of vaccinated people, $\sigma$ rate of vaccinated individuals become susceptible again, $\beta$ disease transmission rate,$\gamma$ reduced the rate of the incubation period, $\rho _1$ and $\rho _2$ recovery rate from acute and chronic stages naturally, $\gamma _2$ and $\gamma _3$ recovery rate of the acute and chronic stage, d untreated death rate, $d_1$ disease-related death rate, $\phi$ a treated person could regain into susceptible since vaccination rate is not perfect and $\gamma _1$ people in the acute stage are moved into chronic stage.

The proposed model leverages some key assumptions grounded in biological evidence to illuminate infectious disease dynamics under optimal control strategies. One critical assumption is that the infectious disease system exhibits backward bifurcation, implying complex and non-linear interactions that contribute to sustained transmission. This assumption aligns with empirical observations in infectious disease ecology, where factors such as host heterogeneity and co-infection dynamics often lead to intricate and non-trivial system behavior. Moreover, the incorporation of optimal control strategies into the model reflects the imperative of intervention measures in disease management. Biological evidence supporting this assumption can be drawn from successful instances of interventions like vaccination campaigns and quarantine measures that have demonstrated efficacy in mitigating disease spread. While the model's novelty or modification status is not explicitly stated, its integration of biological evidence and optimal control strategies contributes to advancing our understanding of infectious disease dynamics and intervention planning³⁵.

Therefore, based on the aforementioned considerations, the interaction flow pattern between infectious disease states is depicted in Fig. 1.

Thus, the whole human population at any time t is given by

$$\begin{aligned} N(t) = {\mathbb {S}}(t)+ {\mathbb {I}}_1(t)+ {\mathbb {I}}_2(t)+ {\mathbb {R}}(t)+ {\mathbb {V}}(t). \end{aligned}$$

The infectious disease model is depicted through a non-linear system of ordinary differential equations while connecting considered compartments in deterministic formation such as.

$$\begin{aligned} \begin{aligned}{}&\frac{d{\mathbb {S}}}{dt}=(1-u)\alpha -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2){\mathbb {S}}-d{\mathbb {S}}+\phi {\mathbb {R}}+\sigma {\mathbb {V}}+\rho _1\gamma _2{\mathbb {I}}_1+\rho _2\gamma _3{\mathbb {I}}_2,\\&\frac{d{\mathbb {I}}_1}{dt}=\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2){\mathbb {S}}-(\gamma _1+\gamma _2+d){\mathbb {I}}_1,\\&\frac{d{\mathbb {I}}_2}{dt}=\gamma _1{\mathbb {I}}_1-(d+d_1+\gamma _3){\mathbb {I}}_2,\\&\frac{d{\mathbb {R}}}{dt}=(1-\rho _1) \gamma _2 {\mathbb {I}}_1+(1-\rho _2) \gamma _3 {\mathbb {I}}_2-(\phi +d) {\mathbb {R}},\\&\frac{d{\mathbb {V}}}{dt}=u\alpha -(\sigma +d) {\mathbb {V}}.\\ \end{aligned} \end{aligned}$$

(1)

The initial conditions are defined as follows,

$$\begin{aligned} {\mathbb {S}}(0)={\mathbb {S}}_0\ge 0,\,\, {\mathbb {I}} _1(0)={\mathbb {I}}_{1,0}\ge 0, \,\,{\mathbb {R}}(0)={\mathbb {R}}_0\ge 0,{\mathbb {I}} _2(0)={\mathbb {I}}_{2,0}\ge 0, \,\,{\mathbb {V}}(0)={\mathbb {V}}_0\ge 0.\\ \end{aligned}$$

Furthermore, the details of each term of the equations influencing the system given in the model (1) are documented in Table 1.

Table 1 Description of each term of the equations of the model (1).

Full size table

Qualitative analysis of the model

Here, in this section, some significant results of the proposed model, namely positivity, invariant region, equilibrium point, and basic reproduction number are given.

Positivity

Positivity is enforced upon the initial conditions or parameters inherent in the model. It ensures that the values of the initial conditions and the parameters in the proposed model are endured greater than zero or non-negative. Therefore, reorganize the model (1) in terms is explained by

$$\begin{aligned} \dot{\eta }(t) = G(\eta (t)), \end{aligned}$$

(2)

where$\,\,\,\eta (t)=(\eta _1,\eta _2,\eta _3,\eta _4,\eta _5)^T:=\big ({\mathbb {S}},{\mathbb {I}}_1, {\mathbb {I}}_2,{\mathbb {R}},{\mathbb {V}})^T, \eta (0)=({\mathbb {S}}(0),{\mathbb {I}}_1(0),\mathbb { I}_2(0),{\mathbb {R}}(0),{\mathbb {V}}(0)\big )^T\epsilon {R}^5_+$, and

$$\begin{aligned} {G}(\eta ) = \begin{pmatrix} {G}_1(\eta )\\ {G}_2(\eta )\\ {G}_3(\eta )\\ {G}_4(\eta )\\ {G}_5(\eta ) \end{pmatrix} = \begin{pmatrix} (1-u)\alpha -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2){\mathbb {S}}-d{\mathbb {S}}+\phi {\mathbb {R}}+\sigma {\mathbb {V}}+\rho _1\gamma _2{\mathbb {I}}_1+\rho _2\gamma _3{\mathbb {I}}_2\\ \beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2){\mathbb {S}}-(\gamma _1+\gamma _2+d){\mathbb {I}}_1\\ \gamma _1{\mathbb {I}}_1-(d+d_1+\gamma _3){\mathbb {I}}_2\\ (1-\rho _1) \gamma _2 {\mathbb {I}}_1+(1-\rho _2) \gamma _3 {\mathbb {I}}_2-(\phi +d) {\mathbb {R}}\\ u\alpha -(\sigma +d) {\mathbb {V}} \end{pmatrix}. \end{aligned}$$

(3)

The situation is stress-free ${G}_i(\eta )\vert _{\eta _i=0}\ge 0 ,i=1,...,5$. According to the well-known Nagumo³⁶ finding, the model (1) solution with an starting point $\eta _0\in R_+^{5}$, such as $\,\ \eta (t)=\eta (t;\eta _0),$ is such that $\eta (t)\in R_+^{5},\,\,\ \forall \,\ t>0$.

Invariant region

We determined the invariant region where the solution of the proposed model is bounded. Now, consider the total population of the given model as

$$\begin{aligned} N(t) = {\mathbb {S}}(t) + {\mathbb {I}}_1(t) + {\mathbb {I}}_2(t) +{\mathbb {R}}(t) +{\mathbb {V}}(t). \end{aligned}$$

(4)

Differentiate (4) respecting the solution of the model (1) provides

$$\begin{aligned} \frac{dN}{dt} = \frac{d{\mathbb {S}}}{dt} + \frac{d\mathbb { I}_1}{dt}+\frac{d{\mathbb {I}}_2}{dt} +\frac{d{\mathbb {R}}}{dt}+ \frac{d{\mathbb {V}}}{dt}. \end{aligned}$$

(5)

Therefore, by substituting all the state equations from the model (1) in Eq. (5), then one gets

$$\begin{aligned} \frac{dN}{dt} = \alpha - dN. \end{aligned}$$

(6)

Taking the integration on both side of (6) with $t \rightarrow \infty$, one obtain

$$\begin{aligned} \Omega = \biggl \{{\mathbb {S}},{\mathbb {I}}_1, \mathbb { I}_2,{\mathbb {R}},{\mathbb {V}} \epsilon R_{+}^{5}: {\mathbb {S}}+ \mathbb { I}_1+{\mathbb {I}}_2+{\mathbb {R}}+{\mathbb {V}} +\le \frac{\alpha }{d}\biggl \}. \end{aligned}$$

Clearly, $\Omega$ is positively invariant, inside which the model is considered to be epidemiological meaningful and mathematically well-posed.

Equilibrium points

Equilibrium points indicate a stable condition in the progression of the disease, where the occurrence of new infections is offset by the combined occurrences of recoveries and deaths. This section outlines the calculation of the disease-free equilibrium point and the endemic equilibrium point of the proposed system. The equilibrium points associated with the model (1) by solving $\frac{d{\mathbb {S}}}{dt} = \frac{d{\mathbb {I}}_1}{dt}= \frac{d{\mathbb {I}}_2}{dt} = \frac{d{\mathbb {R}}}{dt} = \frac{d{\mathbb {V}}}{dt} = 0$. Accordingly, the disease-free equilibrium point $F_0$, is attained by letting for ${\mathbb {I}}_1=0,\, {\mathbb {I}}_2=0,\, {\mathbb {R}}=0,\, {\mathbb {V}}=0$ such as,

$$\begin{aligned} F_0 =\big ({\mathbb {S}}_0,\,0,\,0,\,0,\,0\big )=\left( \frac{(1-u) \alpha }{\beta \gamma +d},\,0,\,0,\,0,\,0\right) , \end{aligned}$$

Now, the endemic equilibrium point is denoted as $F^*=\big ({\mathbb {S}}^*,\,{\mathbb {I}}_1^*,\,{\mathbb {I}}_2^*,\,{\mathbb {R}}^*,\,{\mathbb {V}}^*\big )$, where,

$$\begin{aligned} \begin{aligned}{}&{\mathbb {S}}^*=\frac{(1-u)d\xi +\phi \xi _1((1-\rho _1)\gamma _2{\mathbb {I}}_1^*+(1-\rho _2)\gamma _3{\mathbb {I}}_2^*)+\sigma u_1 \alpha \xi _1+\rho _2\gamma _3[(\phi +d)(\sigma +d)]}{\psi {\mathbb {I}}_2^*\xi (\beta {\mathbb {S}}-\eta )},\\&{\mathbb {I}}_1^*=\frac{ \beta \gamma {\mathbb {I}}_2 S}{\beta {\mathbb {S}}-(\gamma _1+\gamma _2+d)},\\&{\mathbb {I}}_2^*=\frac{\gamma _1 {\mathbb {I}}_1}{(d+d_1+\gamma _3)},\\&{\mathbb {R}}^*=\frac{(1-\rho _1)\gamma _2 {\mathbb {I}}_1+(1-\rho _2)\gamma _3 {\mathbb {I}}_2}{(\phi +d)},\\&{\mathbb {V}}^*=\frac{u \alpha }{\sigma +d}, \end{aligned} \end{aligned}$$

where $\xi =(\sigma +d)(\phi +d)(d+d_1+\gamma _3),\quad \xi _1=(\sigma +d)(d+d_1+\gamma _3),\,\,\,\text {and}\,\,\, \eta = \gamma _1+\gamma _2+d, \psi = \rho _1\gamma _1\beta \gamma .$

The basic reproduction number

The basic reproduction number $R_0$ is a prevailing metric for evaluating the contagiousness or transmissibility of a communicable disease. It signifies the average number of individuals that are infected by one infectious individual in a susceptible population. Biologically, it reflects the contagiousness and transmissibility of a pathogen. If $R_0>1$, it indicates that the disease has the potential to sustain transmission within the population, leading to an outbreak. Conversely, if $R_0<1$, the infection is unlikely to establish a self-sustaining chain of transmission, and it may eventually die out in the population. Moreover, the reproduction number is estimated by using the method of Darraish and Wathmough³⁷, and with the principle of the next-generation matrix, we have

$$\begin{aligned} F= \begin{pmatrix} -\beta {\mathbb {S}}&{} \beta \gamma {\mathbb {S}}\\ 0 &{}0\\ \end{pmatrix} \,\,\,\, \text {and}\,\,\,\, V= \begin{pmatrix} -(\gamma _1 + \gamma _2+d) &{}0\\ \gamma _1 &{}-(d+d_1+\gamma _3)\\ \end{pmatrix}. \end{aligned}$$

Therefore, the reproduction number $R_0$ for our proposed model (1) is the spectral radius of the next generation matrix $F V^{-1}$ and calculated as,

$$\begin{aligned} FV^{-1} = \begin{pmatrix} \Big (\frac{\beta {\mathbb {S}}}{\gamma _1+\gamma _2+d}- \frac{\beta {\mathbb {S}}\gamma \gamma _1}{(\gamma _1+\gamma _2+d)(d+d_1+\gamma _3)}\Big ) &{} \frac{\beta \gamma {\mathbb {S}}}{d+d_1+\gamma _3}\\ 0 &{} 0 \end{pmatrix}. \end{aligned}$$

(7)

Therefore, from (7), the reproduction number $R_0$ is calculated as,

$$\begin{aligned} R_0 = \frac{\beta \, {\mathbb {S}}_0\, \eta _2 + \beta \, {\mathbb {S}}_0\, \gamma \, \gamma _1}{\eta _1\, \eta _2}, \end{aligned}$$

(8)

where $\eta _1 = (\gamma _1+\gamma _2+d )$,  $\eta _2 = d + d_1 +\gamma _3$. The disease will spread in the population if the basic reproduction number $R_0 > 1$.

Sensitivity analysis

The degree of influence of input parameters over the dynamics of the infectious disease model is an asset through the launch of sensitivity analysis. The sensitivity indices of the infectious disease model given in (1) are deduced by using the approach of Chitnis et al.³⁸. We calculate the normalized forward sensitivity indices of a parameter x of $R_0$. Let

$$\begin{aligned} \Delta _x^{R_0} = \frac{\partial R_0}{\partial x} \frac{x}{R_0}. \end{aligned}$$

The estimated sensitivity projected in reproduction number with respect to various parameters is given as

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _\beta ^{R_0} = \frac{\partial R_0}{\partial \beta } \frac{\beta }{R_0} = 1, \ \, \, \, \, \, \, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\Delta _\gamma ^{R_0} = \frac{\partial R_0}{\partial \gamma } \frac{\gamma }{R_0} = 0.270,\\ \Delta _{\gamma _2}^{R_0} = \frac{\partial R_0}{\partial \gamma _2} \frac{\gamma _2}{R_0} = -0.5, \ \,\,\,\,\,\,\,\,\,\,\,\,\Delta _{\gamma _3}^{R_0} = \frac{\partial R_0}{\partial \gamma _3} \frac{\gamma _3}{R_0} = -0.876,\\ \Delta _d^{R_0} = \frac{\partial R_0}{\partial d} \frac{d}{R_0} = 0.185, \ \, \, \, \, \, \,\,\,\,\,\Delta _{d_1}^{R_0} = \frac{\partial R_0}{\partial d_1} \frac{d_1}{R_0} = -0.239,\\ \Delta _{\gamma _1}^{R_0} = \frac{\partial R_0}{\partial \gamma _1} \frac{\gamma _1}{R_0} = 0.439. \end{array}\right. } \end{aligned}$$

Table 2 Sensitivity indices and parameters of the reproduction number $R_0$.

Full size table

Table 2 comprehends the directional projection of the parameters involved in the considered model explaining the viral infectious disease transmission flow. The existence of a positive relationship is voted through the plus sign whereas the negative sign avows the showing of the negative relationship between parameters and the transmission rate. One may be noted that an increase in the reproduction number remains associated with an increase in parameters such as disease transmission rate, reduced incubation period, rate of transfer from the acute stage to the chronic stage, and untreated death rate. Whereas, the reproduction number is estimated to be negatively associated with parameters such as disease-related death rate, and recovery rates belonging to the acute stage and chronic stage.

Figure 2 shows the dynamical behavior of the sensitivity analysis of the basic reproduction number $R_0$, in the context of the disease transmission. Moreover, this analysis helps us to understand the significance of various factors that play a vital role in the transmission of infectious diseases. We plotted the reproduction number $R_0$ as a function of $\gamma _3, d_1, \gamma _1, \gamma _2, \beta$ and d. Figure 2a shows the effect of effectual disease-related death rate and recovery rate of chronic infected patients on the reproduction number, we can see that these two parameters may increase to $R_0$ greater than unity if the value of $\gamma _3$ remain up to greater than 1 and the recovery rate also remains greater than 1. The combined effect of transmission and recovery rate on $R_0$ is shown in Fig. 2b. As one parameter $\beta$ is directly proportional and $\gamma _1$ is inversely related to $R_0$, thus we can see that the basic reproduction number value may be reduced below unity even if the transmission rate $\beta$ approaches 1 and the recovery rate of acute and chronic infected increases. Figure 2c shows the effect of transmission rate $\beta$ and d. We can see that these two parameters have a considerable impact in reducing or increasing the value of the basic reproduction number $R_0$. Furthermore, Fig. 2d shows the effect of recovery rate $gamma_2$ and disease-related death rate d, and also Fig. 2e shows the inverse relationship between $R_0$ and parameters. Similarly, Fig. 2f shows the impact of reducing or increasing the value of reproduction number $R_0$.

Stability analysis

In this section, we analyze the local and global stabilities of the proposed model at equilibrium points. Local stability is often determined by examining the eigenvalues of the Jacobian matrix evaluated at the equilibrium point. If all eigenvalues have negative real parts, the equilibrium is locally stable. Global stability considers the system's behavior across its entire range, often requiring more advanced mathematical techniques such as Lyapunov analysis.

Local stability analysis

The aim of conducting local stability analysis is to determine whether a minor disturbance in the disease system would lead to the persistence or elimination of the disease. In the realm of epidemiological models, comprehending the concept of local stability at equilibrium points holds significance. It provides insights into the dynamics of infectious diseases and their transmission within a population. The endemic equilibrium point and disease-free equilibrium point, which are locally asymptotically stable, are illustrated in the following theorem.

Theorem 4.1

The disease-free equilibrium point $F_0$ is locally asymptotically stable when $R_0$ < 1, however, when $R_0>1$, it is unstable.

Proof

The Jacobin matrix of disease-free equilibrium point $F_0$ for the proposed model is given as

$$\begin{aligned} J(F_0)= \begin{pmatrix} -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2)-d &{}- \beta {\mathbb {S}}_0+\rho _1\gamma _2 &{} \beta \gamma {\mathbb {S}}_0+\rho _2 \gamma _3 &{} \phi &{}0\\ -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2) &{} \beta {\mathbb {S}}_0-(\gamma _1+\gamma _2+d) &{} \beta \gamma {\mathbb {S}}_0 &{} 0&{}0 \\ 0 &{} \gamma _1 &{} -(d + d_1 +\gamma _3)&{}0&{}0\\ 0 &{} (1-\rho _1)\gamma _2 &{} (1-\rho _2)\gamma _3 &{} -(\phi +d)&{}0 \\ 0&{}0&{}0&{}0&{}-(\sigma +d) \end{pmatrix}. \end{aligned}$$

(9)

The evaluation of the jacobian at drinking-free equilibrium provides

$$\begin{aligned} J(F_0)= \begin{pmatrix} -d &{}- \beta {\mathbb {S}}_0+\rho _1\gamma _2 &{} \beta \gamma {\mathbb {S}}_0+\rho _2 \gamma _3 &{} \phi &{}0\\ 0 &{} \beta {\mathbb {S}}_0-(\gamma _1+\gamma _2+d) &{} \beta \gamma {\mathbb {S}}_0 &{} 0&{}0 \\ 0 &{} \gamma _1 &{} -(d + d_1 +\gamma _3)&{}0&{}0\\ 0 &{} (1-\rho _1)\gamma _2 &{} (1-\rho _2)\gamma _3 &{} -(\phi +d)&{}0 \\ 0&{}0&{}0&{}0&{}-(\sigma +d) \end{pmatrix}. \end{aligned}$$

(10)

With the aid of the Jacobin matrix of the disease-free equilibrium point, the resultant eigenvalues are $\lambda _1 = -\beta (\mathbb { I}_1+\gamma {\mathbb {I}}_2)< 0, \lambda _2 = -(\sigma +d) < 0,$ whereas,

$$\begin{aligned} \begin{vmatrix} J(F_0)-\lambda I \end{vmatrix} = \begin{vmatrix} \beta {\mathbb {S}}_0-(\gamma _1+\gamma _2+d)-\lambda&\beta \gamma {\mathbb {S}}_0&0\\ \gamma _1&-(d+d_1+\gamma _3)-\lambda&0\\ (1-\rho _1)\gamma _2&(1-rho_2)\gamma _3&-(\phi +d)-\lambda \end{vmatrix}=0, \end{aligned}$$

$\lambda _4 =-(d+d_1+\gamma _3),\, \lambda _5 = -(\phi +d), \text {and} \, \lambda _3= \beta {\mathbb {S}}_0-(\gamma _1+\gamma _2+d) <0$. $\lambda _1, \lambda _2, \lambda _4, \text {and} \lambda _5$ are real and negative. Further, $\lambda _3$ is negative and real if $R_0 < 1$. Using the Routh-Hurwitz criterion³⁹, it is verified that, each eigenvalue of the polynomial equation contains a non-positive real parts when $R_0<1$. As a result, the disease-free equilibrium point $F_0$ is locally asymptotically stable. $\square$

Theorem 4.2

The endemic equilibrium point $F^*$ will be locally asymptotically stable if the value of $R_0$ > 1, however, when $R_0 < 1$, it is unstable.

Proof

The determination of the Linearization for model (1) at the endemic equilibrium point $F^*$ is carried out by

$$\begin{aligned} J(F^*)= \begin{pmatrix} -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2)-d &{}- \beta {\mathbb {S}}+\rho _1\gamma _2 &{} \beta \gamma {\mathbb {S}}+\rho _2 \gamma _3 &{} \phi &{}0\\ -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2) &{} \beta {\mathbb {S}}-(\gamma _1+\gamma _2+d) &{} \beta \gamma {\mathbb {S}} &{} 0&{}0 \\ 0 &{} \gamma _1 &{} -(d + d_1 +\gamma _3)&{}0&{}0\\ 0 &{} (1-\rho _1)\gamma _2 &{} (1-\rho _2)\gamma _3 &{} -(\phi +d)&{}0 \\ 0&{}0&{}0&{}0&{}-(\sigma +d) \end{pmatrix}. \end{aligned}$$

By using row matrix transformation the following matrix is given by:

$$\begin{aligned} J(F^*)= \begin{pmatrix} -\mu &{}- 0 &{} 0 &{} 0 &{}0\\ -\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2) &{} -\mu _1 &{} 0 &{} 0&{}0 \\ 0 &{} \gamma _1 &{} -\mu _2&{}0&{}0\\ 0 &{} (1-\rho _1)\gamma _2 &{} (1-\rho _2)\gamma _3 &{}-\mu _3 &{}0 \\ 0&{}0&{}0&{}0&{}-\mu _4&{} \end{pmatrix}. \end{aligned}$$

(11)

where, let $\mu =\beta ({\mathbb {I}}_1+\gamma {\mathbb {I}}_2)-d, \ \,\, \mu _1=(\gamma _1+\gamma _2+d), \ \,\, \mu _2 = (d + d_1 +\gamma _3), \ \,\, \mu _3 =(\phi +d), \ \,\,\text {and}\ \,\, \mu _4=(\sigma +d).$ It is clear that all of eigenvalues values of $J(F^*)$ have negative and real if $R_0 > 1$. $\square$

Global stability analysis

Global stability in a disease model examines whether, over the entire range of conditions, the system converges to a stable state in terms of disease behavior. It is crucial for understanding the persistent, long-term dynami...

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