Analysis of Dengue Transmission Dynamic Model by Stability and Hopf Bifurcation with Two-Time Delays

Background : Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. Methods : The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. Results : Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number ( R 0 ) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R 0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. Conclusions : The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.


Introduction
Dengue fever is a critical feverish illness caused by mosquito-borne dengue fever viruses (DENV) [1], which are Flaviviridae flaviviruses. When an infected Aedes aegypti mosquito infects a person, many viruses are transmitted. The transmission of the dengue virus is mostly carried out by humans. A virus-infected mosquito feeds on the blood of a virus-infected person. The virus spreads to tissues such as the duct gland from the mosquito's gut in 8-10 days. The virus appears to have no negative influence on the mosquito. When a DENV-carrying mosquito bites a human, both the virus and the mosquito's secretions are injected into the skin. As it circulates throughout the body, it clings to and enters white blood cells, multiplying in them. White blood cells produce a number of signal proteins in response, including interferon, which can lead to symptoms including fever, flu-like symptoms, and excruciating pains.
A variety of organs, including the liver and bone marrow, are regularly damaged by serious infections. They also increase the body's production of viruses and frequently cause fluid to leak from the circulation into internal organ cavities via the membranes of tiny blood vessels. As a consequence, blood pressure drops to a level where many organs are unable to get enough blood due to decreased blood flow in the blood vessels. The other major effect of dengue fever is bleeding, which was made more likely by bone marrow diseases that decreased the quantity of platelets needed for effective blood coagulation [2].
We provide an associative model that supports the susceptible-infective-recovery-susceptible (SIRS) in the human population and the susceptible-infective (SI) in the vector (mosquito) population while avoiding delays in a trial to examine the dynamics of infection across considerable time periods when susceptible people are born and immunity is lost. Because of vector dynamics, the vector population is typically regarded as being in equilibrium with the human population [3,4]. The analysis of the delay differential equation in the epidemic model was studied in [5][6][7].
The transmission process between humans and mosquitoes takes a long time because of the different parasites' incubation times [2,8]. The dynamics of dengue virus transmission with a delayed SIVA model are studied in [9]. Sefidgar et al. [10] discussed the nonlinear system of fractional differential equations that appear in a model of HIV infection of CD4+T cells and proposed the LAM for solving the system. The results showed the effectiveness and efficiency of the method. Wei et al. [11] developed a system for a vectorborne disease with a direct method of transmission, but they also showed how the addition of a latency inside the host-tovector transmitter term will make the system unstable and cause periodic solutions through Hopf bifurcation.
Omame et al. [12] formulated a mathematical model for the co-infection, COVID-19 and dengue transmission dynamics with optimal control and cost-effectiveness analysis. The results showed that the strategy for implementing control against incident dengue infection is the most cost-effective in controlling dengue and COVID-19 coinfection. Omame et al. [13] design a non-integer ordered model for SARS-CoV-2, dengue, and HIV co-dynamics to assess the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV through fractional derivatives. They showed, using numerical simulations, that keeping the spread of SARS-CoV-2 low would have a significant impact on reducing the co-infections of SARS-CoV-2 and dengue or SARS-CoV-2 and HIV. Based on SIRS-SI in the human and vector populations, Wan and Cui [14] presented a model that utilized two latencies for communication between humans and vector populations with the standard incidence rate, loss of immunity, and rate of recovery from infectiousness. To explore the possibility of equilibrium stability and dynamic behavior, Xu and Zhou [15] projected the dynamics of delayed vector-borne transmission with reinfection. An epidemic model with vaccination and numerous time delays is taken into account in [16] along with the stability and Hopf bifurcation analysis. Baleanu and Babak [17] studied a terminal value problem for nonlinear systems of generalized fractional differential equations and formulated a classical operator and a related weighted space with a generalized fractional operator. The results showed the effects of various choices of weight function on modeling with a TVP. Nowadays, many authors do their research on epidemic models [18][19][20] with various strategies. To take into consideration the amount of time required for a viral infection to spread to host and vector populations, Yanxia et al. [21] design an associated upgraded vector-borne epidemic model with two latency periods and reinfection. We created a new model using the information in this article and included three new parameters, including loss of immunity, recovery from infectiousness, and partial protection of the human population.
In this paper, the basic reproduction number for the developed model was determined, and the existence of equilibrium was also examined. The aim of the paper is to explore the stability and Hopf bifurcation of a dynamic model of dengue transmission that incorporates two delays. The numerical simulations were described, and the main conceptual outcomes were exhibited. Fig. 1 provides a schematic overview of this model.

Model Formulation
The model considers a uniform mix of human and mosquito populations, ensuring that every mosquito bite provides the same possibility of spreading the virus (or transmitting the virus from an infected human). Because mosquitoes cannot recover from infection, their infection period ends when they die, owing to their incredibly short lifespan. As a result, the mosquito population has a relatively low immunity class, and mortality rates are comparable across all categories.
H N (t) represents the whole human population at time t, which is divided into three compartments: susceptible humans S H (t), infectious humans I H (t), and recovered humans R H (t). So, the entire human population is Similarly, V N (t) represents the whole mosquito population at time t, which is divided into two compartments: susceptible vector S V (t) and infectious vector I V (t). So, the entire vector population is V N (t) = S V (t) + I V (t). Assumptions of the model: (a) The human and vector total population sizes are considered to be constant. It is expected that new humans will enter the human population at any time at Ω H rate through birth or immigration, and those susceptible mosquitoes will be recruited at a constant rate Ω V .
(b) Depending on the sickness, people shift from one class to another as their health improves. This infusion is not contagious since there is no vertical transmission or immigration of affected persons. When the Aedes aegypti mosquito bites the host, all people get infected, and dengue development begins.
(c) Natural death occurs at a rate of d h and d v [21] (according to their limited life span) for all humans and mosquitoes respectively, regardless of condition.
(d) Individuals who have recovered in the human population acquire partial immunity (σ) or loss of immunity (ρ) [21].
(e) Illness mortality rates for humans and mosquitoes are µ h and µ v , respectively.
(f) The term e −d h τ 1 and e −dvτ 2 is the human and mosquito survival rate [22].
(g) Mosquitoes do not die or become infected by infection.
The system of non-linear differential equations for the dengue model is with the following initial conditions: The systems Eqn. 1 to Eqn. 5 is reformulated as The parameters of the above system Eqn. 7 is described in Table 1.

Positivity and Boundedness Solution of Dengue Transmission Dynamic Model
The solution of the system Eqn. 7 is usually positive for positive initial values of the data at all times t ≥0. Especially, the feasible region is

Theorem 1
The is positively-invariant for the system Eqn. 7.

Proof
According to [23,24], it is easy to show that the solution of the Eqns. 1 to 5 with the initial conditions Eqn. 6 is distinct and nonnegative for all t ≥0, based on the fundamental theory of differential equations.
From Eqns. 1 to 3, the rate of change of whole human population is given by Without loss of generality, this equation can be expressed as an inequality as .
From Eqns. 4 to 5, the rate of change of whole mosquito population is given by Without loss of generality, this equation can be expressed as an inequality as Then, lim Then for any, ϵ >0, is positively-invariant for the system (Eqn. 7).

Basic Reproduction Number of the Model
The illness-free equilibrium, E 0 = {S 0 H ,I 0 H ,I 0 V } is the model's steady state in the lack of infection or illness. All the components of E 0 are determined from the first three equations of system Eqn. 7 by putting the RHS equal to zero and assuming that I 0 H = 0 and I 0 V = 0, where I 0 H and I 0 V refer to the equilibrium points. Thus, The next generation matrix technique, as designated by Diekmann et al. [25], was used to determine the basic reproduction number R 0 .
From system Eqn. 7, the illness states A and the transfer states B are given by 2 and B= Using Jacobian matrix, the partial derivatives of A and B with respect to I H and I V at the illness-free equilibrium Eqn. 10, are given by Now the next generation matrix FV −1 can be calculated as The eigenvalues of Eqn. 11 are used to derive the basic reproduction number, which is

Endemic Equilibrium Existence of the Model
The endemic equilibrium E * , is the model's stable state in which the illness continues. All the components of E * are obtained from the system Eqn. 7 by framing the right -hand side equal to zero. Thus, where I * H is the positive root of the quadratic equation shown below where be the roots of Eqn. 14. Clearly, we have X 1 >0, X 2 >0 and X 3 >0 if R 0 <1, and X 1 >0, X 3 <0 if R 0 >1. It is obvious that I 1 and I 2 are negative roots if R 0 <1, and that I 1 is positive roots if R 0 >1. The following result is drawn from the relationship between the roots Eqn. 14 and the equilibrium of model Eqn. 7.

Theorem 2
For τ 1 , τ 2 ≥0, illness-free equilibrium E 0 is locally asymptotically stable if R 0 <1 and is unstable if R 0 >1. Proof The Jacobian matrix of the system Eqn. 7 at E 0 is The characteristic equation of Eqn. 15 is given by (16) One negative eigen value is λ = −(d h + ρ) and the remaining eigen values of the characteristic equation are The quadratic equation is the same in the ODE case for τ 1 = τ 2 = 0. In that case, the real component of every eigenvalue of the characteristic equation is already negative. The Hurwitz criterion states that when τ 1 = τ 2 = 0, the illness-free equilibrium E 0 at R 0 <1 is locally asymptotically stable, however, when R 0 >1, it is unstable.
Let us first consider R 0 >1. It is simple to demonstrate that Eqn. 17 has a real positive root. Rearranging Eqn. 17 in the form (18) Suppose λ is real. From Eqn. 18, the LHS and RHS are denoted by A(λ) and B(λ) respectively. We have A(0) = 0 and lim λ→∞ A(λ) = ∞.
Here, B(λ) is a decreasing function of λ As a result, Eqn. 17 has a non-negative real root for any τ 1 ≥0 and τ 2 ≥0, and the illness-free equilibrium is destabilizing.
Hence E 0 is locally asymptotically stable in system Eqn. 7 if R 0 <1 and is unstable if R 0 >1.

Proof
The Jacobian matrix of the system Eqn. 7 of E * is (19) The characteristic equation of Eqn. 19 is given by where When τ 1 = 0 and τ 2 = 0 The characteristic Eqn. 20 becomes where, According to the Routh-Hurwitz criterion, the roots of the characteristic Eqn. 21 does not have a positive real part if and only if the coefficients of A i are non negative and matrix H i >0, for i = 0, 1, 2, 3. From this Hence the endemic equilibrium E * is locally asymptotically stable when τ 1 = τ 2 = 0.

Numerical Simulations
Theorems 2, 3, 4, and 5 explore the stability of an endemic equilibrium, which is important from an epidemiological perspective. The parameter values of the model is showed in Table 2 (Ref. [3,21]). By Theorem 3, when , τ 1 = τ 2 = 0 the stability of the endemic equilibrium E * (267.781, 210.142, 196.792) is converging towards being locally asymptotically stable (Fig. 2). It is tougher to control the disease as R 0 = 3.17647 > 1 and dengue fever persists in both human and vector populations.
In every instance, the stability of the endemic equilibrium E * is comparatively higher than that of the existing literature [21], which strengthens the model we have created.

Conclusions
In this study, positivity and boundedness were verified for the solution of the dengue transmission dynamic model. The basic reproduction number R 0 was chosen to ensure the model's stability. It was established that the two delayed models' endemic equilibrium E * and illness-free equilibrium E 0 both existed. The steadiness of illness-free equilibrium was determined for τ 1 , τ 2 ≥0 in terms of R 0 . The steadiness of endemic equilibrium was determined for τ 1 = τ 2 = 0; τ 1 >0 and τ 2 = 0; τ 1 = 0 and τ 2 >0 in terms of positivity of R 0 . The local asymptotic stability of the endemic equilibrium E * occurs when τ 1 = τ 2 = 0. The endemic equilibrium becomes unstable and undergoes Hopf bifurcation if τ 2 = 0 and τ 1 > τ * 1 . Similarly, the endemic equilibrium becomes unstable and undergoes Hopf bifurcation if τ 1 = 0 and τ 2 < τ * 2 . As a result, the endemic equilibrium is locally asymptotically stable if two delays are greater than zero. Our numerical simulations clearly demonstrate that, whereas susceptible and infected population levels are initially unstable, they become stable as time moves on. The length of the delay has no effect on the stability of the illness-free equilibrium. The Hopf bifurcation might occur nonetheless, depending on how much the delay affects the underlying equilibrium's stability. Future research will expand on this assessment to examine the effects of a few control measures built into our model. The best way to control the disease will be examined as well, taking into account a variety of prevention strategies like self-defense, medical attention, and insecticide spraying.

Medical Implication of Mathematical Study
In this paper, mosquito-borne dengue fever viruses (DENV) were studied with the help of two delays. We have observed the following medical implications as mathematical observations.
Since the basic reproduction number is given by R 0 , if R 0 <1, the disease does not survive, and if R 0 >1, the disease keeps spreading, the infection rate will increase. The results are mostly favorable to R 0 >1, but it requires good medical treatment, and the patients are also required to have the place very clean, free from stagnant water and unwanted materials around the shelter, to reduce the reproduction of dengue-spreading mosquitoes. Bifurcation in mathematical theory is used to verify the topological structure of the solutions of the system of differential equations. This is not a quantitative result, but it provides a qualitative assessment of whether the system is stable or not after some transition time. Medically, the bifurcations can be understood as a visualization of the possibility of a group of infective hosts becoming healthy or not. By our numerical simulations, we can easily see that initially the populations of susceptibles and infected were not stable, but as time increased, they be-came stable. It clearly shows that, medically, it is possible to bring society back to being free from dengue.
Other than the basic reproduction number, stability analysis, and bifurcation analysis, we also like to mention a few points shared by the World Health Organization (WHO) [28]. DENV is caused by females of the mosquito species Aedes aegypti and rarely by Albopictus.
Severe dengue is a very deadly disease, as it causes serious illness and death in some Asian and Latin American countries. It requires management by medical experts. Though Dengue has various epidemiological structures (DENV-1, DENV-2, DENV-3 and DENV-4) produced by the Flaviviridae family of viruses, there is a strong belief that once recovered from this disease, one will have a lifelong immunity against the same [29].
As we have mentioned, medically, there is no proper treatment for the later stages of dengue. Therefore, mathematical modeling is useful to provide qualitative analyses for the chance of recovery for a large infected community.

Availability of Data and Materials
All the datasets and materials from which the entire study was done are available within the manuscript.

Author Contributions
Conceptualization -PRM, VA, VS; Supervision -VA, VS, DB; Writing -original draft -PRM, PBD, DB; Numerical results -PRM, VS, PBD; Analysis & Verification -PRM, VA, DB; Final draft-proof reading -PRM, VA, VS, PBD, DB. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript. All authors read and approved the final manuscript. All authors have participated sufficiently in the work and agreed to be accountable for all aspects of the work.

Ethics Approval and Consent to Participate
Not applicable.