Int. J. Anal. Appl. (2022), 20:57 Received: Sep. 28, 2022. 2010 Mathematics Subject Classification. 34D20, 92D30, 93D05, 97M60. Key words and phrases. basic reproduction number (R0); dengue fever; Lyapunov; numerical simulation; sensitivity; stability. https://doi.org/10.28924/2291-8639-20-2022-57 © 2022 the author(s) ISSN: 2291-8639 1 Analysis of Vector-host SEIR-SEI Dengue Epidemiological Model Md Rifat Hasan1,2,*, Aatef Hobiny1, Ahmed Alshehri1 1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2Department of Applied Mathematics, Faculty of Science, Noakhali Science and Technology University, Noakhali 3814, Bangladesh *Corresponding author: rifatmathdu@gmail.com ABSTRACT. Approximately worldwide 50 nations are still infected with the deadly dengue virus. This mosquito-borne illness spreads rapidly. Epidemiological models can provide fundamental recommendations for public health professionals, allowing them to analyze variables impacting disease prevention and control efforts. In this paper, we present a host-vector mathematical model that depicts the Dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (SEIR) model for the human interacting with a susceptible-exposed-infected (SEI) model for the mosquito. Using the Next Generation Technique, the basic reproduction number of the model is calculated. The local stability shows that if R0<1 the system is asymptotically stable and the disease dies out, otherwise unstable. The Lyapunov function is also used to evaluate the global stability of disease-free and endemic equilibrium points. To analyze the effect of the crucial aspects of the disease's transmission and to validate the analytical findings, numerical simulations of a variety of compartments have been constructed using MATLAB. The sensitivity analysis of the epidemic model is performed to establish the relative significance of the model parameters to disease transmission. 1. Introduction Dengue fever incidence has risen dramatically in the recent two decades [1]. Approximately half of the earth’s population may be in danger of contracting one of the nearly 390 million new infections that are thought to arise annually. Dengue viral disease is transmitted from mosquitoes https://doi.org/10.28924/2291-8639-20-2022-57 2 Int. J. Anal. Appl. (2022), 20:57 to humans by mosquitoes, spreading rapidly expanding over the world by its four serotypes. Prior to relatively recently, the female Aedes aegypti mosquito which identifies as a primary vector for dengue was mostly found in tropical and subtropical regions [2]. Due to the highly adapted ability in urban regions of the Aedes aegypti mosquito, dengue spread all over the world. However, the full extent of the disease's effects remains unclear, and new monitoring strategies are needed, according to issues with underreporting and case misidentification. Other arboviruses, chikungunya as well as zika have lately emerged, posing additional issues for surveillance and management, particularly in South Asia [3]. A wide range of factors (including people, mosquitoes, and the virus) interact with one another to spread the dengue virus in a diverse environment [4]. Studies on dengue transmission face several obstacles due to the space's inherent complexity. A variety of causes are related to the current epidemic. These included worldwide host and vector mobility (which accelerated viral circulation), urban congestion (which encouraged various transmissions through a single infected vector), and the loss of previously effective vector control measures. Temperature, precipitation, and humidity all affect vector development at all stages, from egg viability through adult longevity and dispersion, among other aspects of dengue transmission. Unplanned development, high inhabitants’ density, and the instability of rubbish collection all of which support the growth of mosquito breeding sites, which lead to increasing dengue occurrence. In recent years, epidemiology research and disease control have benefited greatly from the use of mathematical modeling. To understand the disease's nature as well as taking appropriate decisions regarding disease management strategies/interventions and processes, mathematical modeling has become a useful tool. Many scholars studied a deterministic model to study the influence of numerous biological parameters on disease dynamics. Prasad et al. [5] studied a systematic review of deterministic mathematical models for vector-borne viral infections. Bhuju et al. [6] described the fuzzy epidemic SEIR-SEI compartmental model with bed nets and fumigation intervention to simulate the transmission dynamics of dengue disease. Tay et al. [7] constructed a transmission model of SI-SIR dengue epidemiological characteristics model to control dengue in Malaysia. Abidemi et al. [8] analyzed the effect of single vaccine usage and its combination with treatment and adulticide measures on dengue population dynamics in Johor, Malaysia. Aleixo et al. [9] gave a clear explanation of a machine learning model that is used to predict the frequency of dengue outbreaks in Rio de Janeiro. Sow et al. [10] developed a computational Zika Dynamics model to examine the effects of vertical transmission between the 3 Int. J. Anal. Appl. (2022), 20:57 vector population and the host population. Sweilam et al. [11] introduced a unique variable-order nonlinear model of the dengue virus that minimizes intervention dosage and duration through optimum bang-bang management. Abidemi et al. [12] developed and analyzed a two-strain deterministic dengue model based on the SIR modeling framework for the spread of the disease and its management in an area with two coexisting dengue virus serotypes. Asamoah et al. [13] investigated an ideal dengue infection control model with partly immune and asymptomatic patients. Linda et al. [14] examined the discrete-time versions of the SIS and SIR models that are stochastic in nature. To assess the influence of raising awareness through the press on the spread of vector-borne illnesses, a non-linear mathematical model was suggested by Misra et al. [15]. The dynamic SIR model with climatic parameters was discussed by Nur et al. [16] for the features of dengue disease transmission in a closed community. 2. Dengue Transmission Model The Ross-Macdonald model, which was first designed for malaria, is a classic mathematical model for vector-borne illnesses that monitors infections in humans as well as mosquitos. In this research, we present a compartmental host-vector mathematical model [17] that depicts the Dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (SEIR) model for the human interacting with a susceptible-exposed-infected (SEI) model for the mosquito. The host-vector mathematical model categorizes the overall human (host) population into four classes: susceptible (𝑆ℎ), exposed (𝐸ℎ), infectious (𝐼ℎ), and recovered (𝑅ℎ), whereas the mosquito (vector) population is divided into three classes: susceptible (𝑆𝑚), exposed (𝐸𝑚), and infectious (𝐼𝑚). Thus, the total human(host) population denoted by 𝑁ℎ is given as Nh(t)=Sh(t)+Eh(t)+Ih(t)+Rh(t) And total mosquitoes’(vector) population is given by: Nm(t)=Sm(t)+Em(t)+Im(t) 4 Int. J. Anal. Appl. (2022), 20:57 Fig 1. Dengue Virus Transmission Dynamics in Different Population Stages In our suggested model, we attempt to provide a fresh direction by taking panic, tension, or anxiety into account in the susceptible, exposed, and infected classes to host population. The influence of panic as well as stress, or anxiety on these clusters is discussed in this work. Panic, stress, and anxiety are all harmful to human’s health. Anxiety may raise insulin levels, which can have an impact on heart health, diabetes, and blood pressure. At the same time, stress can have a negative impact on human immune system. Extreme stress can impair immunity as well as chronic stress might jeopardize a major health condition. People suffering from panic attacks are more likely to get infected, and the death rate among infected people rises. Therefore, we anticipate that the amount of susceptible, exposed, and infected, is decreasing, i.e., moving to death due to panic, stress, or anxiety. Figure 1 depicts the suggested model's flow diagram as well as the nonlinear system of differential equations that represents the dynamics of host-vector dengue disease, which is represented by: Human population (h) dSh dt =Λ1-β1ShIm-β2ShIh-μ1Sh-α1Sh dEh dt =β2ShIh-μ1Eh-α1Eh dIh dt =β1ShIm-β3Ih-μ1Ih-α1Ih dRh dt =β3Ih-μ1Rh 5 Int. J. Anal. Appl. (2022), 20:57 Vector population (𝑚) dSm dt =Λ2-β4SmIh-μ2Sm dEm dt =β4SmIh-β5Em-μ2Em dIm dt =β5Em-μ2Im (1) with the initial conditions Sh(0)≥0,Eh(0)≥0,Ih(0)≥0,Rh(0)≥0,Sm(0)≥0,Em(0)≥0, and Im(0)≥0, where the biological descriptions of parameters is presented in Table 1. Table 1. Values for baseline parameters with definitions and biological descriptions of Dengue model Parameter Biological descriptions 𝜦𝟏 Recruitment rates of human population 𝜷𝟏 Infectious rate from vector to host 𝜷𝟐 Infectious rate within host 𝝁𝟏 Humna’s natural death rate 𝜶𝟏 Panic/tension/anxiety rate of human 𝜷𝟑 Recovery rate of infected human 𝚲𝟐 Recruitment rates of vector population 𝜷𝟒 Infection rate from human to vector 𝜷𝟓 Extrinsic incubation of vector 𝝁𝟐 Natural death rate of vector population 3. Positivity and boundedness of solutions The positivity and boundedness of the solutions are crucial features of an epidemiological model. As a result, it is critical to demonstrate that all variables are non-negative for all time 𝑡 ≥ 0, implying that any solution with positive beginning values will remain positive for all time 𝑡 ≥ 0. So, positivity indicates that the population will survive for a long period. The dynamical model of the transmission shall be investigated into the biologically feasible regions Θ ⊂ ℝ7 +, Such that Θ={Sh(t),Eh(t),Ih(t),Rh(t),Sm(t),Em(t), Im(t)ϵR7 +:Nh(t)≤ Λ1 μ1 ,Nm(t)≤ Λ2 μ2 } 6 Int. J. Anal. Appl. (2022), 20:57 Theorem 3.1. The feasible region is positively invariant for the model (1) with the initial condition defined by Θ⊂R7 + . Proof. Let Nh(t)=Sh(t)+Eh(t)+Ih(t)+Rh(t), then dNh(t) dt = dSh(t) dt + dEh(t) dt + dIh(t) dt + dRh(t) dt Hence dNh(t) dt =Λ1-μ1Nh(t)-α1Sh-α1Eh-α1Ih dNh(t) dt ≤Λ1-μ1Nh(t) dNh(t) dt +μ1Nh(t)≤Λ1 Nh(t)≤ Λ1 μ1 Nh(0)e -μt Thus, 𝑁ℎ(𝑡) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in Θ with starting conditions. Again, Nm(t)=Sm(t)+Em(t)+Im(t), then dNm(t) dt = dSm(t) dt + dEm(t) dt + dIm(t) dt Hence dNm(t) dt =Λ2-μ2Nm(t) dNm(t) dt ≤Λ2-μ2Nm(t) dNm(t) dt +μ2Nm(t)≤Λ2 Nm(t)≤ Λ2 μ2 Nm(0)e -μt Thus, Nm(t) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in Θ with starting conditions. 7 Int. J. Anal. Appl. (2022), 20:57 Therefore, the feasible region Θ is positively invariant, attracting all solutions in ℝ7 +. Theorem 3.2. The solution of the system (1) is positive and bounded for all Sh(t),Eh(t),Ih(t), Rh(t),Sm(t),Em(t), Im(t)ϵR7 + . for all 𝑡 > 0 . Proof. To demonstrate the solution's positivity, we need to show that on any hyperplane enclosing the positive vector space ℝ7 + from the system (1), we have dSh dt | Sh=0 =Λ1≥0 dEh dt | Eh=0 =β2ShIh≥0 dIh dt | Ih=0 =β1ShIm≥0 dRh dt | Rh=0 =β3Ih≥0 dSm dt | m=0 =Λ2≥0 dEm dt | Em=0 =β4SmIh≥0 dIm dt | m=0 =β5Em≥0 So, the system (1) solution is positive. 4. Qualitative analysis of model In this section qualitative analysis of the dengue system (1) by calculating disease free equilibrium (DFE) and the endemic equilibrium (EE) with help of basic reproduction number (E0). 4.1. Disease-free equilibrium To calculate the disease-free equilibrium (DFE) E0 of the dengue system (1), we set the right-hand side of equals to zero and obtain the following expression’s E0=(Sh 0 ,Eh 0,Ih 0,Rh 0,Sm 0 ,Em 0 ,Im 0 )=( Λ1 μ1+α1 ,0,0,0, Λ2 μ2 ,0,0) 8 Int. J. Anal. Appl. (2022), 20:57 4.2. Basic reproduction number For the purpose of assessing an infectious disease, a crucial threshold parameter is the basic reproduction number R0. It decides whether the disease will disappear or stay in the community throughout time. R0 is the secondary infections number which may be caused by a single primary infection whereas the population is susceptible. Assume R0 > 1, and one primary infection can generate in several secondary infections. Therefore, the disease-free equilibrium (DFE) is unstable, also an epidemic occurs. The reproduction number for the Dengue system is calculated utilizing the next generation matrix approach [18]. We look at the F* and V* matrices, are designated for the new infections’ development and classified migration of infective partitions. F*=( β2ShIh β1ShIm β4SmIh 0 ) V*= ( (μ1+α1)Eh (β 3 +μ1+α1)Ih (β5+μ2)Em -β5Em+μ2Im ) The Jacobian are calculated by taking the partial derivatives of F and V at DFE point E0 and are as follows: F=( 0 β2Sh 0 0 0 0 0 β1Sh 0 β4Sm 0 0 0 0 0 0 ) V= ( μ1+α1 0 0 0 0 β3+μ1+α1 0 0 0 0 β5+μ2 0 0 0 -β5 μ2) The basic reproduction of the dengue system is calculated by using the spectral radius of the matrix 𝑅0 = 𝜌(𝐹𝑉 −1), which is provided by the following equation R 0 2 = β1β4β5Λ1Λ2 μ2 2(μ1+α1)(β5+μ2)(β3+μ1+α1) 4.3. Endemic equilibrium The endemic equilibrium point of the dengue dynamical system (1) 9 Int. J. Anal. Appl. (2022), 20:57 E1=(Sh * ,Eh *,Ih *,Rh *,Sm * ,Em * ,Im * ) Where, Sh * = Λ1 β1Im+β2Ih+μ1+α1 , Eh *= β2ShIh β5+μ2 , Ih *= β1ShIm β3+μ1+α1 Rh *= β3 μ1 ,Sm * = Λ2 β4Ih+μ2 , Em * = β4SmIh β5+μ2 , Im * = β5Em μ2 5. Stability analysis The stability study of disease-free and endemic equilibrium is performed in this part. The basic reproduction number (R0) is used to observe the equilibrium point’s stability of the local as well as global. The Jacobian matrix which gives the eigenvalues can be used to do stability analysis. 5.1. Local stability around equilibrium point Theorem 5.1. For 𝑅0 < 1 the disease-free equilibrium (E0) of the system (1) is locally asymptotically stable and unstable if 𝑅0 > 1. Proof. The Jacobian matrix of the model (1) at the disease-free equilibrium point (E0) is J(E0)= ( -(μ1+α1) 0 -β2Sh 0 0 0 0 -β1Sh 0 0 -(μ1+α1) β2Sh 0 0 0 0 0 0 0 -(β 3 +μ1+α1) 0 0 0 β1Sh 0 0 0 β3 -μ1 0 0 0 0 0 -β4Sm 0 0 -μ2 0 0 0 0 β4Sm 0 0 0 -(β 5 +μ2) 0 0 0 0 0 0 β5 -μ2 ) The four eigenvalues of 𝐽(𝐸0) at the disease- free equilibrium are -μ1, -μ2, -(μ1+α1)(multiplicity 2) and the remaining eigenvalues are given by the following cubic equation λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2-β1β4β5Sh 0 S m 0 =0 Where, v1=β3+μ1+α1 10 Int. J. Anal. Appl. (2022), 20:57 v2=β5+μ2 Now, λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2 (1- β1β4β5Sh 0 S m 0 v1v2μ2 )=0 λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2(1-R0 2 )=0 Here, (v1+v2+μ2)>0 (v 1 v 2 +v1μ2+v2μ2)>0 v1v2μ2(1-R0 2 )>0, if R0<1 And (v1+v2+μ2)(v1v2 +v1μ2+v2μ2)>v1v2μ2(1-R0 2 ) (Since (v1+v2+μ2)(v1v2 +v1μ2+v2μ2) v1v2μ2 >9>(1-R0 2 )) Therefore, if 𝑅0 < 1, all of the preceding requirements are satisfied. As a result, the disease-free equilibrium point 𝐸0 is locally asymptotically stable according to the Routh-Hurwitz criteria; otherwise, it is unstable. 5.2. Global stability around equilibrium point In this segment, we will evaluate equilibrium points E0 and E1 stability. The next two theorems show the results of the stability analysis of these equilibrium sites. Theorem 5.2. If 𝑅0 < 1, the disease-free equilibrium (E0) is globally asymptotically stable. Proof. We consider the Lyapunov function of the form in G(t)=(Sh-Sh 0 lnSh)+Eh+Ih+Rh+(Sm-Sm 0 ln Sm)+Em+Im Differentiating w.r.t t, we get: G'(t)=(1- Sh 0 Sh ) Sh ' +Eh ' +Ih ' +Rh ' +(1- Sm 0 Sm )Sm ' +Em ' +Im ' 11 Int. J. Anal. Appl. (2022), 20:57 G '(t)= (1- Sh 0 Sh )(Λ1-β1ShIm-β2ShIh-μ1Sh-α1Sh)+β2ShIh-μ1Eh-α1Eh +β1ShIm-β3Ih-μ1Ih-α1Ih+β3Ih-μ1Rh+(1- Sm 0 Sm )(Λ2-β4SmIh-μ2Sm) +β4SmIh-β5Em-μ2Em+β5Em-μ2Im On solving further get: =(1- Sh 0 Sh )Λ1+(μ1+α1)(1- Sh Sh 0 ) Sh 0 +β1Sh 0 Im+β2Sh 0 Ih-μ1Eh-α1Eh -μ1Ih-α1Ih-μ1Rh+(1- Sm 0 Sm )Λ2+μ2 (1- Sm Sm 0 )Sm 0 +β4Sm 0 Ih-μ2Em-μ2Im Using the equilibrium condition (μ1+α1)Sh 0 =Λ1 and μ2Sm 0 =Λ2 into the above equation G '(t)=(2- Sh 0 Sh - Sh Sh 0 )Λ1+(2- Sm 0 Sm - Sm Sm 0 )Λ2-Im(μ2-β1Sh 0 )-Ih(μ1+α1-β2Sh 0 -β4Sm 0 ) -μ1Eh-α1Eh-μ1Rh-μ2Em =-Λ1 (Sh-Sh 0 ) 2 ShSh 0 -Λ2 (Sm-Sm 0 ) 2 SmSm 0 -μ1Eh-α1Eh-Ih(μ1+α1-β2Sh 0 -β4Sm 0 ) -μ1Rh-μ2Em-Im(μ2-β1Sh 0 ) The above equation shows that 𝐺′(𝑡) ≤ 0 𝑎𝑛𝑑 G '(t)=0 for Sh=Sh 0 , Eh=0, Ih=0, Rh=0,Sm=Sm 0 , Em=0, Im=0. So, the largest invariance set is the singleton set {𝐸0}. Therefore, by using the principle of LaSalle’s invariance the disease-free equilibrium (𝐸0) is globally asymptotically stable. Theorem 5.3. If 𝑅0 > 1, the endemic equilibrium (E1) is globally asymptotically stable. Proof. We consider the Lyapunov function of the form in W(t)= 1 2 (Sh-Sh * ) 2 + 1 2 (Eh-Eh * ) 2 + 1 2 (Ih-Ih * ) 2 + 1 2 (Rh-Rh * ) 2 + 1 2 (Sm-Sm * ) 2 + 1 2 (Em-Em * ) 2 + 1 2 (Im-Im * ) 2 Differentiating with respect to time t, we get: W'(t)=(Sh-Sh * )Sh ' +(Eh-Eh * )Eh ' +(Ih-Ih * )Ih ' +(Rh-Rh * )Rh ' +(Sm-Sm * )Sm ' +(Em-Em * )Em ' +(Im-Im * )Im ' 12 Int. J. Anal. Appl. (2022), 20:57 =(Sh-Sh * )(Λ1-β1ShIm-β2ShIh-μ1Sh-α1Sh)+(Eh-Eh * )(β2ShIh-μ1Eh-α1Eh) +(Ih-Ih * )(β1ShIm-β3Ih-μ1Ih-α1Ih)+(Rh-Rh * )(β3Ih-μ1Rh)+(Sm-Sm * )(Λ2-β4SmIh-μ2Sm) +(Em-Em * )(β4SmIh-β5Em-μ2Em)+(Im-Im * )(β5Em-μ2Im) Using the equilibrium conditions Λ1=μ1Sh * +μ1Eh *+μ1Ih *+μ1Rh *+α1Sh * +α1Eh *+α1Ih * and Λ2=μ2Sm * +μ2Em * +μ2Im * into the above equation W'(t)=(Sh-Sh * )(μ1Sh * +μ1Eh *+μ1Ih *+μ1Rh *+α1Sh * +α1Eh *+α1Ih * -β1ShIm-β2ShIh-μ1Sh-α1Sh) +(Eh-Eh * )(β2ShIh-μ1Eh-α1Eh)+(Ih-Ih * )(β1ShIm-β3Ih-μ1Ih-α1Ih)+(Rh-Rh * )(β3Ih-μ1Rh) +(Sm-Sm * )(μ2Sm * +μ2Em * +μ2Im * -β4SmIh-μ2Sm)+(Em-Em * )(β4SmIh-β5Em-μ2Em) +(Im-Im * )(β5Em-μ2Im) =-μ1 (Sh-Sh * ) 2 -α1 (Sh-Sh * ) 2 +(μ1+α1)Eh * (Sh-Sh * )+(μ1+α1)Ih * (Sh-Sh * ) -β1ShIm (Sh-Sh * )-β2ShIh (Sh-Sh * )+β2ShIh(Eh-Eh * )-(μ1+α1)Eh * (Eh-Eh * ) +β1ShIm(Ih-Ih * )-(β3+μ1+α1)Ih(Ih-Ih * )+β3Ih(Rh-Rh * )-μ1Rh(Rh-Rh * ) -μ2 (Sm-Sm * ) 2 +μ2Em * (Sm-Sm * )+μ2Im * (Sm-Sm * )-β4SmIh (Sm-Sm * ) +β4SmIh(Em-Em * )-(Em-Em * )Em(β5+μ2)-β4SmIh (Sm-Sm * ) = -μ1 (Sh-Sh * ) 2 -α1 (Sh-Sh * ) 2 -(μ1+α1){Eh * (Eh-Eh * )-Eh * (Sh-Sh * )} -(μ1+α1){Ih(Ih-Ih * )-Ih * (Sh-Sh * )-β1ShIm (Sh-Sh * -Ih+Ih *) -β2ShIh (Sh-Sh * -Eh+Eh *)-β3Ih(Ih-Ih *-Rh+Rh * )-μ1Rh(Rh-Rh * )-μ2 (Sm-Sm * ) 2 -μ2 {(Em-Em * )Em-Em * (Sm-Sm * )} -μ2 {(Im-Im * )Im-Im * (Sm-Sm * )} -β4SmIh (Sm-Sm * -Em+Em * ) -β5Em(Em-Em * -Im+Im * ) 13 Int. J. Anal. Appl. (2022), 20:57 The above equation shows that 𝑊′(𝑡) ≤ 0 𝑎𝑛𝑑 W'(t)=0 for Sh=Sh * , Eh=Eh *, Ih=Ih *, Rh=Ih *,Sm=Sm * , Em=Em * , Im=Im * . So, the largest invariance set is the singleton set {𝐸1}. Therefore, by using the principle of LaSalle’s invariance the endemic equilibrium 𝐸1 is globally asymptotically stable. 6. Sensitivity analysis of the system Sensitivity analysis identifies the most effective model parameters that have effects on the Dengue model system's basic reproduction number [19]. Epidemiologists may forecast the important factors that play a significant part in virus-spreading dynamics using such analyses [20]. To avoid or manage the disease's effect, we must first identify the sensitivity induce values, which will give us an idea of which parameters for model would be maintained or monitored. In the current context, Dengue virus infection is spreading globally at a rapid rate, and this hazardous virus poses a serious threat to the human population. To inhibit the transmission of infection, we must first discover which model parameters are critical to disease transmission. To detect model's such parameters, we must need to evaluate the basic reproduction number variation which depends upon the model parameters; Alternatively, we must compute the normalized forward sensitivity index of the basic reproduction number 𝑅0 with respect to various parameters of the model. Our goal here is to approximate important model parameters that govern the basic reproduction number 𝑅0. To analyze the sensitivity, we utilize the normalized forward sensitivity index of the basic reproduction number 𝑅0 with regard to the system (1) parameter 𝜌, which is signified by ΓR0 ρ = ∂R0 ∂ρ . ρ R0 Table 2: Sensitivity indices of R0 evaluated at the baseline parameter values of the model. Parameter Sensitivity index 𝚲𝟏 +0.5 𝚲𝟐 +0.5 𝜷𝟏 +0.5 𝝁𝟏 -0.4859780904 𝜶𝟏 -0.0172362471 𝜷𝟑 -0.4967856625 𝜷𝟒 +0.5 𝜷𝟓 +0.1412037037 𝝁𝟐 -1.1412037037 14 Int. J. Anal. Appl. (2022), 20:57 On basic reproduction number 𝑅0, the parameters higher sensitivity index indicates the more influence sensitive parameter. The system parameter's sensitivity index with positive sign suggests that the basic reproduction number 𝑅0 increases when the parameter increases, and vice versa. In Table 2 and fig. 2, we applied a sensitive index to R0 in relation to each parameter. According to our research, the most important model parameters are recruitment rates of human population (𝛬1), recruitment rates of vector population (𝛬2), rate of infectious from vector to host ( 𝛽1), natural death rate of human ( 𝜇1), panic/tension/anxiety rate of human ( 𝛼1), recovery rate of infected human ( 𝛽3), infection rate from human to vector ( 𝛽4), extrinsic incubation of vector ( 𝛽5), and natural death rate of vector population ( 𝜇2). The most significant sensitivity index of the system is the natural death rate of vector population 𝜇2. Fig 2. Sensitivity indices of R0 7. Numerical results In the numerical part, the suggested model simulation is performed with the assist of MATLAB software. For numerical simulation, the parameter for the system (1) are given in Table 3. Table 3. System (1) parameters values Parameter Values Units Reference 𝚲𝟏 .9999 day -1 [21] 𝜷𝟏 .8500 day -1 [22] 𝜷𝟐 .6794 day -1 [23] 𝝁𝟏 .003468 day -1 [24] 𝜶𝟏 .000123 day -1 assumed 𝜷𝟑 .5555 day -1 [22] 𝚲𝟐 .0034 day -1 [25] 𝜷𝟒 .7186 day -1 [26] 𝜷𝟓 .0062 day -1 [8] 𝝁𝟐 .000244 day -1 [25] 15 Int. J. Anal. Appl. (2022), 20:57 Figs. 3–6 depict the dynamical system simulation exhibiting the influences of numerous parameters on the transmission dynamics model, demonstrating how parameters are efficient in inducing epidemics on different human populations as well as vector populations. Figs. 3-4 depict the performance of a susceptible and infected host population, as well as the infectious rate from vector to host (𝛽1) and the rate of panic/tension/anxiety in humans (𝛼1). Fig. 3(A) describes that there is no effects on different values of 𝛽1 between the early stage 0 to 5 days. Also, it indicates that the susceptible humans decrease with an increase in transmission the infectious rate from vector to host (𝛽1) and vise-verse. In fig. 3(B) indicates that panic/tension/anxiety rate in humans (𝛼1) has an impact between 5 to 75 days. Fig 3. (A) Susceptible host population 𝑆ℎ with different values of 𝛽1. (B) Susceptible host population 𝑆ℎ with different values of 𝛼1. It is interesting to see that in fig. 4 the infected host population decreases 0 to 5 days, after that it grows exponentially. When the interaction between vector to host increases the infected host increases in fig. 4(A), whereas in fig. 4(B) panic/tension/anxiety rate has the opposite effects. Fig 4. (A) Infected host population 𝐼ℎ with different values of 𝛽1. (B) Infected host population 𝐼ℎ with different values of 𝛼1. 16 Int. J. Anal. Appl. (2022), 20:57 Infection rates from humans to vectors and extrinsic incubation of the vector are shown in figures 5–6 together with the behavior of a susceptible and infected vector population. In between the first 40 days, the susceptible vector grows after that it decreases. Fig. 5(A) shows when the infection rates from humans to vectors increases to 10%, the susceptible vectors slightly down to the original. The effects on extrinsic incubation of the vector in fig. 5(B) shows after the 40 days. Fig 5. (A) Susceptible vector population 𝑆𝑚 with different values of 𝛽4. (B) Susceptible vector population 𝑆𝑚 with different values of 𝛽5. Figure 6 shows the fluctuation of the infected vector with respect to time t for various values of 𝛽4 and 𝛽5. It is obvious that as the value of grows, so does the infected vector. Fig. 6(A) demonstrates the slightly deviation of different 𝛽4 to original, whereas deviation between different 𝛽5 to original is high. Fig 6. (A) Infected vector population 𝐼𝑚 with different values of 𝛽4. (B) Infected vector population 𝐼𝑚 with different values of 𝛽5. 17 Int. J. Anal. Appl. (2022), 20:57 8. Conclusion The epidemic vector-borne disease has devastated many nations. Form which, the focus of this article was to analyze dynamic dengue fever. We developed a dynamical mathematical model that would represent them and incorporate the impact of panic, tension, or anxiety on the human population. Model's qualitative analysis was calculated, including illness free equilibrium, endemic equilibrium, and basic reproduction number. Numerical simulation through various parameter settings showed the progression of epidemics, the system’s behaviors, and support theoretical results. The maximum sensitivity index was obtained for the vector death rate in the sensitivity study, and this parameter was regarded the most sensitive. 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