Simulations and fractional modeling of dengue transmission in Bangladesh

1.   Introduction

Dengue is a debilitating viral infection spread by the bite of Aedes mosquitoes carrying any one of the four dengue viral serotypes. It is common in urban and semi-urban areas of tropical and subtropical climates around the world. These serotypes of the virus that can cause dengue are closely related but antigenically distinct (DEN-1, DEN-2, DEN-3, DEN-4). While recovery from one virus confers lifetime immunity against that virus, ^[1] notes that recovery from the other three viruses offers no protection against infection. The majority of the world's population, particularly those who live in tropical and subtropical regions like Bangladesh, are at risk. About 390 million dengue infections are estimated to occur annually, of which a quarter of the cases (67–136 million) will manifest clinically ^[2], with the overall incidence of dengue having increased 30-fold over the past 50 years ^[3]. In Asia, a severe dengue outbreak was reported in 1950 in the Philippines and Thailand ^[3], and in 1964 Bangladesh experienced a dengue outbreak for the first time. Between 1964 and 1999, sporadic cases and small outbreaks clinically suggestive of dengue occurred across the country but were not officially reported, even though 5551 people were affected with 93 accounted fatalities ^[4].

Dengue Fever (DF) is marked by an onset of sudden high fever, severe headache, pain behind the eyes, and pain in muscles and joints. Some may also have a rash and varying degrees of bleeding from various parts of the body. Dengue has a wide spectrum of infection outcomes (asymptomatic to symptomatic). Symptomatic illness can vary from DF to more serious dengue hemorrhagic fever (DHF) ^[5,6]. DHF is a more severe form, seen only in a small proportion of those infected. DHF is a stereotypic illness characterized by 3 phases; the febrile phase with high continuous fever usually lasting for less than 7 days; the critical phase lasting 1–2 days usually apparent when the fever comes down, leading to shock if not detected and treated early; convalescence phase lasting 2–5 days with an improvement of appetite. Dengue Shock Syndrome (DSS) is a dangerous complication of dengue infection and is associated with high mortality. Severe dengue occurs as a result of secondary infection with a different virus serotype. Statistics show that Bangladesh reported its first case of mosquito-borne virus infection in 2000, and about 100 people died of the disease from 2000 to 2003.

The first official outbreak of DF in Bangladesh was in 2000, and since then the number of hospitalized patients have exceeded 3000 patients six times: 6232 in 2002, 3934 in 2004, 3162 in 2015, 6060 in 2016, 10,148 in 2018, and 1, 00107 as of Nov 30, 2019, with estimated projections of more than 1, 12,000 cases by the end of 2019 (appendix) ^[4,7,8]. In parallel with the major epidemics in 2018 and 2019's outbreak, 26 deaths and 129 deaths, respectively, have been officially documented by the government surveillance systems with a clear predominance of cases and fatalities during the summer months (July to November), even if the death tally is likely to be much higher because of actual under-reporting (appendix) ^[8,9].

In reality, Bangladesh is suffering from the pain of mosquitoes. On Saturday 15 October 2022, A total of 1,916 people are admitted to different government and private hospitals in Dhaka and 799 outside Dhaka. A total of 23,592 patients were admitted to the hospital from January 1 to October 14 this year. Of them, 20,794 people left the hospital after recovering. The number of dengue patients is increasing every day. The list of deaths is getting longer. So far this year, 83 people have died of dengue. Of these, 28 this month. In two weeks of this month, 7,500 patients were admitted to the hospital with the infection. Experts said pre-monsoon, post-monsoon, and monsoon surveillance are being done every year to see the mosquito situation.

A month-wise patient admission analysis showed that 126 patients were admitted to the hospital in January 2022, 20 in February, 20 in March, 23 in April, 163 in May, 737 in June, 1,571 in July, and 3,521 in August. The highest number of dengue patients were admitted in September and 9,911 people were admitted, 34 people died. In 2019, 1, 01,354 people were admitted to the hospital with dengue. This is the highest number of patients admitted in a year in the country so far. That year, 164 people died. In 2018, 10,148 people were admitted to hospitals with dengue and 26 died. Although dengue infection was not seen much during the corona epidemic in 2020, 28,429 people were infected and 105 people died of dengue across the country in 2021 ^[10]. DF is the fastest-growing infectious disease in the world, causing an average of more than 500,000 potentially fatal infections and about 20,000 deaths each year. As of 17 November, 26,000 dengue cases were reported with 98 deaths ^[11]. So far, the case fatality rate seems higher in this outbreak (98/26,000) as compared to 2019 massive outbreak (179/101,354) (Table 1).

Recently, fractional calculus has gained much interest from researchers. Because it has been widely used in different fields of science, and engineering, and also in different real-world problems. Several scholars have used different types of operator approaches and applied them to many linear and non-linear diseases and complex models ^{[12,13,14,15,16]}. In this paper, the proposed dengue epidemic model is derived using the Caputo fractional derivatives. In recent years, fractional-order calculus is found to be more appealing in modeling for a real-world problem in comparison to a classical integer order as it provides a tool for the description of memory effects and genetic properties of various materials. Recent entomological studies revealed that mosquitoes did not feed randomly on human blood, but they use their prior experience with human location and human defensiveness to select the host to feed on ^[17]. Thus, in dengue transmission, a future state does depend on the history of the transmission. Hence, the fractional-order differential equation is found to be the best approach to model the transmission. The purpose of this study is to propose and study a more accurate mathematical model of dengue transmission using the fractional-order derivative than those previously presented in the literature ^[18,19,20].

This paper is organized as follows: In Section 2, preliminaries and notations; Section 3 describes the model formulation for the fractional-order derivative; Section 4 explains the existence and uniqueness of a non-negative solution; Section 5, presents model equilibria and basic reproduction number; Section 6, stability analysis of equilibrium (DFE and EE state) presents, where the Jacobian matrix uses for disease-free and Lyapunov function uses for endemic equilibria; Section 7 basic reproduction number is presented through sensitivity analysis by calculating relevant parameters; Section 8, the numerical simulations; Section 9, the discussion and conclusion.

2.   Preliminary definitions

The study of derivatives and integrals of any real number, even those of complex order, can be done very well using fractional calculus. Even though we use the CD in our proposed fractional dengue model, there are some fundamental concepts concerning the CD that we must understand first. Other relevant properties are listed below ^{[21,22,23,24,25]}.

Definition 1(^[25]) Suppose $\alpha > 0$ and $f\in L^1([0, b], \mathbb{R})$ where $[0, b]\subset \mathbb{R}_+$. The fractional integral of order $\alpha$ of function $f$ in the sense of Riemann-Liouville is defined as follows:

where $\Gamma(\cdot)$ is the classical gamma function defined by

The initial value problem for Caputo fractional differential equation is

and the corresponding fractional Volterra integral Eq ^[26] is

Definition 2^[22] The Caputo fractional derivative of order $\alpha\in (n-1, n]$ of $f(x)$ is defined as

where $n = [\alpha] + 1$ and $[\alpha]$ represent the largest integer that is less or equal to $\alpha$.

Definition 3^[27] The Laplace transform of an n-th derivative operator is obtained as

Similarly for $\alpha\in(n-1, n]$ we obtain the Laplace transform of the Caputo fractional operator as

Definition 4^[23]. An entire function called Mittag-Leffler is defined in the form of power series as

and

The notion of convergence of mittag-Leffler function is fully discussed in ^[21].

Theorem 1 ^[28,29]. Let us consider a fractional non-autonomous system like (4) with $x^*$ say an equilibrium point and $X\in R^n$ a domain containing $x^*$ and let $H : [0, \infty) \times X \rightarrow R$ be a continuous and differentiable function, such that

and

$\forall \alpha\in(0, 1)$ and all $x\in X$, where $V_1(x)$, $V_2(x)$ and $V_3(x)$ are positive definite continuous functions of $X$, then the equilibrium point of system (4) is uniformly asymtotically stable ^[30].

The Lyapunov function described above will be used to investigate the global stability of the proposed fractional dengue model.

Lemma 1^[31]. For a continuous and differentiable function $H(t)\in R_+$ and $\alpha \in (0, 1)$, then for any time $t\geq t_0$ we have

3.   The model formulation

In this paper, we investigate the $S_hI_hH_hR_hS_mI_m$ human-mosquito fractional model, which comprises of two distinct populations, including human populations and mosquito populations. Three epidemiological states of humans are included in the proposed fractional-order dengue model: $S_h(t)$ susceptible (individuals who can contract the virus), $I_h(t)$ infected (individuals who can transmit the virus to others), $H_h(t)$ hospitalized human (the compartment of people who are hospitalized after infection), and $R_h(t)$ recovered (individuals who have required immunity). Since $N_h$ is assumed to be constant, $N_h(t) = S_h(t) + I_h(t) + H_h(t)+ R_h(t)$. For the mosquito model, we only consider the susceptible and infected mosquitoes, since the mosquito does not enter the recovery phase after being infected due to its shortened lifespan. On the other hand, the mosquito population is divided into two compartments, susceptible $(S_m)$, and infectious $(I_m)$ with $N_m(t) = S_m(t) + I_m(t),$ where $\alpha\in (0, 1]$ is the order of the fractional derivative. All model parameters are assumed to be non-negative. The following Table 1 lists the parameters that are used in our model. The fractional derivatives use in model (3.1) are all in the Caputo sense. So the model diagram of human-mosquito transmission dynamics of the disease is given below in Figure 1:

Now we present a fractional model with CD given by,

With the following non-negative initial conditions

where $0 < \alpha\leq1$ and ${}^CD^{\alpha}_t$ is the Caputo fractional derivative of order $\alpha$. The size of the entire human population and mosquito population is represented by the $N_h$ and $N_m$ so that, $N_h(t) = S_h(t) + I_h(t) + H_h(t) + R_h(t)$ and $N_m(t) = S_m(t)+ I_m(t).$ The birth rate of human and mosquito populations are denoted as $\Lambda_h$ and $\Lambda_m$ respectively. The natural death rate for humans and mosquitoes is described by the parameters $\gamma_h$ and $\gamma_m$ and disease related death rate of human is denoted by $\epsilon_h$. We assume that biting rate for humans and mosquitoes are $\phi_h$ and $\phi_m$, respectively. $\beta_h$ is the transmission probability from infected human to susceptible mosquito, $\beta_m$ is the transmission probability from infected mosquito to susceptible human, $\tau_h$ represents natural recovery rate of infected human, $\mu_h$ is the rate of hospitalized infected humans and $\upsilon_h$ is the recovery rate of hospitalized infected human.

The different parameters used in this fractional model with their values and references are given below in Table 2:

4.   Fundamental solution procedure of q-HATM

In this section, we present the solution procedure of q-HATM ^[35] by using LT with q-HAM. Soon after, it is employed by number of authors to evaluate the solution for numerous families of differential equations exemplifying diverse phenomena including economic growth, biological models, human disease, chaotic behaviour, chemical reaction, optics, fluid mechanics and others ^[35,36,37] and further derived some fascinating consequences in comparison of other modified and classical algorithms.

Here, to present the procedure of q-HATM we hire the following differential equation of fractional order

with the initial condition

where ${}^CD^{\alpha}_t v(x, t)$ denotes the CD of v(x, t). On employing LT on Eq (4.1), we obtain

For $\phi(x, t; q)$, N is contracted as follows

where $q \in[0, \frac{1}{n}]$. Then, we present homotopy with the embedding parameter q and non-zero auxiliary parameter by HAM as

where L is signifying LT. For $q = 0$ and $q = \frac{1}{n}$, the following conditions satisfies

With the help of Taylor theorem, we have

where

After differentiating Eq (4.7) m-times with q and multiplying by $\frac{1}{m!}$ and substituting $q = 0$, one can get

where the vectors are defined as

Eq (4.9) reduces after employing inverse LT to

where

and

Here, $H_m$ is homotopy polynomial and presented as

By using Eqs (4.11) and (4.12), we get

The series solution by projected algorithm is defined as

5.   q-HATM solution for considered model

Here, we evaluate the solutions for model (3.1) with different parameters. Consider the system of the equation describing the fractional-order $S_hI_hH_hR_hS_mI_m$ epidemic model of dengue disease

With the following initial conditions

Taking LT on Eq (5.1) and then using the initial conditions, we get

Now, the non-linear operator N presented as:

By applying the considered method and for $H(t) = 1$, the n-th order deformation equation is presented as

where

Now using inverse LT on Eq (5.4), we get

Using the initial conditions and the above system, we have

We can get the rest of the term in a similar way. The q-HATM series solution for the FBM equation considered in Eq (5.1) is given by

5.1.   Boundedness of the solutions

In this section the total population is denoted as

and the total mosquito population is denoted as

The linearity of the Caputo operator in the above two different populations becomes,

and

We apply the Laplace transform method ^[27] to solve the Gronwall's inequality in (5.8) and (5.9) with initial condition $N(t_0)\geq0$

and

The linearity property of the Laplace transform gives

and

Splitting (5.10) and (5.11) to partial fraction gives the following:

and

According to Taylor series, we get

Therefore

and

using the inverse Laplace transform of (5.12) and (5.13), we have

and

According to Laplace formula,

or

Thus

and

Substituting the Mittag-Leffler function we get,

and

where $E_1(-\gamma_h^\alpha t^\alpha)$, $E_{k+1}(-\gamma_h^\alpha t^\alpha)$, and $E_1(-\gamma_m^\alpha t^\alpha)$, $E_{k+1}(-\gamma_m^\alpha t^\alpha)$ are the series of Mittag-Leffler function (as in definition 4), so we say that the solution to the model is bounded. Thus,

and

5.2.   Uniqueness of the solution

In this section we will prove the uniqueness of (3.1). Consider the system (3.1) written as

Theorem 2. System (5.18) satisfies Lipschitz continuity

Proof. Since

It is clear that F is continuous and bounded function. Using Picard-Lindelof theorem ^[29] we establish the following theorem.

Theorem 3. Let $0 < \alpha < 1, I = [0, h^*]\subseteq \mathbb{R}$ and $J = |x(t)-x(0)|\leq k$ and let $f: I\times J\rightarrow \mathbb{R}$ be continuous bounded function, that is $\exists M > 0$ such that $|f(t, x)|\leq M,$ Since f satisfies Lipschitz conditions. If $Lk < M,$ then there exists a unique $x\in C^\alpha[0, h^*]$ that holds for the initial value problem (5.18). Where $h^* = \min \{h, (\frac{k{\Gamma(\alpha+1)}}{M})^{\frac{1}{\alpha}}\}.$

Proof. Suppose $T = {x\in C[0, h^*]:||x(t)-x(0)||\leq k},$ Since $T\subseteq\mathbb{R}$ and its closed set, then T is complete metric space. The continuous system (5.18) can be transformed to equivalent equations as;

Equation (5.21) is equivalent to Volterra integral equation that solves (5.18). Define an operator F in T

Now we need to proof that (5.22) satisfies the hypothesis of contradiction mapping principle. First to show $F:T\rightarrow T,$

Or, $x(0) - k\leq F[x](t)\leq x(0) + k, \forall t\in [0, h^*].$ Hence the operator F maps T onto itself. Secondly, to show that T is a contradiction, we have

Since by hypothesis $\frac{Lk}{M} < 1$, then T is a contradiction and has a unique fixed point. Thus, system (5.18) has a unique solution in $\bar\Omega$.

Theorem 4. The closed set $\Omega = \{(x_1, x_2, x_3, x_4)\in R_+^4 : 0\leq x_1+x_2+x_3+x_4\leq M_1, 0\leq x_5+x_6\leq M_2\}$ is a positive invariant set for the proposed fractional order system (3.1).

Proof. To prove that the system of Eq (3.1) has a non-negative solution, the system of Eq (3.1) implies

Thus, the fractional system (3.1) has non-negative solutions. In the end, from the first four equations of the fractional system (3.1), we obtain ${}^C_0D^{\alpha}_t (x_1+x_2+x_3+x_4)\leq \Lambda_h^\alpha-\gamma_h^\alpha(x_1+x_2+x_3+x_4).$ Solving the above inequality, we obtain

so by the asymptotic behavior of Mittag-Leffler function ^[25], we obtain $(x_1(t)+x_2(t)+x_3(t)+x_4(t))\leq \frac{\Lambda_h^\alpha}{\gamma_h^\alpha}\cong M_1.$ Taking the same steps for the last two equations of system (3.1), we get

Hence, the closed set $\Omega$ is a positive invariant region for the fractional-order dengue model (3.1).

6.   Model equilibria and basic reproduction number

By getting the model equilibria from the fractional dengue model (3.1) we obtain and by setting

So from some algebraic solution of (3.1) we get,

The fractional dengue model (3.1) obtain the following two equilibrium points: (1) The DFE for the model (3.1) is given by,

(2) The EE for the model (3.1) is given by,

where

where, $\tau^*_h = \frac{\phi_m^\alpha\beta_m^\alpha I_m^*}{N_h^*}$ and $\tau_m^* = \frac{\phi_h^\alpha\beta_h^\alpha I_h^*}{N_m^*}.$ Observe that $S_h^*,$ $I_h^*,$ $H_h^*,$ $R_h^*,$ $S_m^*,$ $I_m^*$ are positive if and only if $\left[\phi^\alpha_h\phi^\alpha_m\beta^\alpha_h\beta^\alpha_m S_h S_m - N_h N_m\gamma^\alpha_m(\mu^\alpha_h+\tau^\alpha_h+\gamma^\alpha_h)\right] > 0.$ Calculate the reproduction number of the fractional model (3.1) by using the method of next generation matrix and the basic reproduction number present in ^[38]. Let us define a vector, $\mathrm{X} = [I_h, H_h, I_m]^{T}$, And

Thus the basic reproduction number of the model (3.1) is

It is easy to prove that $S_h^*,$ $I_h^*,$ $H_h^*,$ $R_h^*,$ $S_m^*,$ $I_m^*,$ and $R^* > 0$ if and only if ${R}_{0} > 1.$

7.   Stability analysis

7.1.   Analysis of DFE

Simplifying the stability of the DFE, Suppose DFE is $E_0 = \left(S_{h_{0}}, I_{h_{0}}, H_{h_{0}}, R_{h_{0}}, S_{m_{0}}, I_{m_{0}} \right)$ = $\left(\frac{\Lambda_h}{\gamma_h}, 0, 0, 0, \frac{\Lambda_m}{\gamma_m}, 0 \right)$ and the Jacobian matrix of the system in (3.1) can be written as

Now calculating the Jacobian matrix J at DFE point $E_0$ and solved $\mathrm{det}(J-\lambda I)$, we obtain, $P_{j}(x) = (\lambda+\gamma^\alpha_h)^2(\lambda+\epsilon^\alpha_h+\upsilon^\alpha_h+\gamma^\alpha_h)(\lambda+\gamma^\alpha_m)(\lambda^2+A\lambda+B),$ Where $A = \mu^\alpha_h+\tau^\alpha_h+\gamma^\alpha_h+\gamma^\alpha_m$ and

It is easy to proof that if $\mathcal{R}_{0} < 1$, then $A > 0$ and $B > 0$. This polynomial $\lambda^ 2 + A \lambda + B$ have two roots with negative real parts. Thats why, $E_{0}$ is locally stable because the real parts of six eigenvalues of the matrix $J(E_{0})$ are all negative. Therefore, overall we can tell the DFE is stable when $B > 0$ and DFE is unstable when $B < 0$. The following theorem is presented for the global stability of the disease free equilibrium case $E_0.$

Theorem 5. The fractional dengue model given by (3.1) for the arbitrary fractional order $\alpha\in (0, 1],$ with $R_0 < 1,$ is globally asymtotically stable.

Proof. The following Lyapunov function is considered for the proof of the global stability of the dengue fractional model (3.1):

where $\eta_i > 0$ for i = 1, 2, 3, 4, are arbitrary constants to be determined later. We consider the result described in sect. 2 and taking the time derivative of G(t), we obtain

Considering the fractional system (3.1), we obtain

Using the values of $S^0_h$ and $S^0_m$ at the DFE and simplifying we obtain

where $\eta_1 = \eta_2 = \gamma^\alpha_m,$ $\eta_4 = \eta_5 = \phi^\alpha_h\beta^\alpha_h.$ Thus, ${}^CD^{\alpha}_t G(t)$ is negative for $R_0\leq 1.$ So it follows from the established results and those given by theorem 1 in ^[24,39] that the fractional dengue model is globally asymptotically stable at the DFE case $E_0$.

7.2.   Analysis of EE

We prove the global stability results for the fractional model (3.1). Firstly we have the following results for the model (3.1) at the constant state:

Now showing the global stability of the model (3.1) in the following theorem.

Theorem 6. If $R_0 > 1$, then the fractional dengue model (3.1) at $E^*$ is globally asymptotically stable.

Proof. We suppose the following Lyapunov function:

The derivative of L(t) with the application of the lemma 1 given in sec. 2 yields

From the fractional dengue model (3.1), we can consider the following way:

By direct calculation, we get the following:

Using the above expressions (7.6) and (7.7) in Eq (7.5), we obtain

From Eq (7.8), we get the following result,

and, in the same way, if

then ${}^CD^{\alpha}_t L(t)\leq0,$ So it observes that, at the EE point $E^*$, the fractional dengue model is globally asymptotically stable at $E^*$, when $R_0 > 1$.

8.   Sensitivity analysis

Sensitivity analysis helps us to identify parameters that have a big impact on the disease transmission. Such information is important not only for experimental design but also for data assimilation and reduction to complex nonlinear models ^[40]. This provides a good strategy to prevent and restrain the disease. The disease will be controlled and mitigated if we can change the value of parameters by the control strategies. Usually, in the epidemiological model, the analysis is used to discover parameters that have greatest influence on the basic reproduction number $R_0$ and should be targeted by the control strategies. The sensitivity indices of the $R_0$ are determined to allow us to measure which parameter has the greatest influence on the changes of $R_0$ and, hence, the greatest effect in determining whether the disease can be eliminated in the population. The normalized forward sensitivity index of a variable ($R_0$) with respect to a parameter is the ratio of the relative change in the variable ($R_0$) to the relative change in the parameter. The systematic description of the sensitivity analysis of the different parameters in $R_0$ for the model is:

So the basic reproduction number is,

It is easy to verify that

Using the parameter values from Table 2, the sensitivity indices of $R_0$ with respect to the parameters are given in Table 3.

We can conclude from Table 3 that the sensitivity indices are sign-related and, $R_0$ is more sensitive to the following parameters ($\phi_h$, $\phi_m,$ $\beta_h$, $\beta_m$) increasing order and a corresponding decrease in $R_0$ as the following parameters ($\mu_h$, $\gamma_h$, $\gamma_m$, $\tau_h$) have a negative impact on $R_0$, that means an increase in these parameters will reduce $R_0$, while ($\phi_h$, $\phi_m$, $\beta_h$, $\beta_m$) has a positive impact, and reducing the value of these parameters will reduce $R_0$. After the above analytical results, we now perform an $R_0$ sensitivity analysis to find exact ways to choose the various parameters in $R_0$. The following can be inferred from the sensitive analysis:

(1). If can reduce the value of the transmission rates $\beta_h,$ $\beta_m$, and biting rate $\phi_h$, $\phi_m$ could be effective control measures to stop the spread of the dengue virus.

(2). If can increase the natural death rate of mosquitoes $\gamma_m$ and natural recovery rate of infected humans $\tau_h$ so that it will not affect other susceptible individuals.

9.   Numerical simulations

In this section, we carry out numerical simulations for system (3.1) by using Euler's algorithm method ^[41]. To have a numerical scheme, we write the model (3.1) in the following form:

where $g = (x_1, x_2, x_3, x_4, x_5, x_6)\in \mathbb{R}_+^6, G(t, g(t))$ is used for a continuous real valued vector function, which additionally satisfies the Lipschitz condition and $g_0$ stands for initial state vector. Taking Caputo integral on both sides of (7.1) we get

To formulate an iterative scheme, we consider a uniform grid on [0, T] with $h = \frac{T-0}{m}$ is the step size and $m\in \mathbb{N}$. Thus, Eq (7.2) gets the structure as follows after making use of the Euler method

Thus, utilizing the above scheme (7.3), we deduced the following iterative formulae for the corresponding classes of the model (3.1).

We used the above approximation for the solution of our fractional system. In Figures (2) to (5), we demonstrate the dynamics of susceptible human ($S_h$), infected human ($I_h$), hospitalized human ($H_h$), recovered human ($R_h$), dead human ($D_h$), new cases of human, reported dengue cases of human, cumulative dengue cases of human, susceptible human with different biting rates, infected human with different biting rates, susceptible mosquitoes ($S_m$), infected mosquitoes ($I_m$), and infected mosquitoes (with different biting rates $b = 0.45, 0.50, 0.68$ with the variation of fractional order $\alpha = 0.75$, integer order and actual values. We noticed that the variation of fractional order has a great influence on the infection level of dengue in both populations. In other words, it can highly reduce the level of DF in the community. We demonstrated the effect of the biting rate $\phi_h$ of the mosquitoes on the dynamics of dengue and observed that the peak of infection can be greatly decreased by decreasing the biting rate $\phi_h$. The mosquito biting and its generation further can be decreased by spraying or wasting the standing water around the home or inside the home, which has a great influence on the population of mosquitoes that end up biting humans. Using bed nets, avoid to visit areas prone to mosquitoes, using mosquito repellent, and covering legs and arms by wearing long sleeves and long pants are useful to prevent biting of mosquitoes. These scenarios predict that the infection can be controlled and prevented by decreasing the index of memory and biting rate of vectors in the community. In this model, we use the real data of Bangladesh from (2012–2022)^[42].

First, we simulate different values of fractional order $\alpha$ with fixed values of the model parameters. In this work, the dengue cases data of 11 years (2012–2022) are used with different parametric values for the numerical simulations based on a case study of Bangladesh cited from the literature; some are fitted, some are estimated and some are referred. We use the total population of Bangladesh, $N_h = 166303494$ ^[43]. The life expectancy in Bangladesh for the year 2022 is 72.87, so we estimate $\gamma_h = 1/72.87$ per year. The parameter $\Lambda_h$ is estimated from $\Lambda_h/\gamma_h = 166303494,$ and assumed that this is to be the limiting population in the disease absence, so $\Lambda_h = 2278130.05$ per year. For the initial values of the model variables, we use the total initial population $N_h(0) = 166303494,$ so that $N_h (0) = S_h(0)$ + $I_h(0)$ + $H_h(0)$ + $R_h(0)$ and $N_m(0) = S_m(0)$ + $I_m(0),$ The initial conditions are assumed as $S_h(0) = 5000, I_h(0) = 1000, H_h(0) = 500, R_h(0) = 100, S_m(0) = 100000,$ $I_m(0) = 80000$ and the parameter values are taken from the literature as given in Table 1. As illustrated in Figures 2 to 8, a smaller fractional order reduces the peak significantly and flattens the progression curve. The dengue cases reported in Bangladesh are shown in the following Figures and the solution obtained by the Euler algorithm method is presented in Figure 2 to 8 for different values of $\alpha$. The number of infected human real data and simulated results will reach the highest level in eight months for $\alpha = 0.75, 1.$ These scenarios predict that the infection can be controlled and prevented by decreasing the index of memory and biting rate of vectors in the community. Fig 6 and 8 describe the behavior of the solutions showing the dynamics of susceptible human, infected human and infected mosquito population for the biting rates 0.45 and 0.68 respectively.

10.   Discussion and conclusions

The objective of this work is to understand, analyze and find the solution to the fractional epidemiological models. In this paper, we analyze a new fractional epidemic model for the transmission of dengue infection with a non-integer derivatives and are analysed using q-HATM. The existence of the solutions of the model is investigated by solving the fractional Grownwall's inequality using the Laplace transform approach. The positivity and boundedness of unique solutions are investigated. The basic reproduction number of the system is calculated by the next-generation method. We establish two equilibrium solutions, disease-free and endemic are obtained. Both local and global stability of the equilibria is investigated to depend on the magnitude of the basic reproduction ratio. Sensitivity analysis of $R_0$ is carried out to know the contribution of input factors in the results of $R_0$ and observed that $\phi_h$, $\phi_m$, $\beta_h$, $\beta_m$ are the most critical parameters that highly contribute to the control and subsequent spread of dengue infection. We showed that the dengue infection is uniformly persistent in the system for $R_0 > 1$. Numerical simulations are carried out and dynamics of the populations are shown to vary for different values of $\alpha$. We obtain feasible results for the dynamics of dengue infection with the variation of memory index $\alpha$ and suggest that the index of memory has a dominant influence on the system. We conclude that the fractional-order model can explore the dengue epidemic disease transmission model rather than the integer-order derivative models. We can suggest that fractional-order (index of memory) $\alpha$ and biting rate for humans and mosquitoes $\phi_h$, $\phi_m$ can remarkably control and greatly decrease the level of disease in society.

DECLARATIONS

Ethics approval and consent to participate:

Not applicable.

Consent for publication:

Not applicable.

Availability of data and materials:

All data and materials used in this study are publicly available.

Competing Interest:

The authors declare that they have no competing interests.

Funding:

This work was supported by the Shanxi Science and Technology innovation team under grant no. (201805D131012-1), and the key projects of the Health Commission of Shanxi Province (2020XM18).

Authors' contributions:

The authors have read and approved the final manuscript.

Acknowledgments

The authors are thankful to the anonymous reviewers for the careful checking and suggestions that improved the presentation of the paper greatly. The authors thank the Chinese Government and the Complex Systems Research Centre, Shanxi University, for their support.