key: cord-0900329-yv3je5a7
authors: Rehman, Attiq ul; Singh, Ram; Agarwal, Praveen
title: Modeling, analysis and prediction of new variants of covid-19 and dengue co-infection on complex network
date: 2021-05-04
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2021.111008
sha: 6764ba01c29dd6488812bedfc4461613c854849e
doc_id: 900329
cord_uid: yv3je5a7

Recently, four new strains of SARS-COV-2 were reported in different countries which are mutants and considered as 70 [Formula: see text] more dangerous than the existing covid-19 virus. In this paper, hybrid mathematical models of new strains and co-infection in Caputo, Caputo-Fabrizio, and Atangana-Baleanu are presented. The idea behind this co-infection modeling is that, as per medical reports, both dengue and covid-19 have similar symptoms at the early stages. Our aim is to evaluate and predict the transmission dynamics of both deadly viruses. The qualitative study via stability analysis is discussed at equilibria and reproduction number [Formula: see text] is computed. For the numerical purpose, Adams-Bashforth-Moulton and Newton methods are employed to obtain the approximate solutions of the proposed model. Sensitivity analysis is carried out to assessed the effects of various biological parameters and rates of transmission on the dynamics of both viruses. We also compared our results with some reported data against infected, recovered, and death cases.

The mathematical modeling of communicable and non-communicable diseases have been attracting the attention of many mathematical modelers [2, 5, 16, 31] since a long time. The first case of the novel Carona virus was identified in Wuhan city of China in December 2019. The rate of infection of the covid-19 was very high therefore, WHO has declared it as a pandemic [35] . Mathematical modeling has been a powerful tool to study the transmission dynamics of covid-19 and other diseases. Many models have been presented by different mathematicians from time to time to get an insight into the dynamics of these diseases [7, 10, 13, 23, 24] . Recently, four new strains of covid- 19 have been reported which are considered as 70% more dangerous than the early existing virus. The covid-19 has not only affected the healths of people but has caused big damage to the financial system of many countries.

Like covid-19, dengue fever is another challenging and very old disease spreading in the tropical and subtropical areas all over the world. The dengue epidemic is a major problem. As per data available in the literature, approximately 50 million people die due to dengue [34] . This epidemic is a mosquito-borne disease transmitted by Aedes albopictus and Aedes aegypti mosquitoes. This fever is caused by four different serotypes, which are DEN (I, II, III, andIV ).It is an RNA virus of the family Flaviviridae. However, a human is infected by only one serotype among these four. A human population is recovering, gaining full immunity to this type of serotype and only minor and transient immunity concerning the other three serotypes. Dengue fever can change from severe to mild. The other severe forms of dengue fever include dengue hemorrhagic fever (DHF) and shock syndrome. The infected populations remain asymptomatic for about three-fourteen days before they begin to experience a sudden onset of dengue fever. There is no specific treatment for this dengue disease; however, hospitalization, bed rest, analgesics, and antipyretics can be obtained for supportive care. People with weak immune systems develop these more serious forms of dengue. As usual, they need to be hospitalized. The full life cycle of epidemic dengue fever involves the role of the hosts and vectors as transmitters as the main source of infection [8, 25] . To prevent dengue fever virus transmission that depends fully on the control of vectors or interruption of host mosquitoes contact with the host, strategies are required at an early stage [30, 34] .

As per the data available in the literature, approximately 0.6 Million people died due to covid-19 and many more infected [35] . It is confirmed that early symptoms of covid-19 are lung infections, breathing problems, fatigue, and cough. Strangely, some cases of gastroenteritis and neurological disorder have come to notice which open new vistas of research in the direction of neurological science [21] . The covid-19 spreads by droplets spreading in air and surface over a susceptible person expose to the droplets gets infected due to covid-19. Through mathematics, we can't make any kind of vaccine for these epidemic diseases, but we can tell them how to prevent these viral diseases through mathematical models [32, 33] . Further, we set different rates by which everyone understands the mathematical model easily. Thus, we use fractional-order derivative in the Caputo sense because it gives a better outcome than the integer-order. Various fractional derivatives operators were developed, but the Riemann-Liouville and the Caputo are mostly used due to their simplicity and sincerity to handle [17, 20, 26] . But at present the other fractional derivatives are in lines namely, Hadamard, Atangana-Baleanu, Caputo-Fabrizio, and many others, [3, 4, 28] . These models have the suitability and efficiency of the Caputo operator. Moreover, Caputo-Fabrizio is the second best since it gives an error rate value of 1.97% for a fractional-order derivative. It is important to mention that fractional-order derivative equations are more fitting than integer order modeling in economic, biological, and social mathematical models where memory effects are important.

We are motivated to study fractional-order differential equations because exponential laws are very traditional approaches to studying the chaotic behavior of a complex dynamical system of population densities and epidemics, but there are certain dynamical systems where dynamical changes undergo faster or slower than exponential laws. In such cases, the Mittag-Leffler function can be used to describe the dynamic changes in such systems. Also, due to the effective memory function of fractional derivative, fractional-order differential equations have been widely used to describe the biological situation. Fractional-order derivatives are useful in studying the chaotic behavior of the dynamical system. Even though fractionalorder is the generalization of an ordinary differential equation to a random order. They have attracted considerable attention due to their ability to deal with more complex systems.

In this paper, we extend the work of the author [1] wherein the authors presented a fractional-order mathematical model in which only dengue class is considered. But in our work, we incorporated the new variants of the covid-19 class in addition to the dengue class and assess the effects of their co-infection. This co-dynamics situation is realistic as some cases were reported in Brazil in which dengue and covid-19 attacked the human population simultaneously [14] . We study a novel hybrid mathematical (SI-SICR) model of co-infection of dengue and covid-19 and address the following questions:-(a) Does the dengue virus act as a launch pad for new SARS-COV-2 strains ? (b) Has new invariants of covid-19 strain possess existing SARS-COV-2 ? (c) What will it take to achieve herd immunity with SARS-COV-2 ? (d) What will be the optimal solution for the mitigation of the dengue and SARS-COV-2 co-infection ?

The rest of the paper is organized as follows. Some basic preliminaries on fractional calculus are presented in the section 2. Section 3 is devoted to the formulation of mathematical modeling. The basic properties of the proposed model are given in the section 4. The stability analysis is discussed in section 5. Optimization analysis is presented in the section 6. The numerical solutions are obtained in the section 7. The results and discussion are provided in the section 8 and finally, the conclusion is drawn in the section 9.

Some basic preliminaries on fractional calculus are given as: Definition 2.1 The Riemann-Liouville fractional integral of the function f : R + → R exists for order α > 0 in two forms, upper and lower. Consider the closed interval [a, b], the integrals are defined as [29] ;

where Γ is the gamma function. Definition 2.2 The Riemann-Liouville fractional derivative of the function f : R + → R is also exists in two forms, upper and lower. This derivative is calculated by using the Lagrange's rule for differential operators. To compute the nth order derivative over the integral of order (n − α), the α order derivative is obtained. It is important to remember n > α, where, n is the smallest integer. Thus the derivatives are defined as [29] ;

Definition 2.3 Due to somedrawback of Riemann-Liouvile derivative an alternatetive definition was given by Caupto [27] , and is defined as below

Definition 2.4 Let us consider a constant point, say c * for the Caupto system that is called its equilibrium point, and is defined as below

is the Sobolev space, of order α = 1 and is defined as

then the Caupto fractional derivative is defined as [18] ;

in which M (α) is the normalization function such that M (0) = M (1) = 1. Definition 2.6 If the function does not belong to the Sobolev space then the new derivative that comes is known as Caupto-Fabrizo fractional derivative and is defined as

If the function f is differentiable then, the new fractional derivative known as Atangana-Baleanu derivative in the Caupto sense and is defined as

.

Here it should be notice that we don't revover the original function when order α = 0, except when at the origin point function is get vinished. To avoid this type of issue, we have the following definition.

Here the function f is not necessary differentiable then, the new fractional derivative known as Atangana-Baleanu derivative in the Riemann-Liouville sense and is defined as

where B(α) is also the normalization function such that

.

The new fractional derivative associted to fractional integral with nonlocal kernel is known as Atangana-Baleunu fractional integral and is defined as [9] ;

If α = 0 we recover the intial function and α = 1 we get the ordinary integral.

In this section, we formulate a deterministic mathematical model of dengue and covid-19 co-infection by dividing the total population into two classes, namely, the vector (mosquito) and host (human). To make the dynamical transmission co-infection model with covid-19 induced death rate φ h and the fraction of covid-19 patient that have already dengue epidemic, the two classes of vector population are considered, namely susceptible vector (mosquito) class, at time t, is denoted by S m (t); infectious vector class (I m (t)). Hence, the total vector population Z m (t) is given by

On the other hand, four classes of host (human) population; susceptible (S h (t)), infectious (I h (t)), covid-19 patients in host population (C h (t)) and recovered(R h (t)) are considered so that the total host population Z h (t) becomes

Notations and parametric values of the variables used in the formulation of dynamical transmission of model are given in the below tables 1 and 2. Let ∆ 1 is the recruitment rate of the vector population, ∆ 2 is the recruitment rate of the host population, b is the vector biting rate, η m is the transmission chance from host to vector, η h is the transmission chance from vector to host, d m is the natural death rate of vector population, d h is the natural death rate of the host population, γ h is the recovery rate of the host population, p is the fraction of covid-19 patient that have already dengue epidemic, and φ h is the covid-19 induced death rate in the host population.

The flows from the susceptible to infected classes of mosquitoes and humans populations depend on the transmission probabilites η m , η h , the bitting rate of the mosquitoes b, and the number of infectious and susceptibles of each species. In S m class, the recuritment rate of mosquiotoes population is ∆ 1 . η m is the rate of infection between S m to I h . Similarly in S h class the recuritment rate of humans population is ∆ 2 . η h is the rate of infection between S h to I m . pI h C h is the rate of flow between I h and C h and γ h + (1 − p)I h C h is the rate of flow between I h and R h respectively.

The transmission dynamics of the co-infection mathematical model is portrayed in figs. 

The classical integer model of co-infection epidemic between mosquito-to-human and viceversa is as follows:

Mosquitoes population Humans population

The intial conditions are as below. 

In this subsection, we have modified the classical integer model (2.1) into fractional-order in the sense of Caupto. The co-infection model system in the form of coupled non-linear fractional-order differential equation is given as:

The Caputo fractional-order co-infection model (3.3) gives the dynamics of host populations, and all state variables and parameters are supposed to be non-negative.

For the well-posedness of the proposed model, its existence and uniqueness theorem is presented below.

In this subsection, the well-posedness of the Caputo-Fabrizio fractional differential co-infection model (3.3) , by the use of fixed point theorems is presented. Thus, we simplify in the following way:

where,

Thus, the co-infection model (3.3) is generalized, in the following.

along with iniatal conditions

where, (.) T shows the transpose operation. By [5] , the integral representation of eqn. (4.3) is equiv. to model (3.3) and is given by

(4.5)

Let E : [0, c] → R with the norm defined by

Next, by the Schauder fixed point theorem, we prove the existence of the solutions of the eqn. (4.3) that is Equiv. to model (3.3) . So, we need the following corollary. 

The dynamical transmission of the Caupto fractional derivative of the co-infection model (3.3) will be analyzed in the following biologically feasible region, Π ⊂ R 6 + , where.

Lemma 4.5 The feasible region Π ⊂ R 6 + is +vely invariant with respect to the intial conditions in R 6 + for the co-infection model (3.3) .

In order to show that the model (3.3) has positive solution, we take R 6 Proposition 4.7 The solution for the model 

(3.3) is non-negative, bounded for all S m (0), I m (0), S h (0), I h (0), C h (0), R h (0) ∈ R 6 + ,

In order to find the equilibrium points let us assume the left hand sides of all the equations of the model (3.3) equal to zero, so that we get the five possible equilibrium points are obtained as follows: (i) The disease-free equilibrium(DFE) point:-

(ii) Infected mosquito free equilibrium point (I m = 0):-

(iii) Infected mosquito free equilibrium point (I h = 0):-

(iv) Covid-19 infected human free equilibrium point (C h = 0):-

(v) The endemic equilibrium(EE) point of the Caupto fractional-order model (3.3) is as follows:-

satisfies,

where,

Here we present the stability of the five possible equilibrium points to study the behaviour of dynamical system.

The Jacobian matrix J of the model (3.3) at E 0 is as follows:

Biologically if R 0 < 1, then the infection will be eliminate, but if R 0 > 1, the infection persist in the system and if R 0 = 1, the bifurcation occur. After some simplification, we can find the threshold quantity that is

On the base of threshold quantity we've the following proposition. In case E 0 is replace by E 1 , the rest of the analysis is similar.

Stability of human free equilibrium point E 2 (I h = 0).

In case E 0 is replace by E 2 , the rest of the analysis is similar.

The Jacobian matrix J E 3 evaluated at the covid-19 free equilibrium point and is given as below:

The characterstic eqn. of the Jacobian matrix J(E 3 ) is

where,

Let D(f ) denotes the disc. of a poly. f (z); then we've

To check the stability of the E 3 , we have the following proposition:

The necessary condition for E 3 , to be locally asymptotically stable, is c 1 > 0.

Theorem 5.4 Let 0 < α < 1, the covid-19 infection free equilibrium point of the Caupto fractional-order dengue fever and covid-19 model (3.3) , is globally asymptotically stable in the closed set Π, whenever R 0 < 1.

Proof:-See Appendix VII.

Now we discuss the stability of the endemic equilibrium point of the model (3.3) for this the Jacobian matrix J E 4 is given as below:

The characterstic eqn. of the Jacobian matrix J(E 4 ) is

where

Let D(g) denotes the disc. of a poly. g(z); then we've

To check the stability of the E 3 , we have the following proposition:

, is +ve & Routh-Hurwitz Criterian are satisfied, i.e., D(g) > 0, a 2 > 0, c 2 > 0, & a 2 b 2 > c 2 , then E 4 is locally asymptotically stable.

(iv) The necessary condition for E 4 , to be locally asymptotically stable, is c 2 > 0.

Since covid-19 is spread via host contact with infected populations [19, 22] . But here in this paper, our aim is to inform people that those who have already been infected by dengue fever have more chances of covid-19 infection as compared to those who have not. Thus, we can put the control parameter on this serious problem to prevent its spreading. The following are the control assumptions: l 1 : To control mosquitoes populations. l 2 : Infectious mosquitoes should be killed. l 3 : covid-19 patients should be quarantined. On the basis of the above assumptions, the objective function is formulated as

where Ω denotes the set of all compartmental variables, W 1 , W 2 , W 3 , W 4 , W 5 , W 6 denotes the positive weight contants for the variables S m , I m , S h , I h , C h , R h respectively. Now, we will find every value of control variables from t = 0 to T s.t.,

where, M is the smooth function for the interval [0, 1]. Therefore, the Langrangian function related to the objective function is given by,

The adjoint eqn. variables, ζ i = (ζ 1 , ζ 2 , ζ 3 , ζ 4 , ζ 5 , ζ 6 ) for the system is calculated by taking the partial derivatives of Θ with respect to each variable.

Hence, this calculation gives us,

Since most of the fractional-order differential eqs. don't have exact analytic solutions, so numerical and approximate methods must obtain these solutions. So, we are constructing a numerical technique, for the fractional model based on the Caupto-fractional derivative, Caupto-Fabrizio and Atangana-Baleanu fractional derivative. For the use of this numerical technique we take some non-linear fractional ordinary differential eqn.:

In this subsection, we deal with the following Cauchy problem

where the derivative is in Caupto fractional derivative. Here, we aim to show the Newton method. For this, we firstly change the eqn. (7.1) into

Since at the point t i+1 = (i + 1)∆t, we have

Now, we replaced the Newton poly. with the eqn. (7.4), we have

. This implies,

This implies

The integrals in the eqn. (7.7) can be written as follows

Use eqn. (7.8) into (7.7). We have the following technique

For simplicity, we write model (3. 3) with C-F fractional derivative is already discussed in eqn. (4.1) . Now the solution for the model (3.3), as followed.

× (i − n + 1) α 2(i − n) 2 + (3α + 10)(i − n) + 2α 2 + 9α + 12 − (i − n) α 2(i − n) 2 +(5α + 10)(i − n) + 5α 2 + 18α + 12 (7.12)

+(5α + 10)(i − n) + 5α 2 + 18α + 12 (7.14)

Now we impose many numerical and analytical techniques have been used to solve the fractional order differential eqs. For the numerical techniques of the fractional model (3.3) one can use the generalized ABM method. In order to find out the approximate solution by using this algorithm, we considered the following non-linear FDE [12] : 

Based on ABM algorithm to integrate eqution (7.17), [11] used the predictor-corrector scheme. On using predictor-corrector scheme to the fractional order dengue fever epidemic and covid-19 model (3.3) and letting t = H/N, h n = nt, & n = 0, 1, 2, 3, ..., N ∈ Z + . Therefore, eqn. (7.17) can be discretized as follows, we've:

1 ≤ k ≤ n, 1, k = n + 1, and the preliminary approxs. S r n+1 , I r n+1 , W r n+1 , X r n+1 , Y r n+1 & Z r n+1 are known as predictor and is given by

where

×(i − n) + 2α 2 + 9α + 12 − (i − n) α 2(i − n) 2 + (5α + 10)(i − n) + 6α 2 + 18α + 12 (7.49 )

(7.52)

In this section, we present results and discussion of co-infection mathematical model ( We have seen that the fractional modeling of biological systems can be advantage and applicable as memory effects are finding in those systems. Although α is not a biological parameter yet, it plays a very significant role in the dynamics of the model (3.3). In figures (3 − 5) , it is observed that the total population increases beginning up to time t = 10(Days) and it decreases afterward. It is also clear that the SARS-COV-2 virus takes at least 10-15 days to show symptoms and attain a peak and afterward, it declines to normal but is still prevalent in the system. Next, the variations of the variables I m (t), S h (t), & I h (t) are demonstrated in figures (6 − 8) with α = 0.97, 0.96, 0.95. In each figure, we have considered different population variables against time. From these three figures, it is learned that if we take the covid-19 infection class is equal to zero, the fractional-order models approach the fixed point over a large period. Figures (9 − 10) exhibited the variation of the variables I m (t), I h (t) & C h (t) for time t and it is seen that as we increase the value of the biting rate of mosquitoes b the line of the graph is slightly increasing. It means that we decrease the value of bitting rate b mitigates the rate of infection which is possible in real life too. Hence, the infected mosquito class is fully dependent on the biting rate. Further, from figures (11 − 12) , one thing we have seen as we increase the value of the fraction rate p of covid-19 patients the graphs of the covid-19 class C h (t) increases. 

Since dengue and covid-19 are very strange infectious diseases that cause havoc all over the world, their mode of transmission is not yet fully understood. In this paper, we investigate the dynamics of dengue and covid-19 co-infection. We computed the reproduction number and established a stability analysis based on some important theorems. It is observed that for the fractional values of α = 0.96, the curve of co-infection of dengue and covid-19 gets flattened well. It means that if we can control the values of fractional order α within suitable intervals (0, 1), the curve of the dengue and covid-19 co-infection reducible to a certain control level as depicted in figure (7) . The effects of fractional-order α and other transmission rates are also depicted graphically in figures. Our results are in good agreement with results which show that dengue fever acts as a launch pad for the SARS-COV-2 virus and causes death. We compared these different models in the sense of Caputo, Caputo-Fabrizio, and Atangana-Baleanue. It is inferred in the numerical simulation that Caputo shows still better results in the form of stability as compared to the other operators. 

The author(s) declare(s) that there has been no conflict of interest.

(11.4) Therefore, R is uniformaly bounded. Next,we will prove that R is equicountinuity. For this, we suppose sup (t,ϕ)∈L×A k |K(t, ϕ(t))| = K * . Then, for any t 1 , t 2 in L s.t. t 2 ≥ t 1 , we have

This implies, |(Rϕ)(t 2 ) − (Rϕ)(t 1 )| → 0, as t 2 → t 1 . Hence, R is equicontinuous and so A k is relatively compact. Therefore, R is completely continuous by the consequence of Arzela-Ascoli theorem. (t − χ) α−1 K(χ, ϕ(χ))dχ ≤ ϕ 0 + a * 1 + a * 2 ϕ Γ(α + 1) c α ≤ ϕ 0 + ℵ(a * 1 + a * 2 ϕ ) ≤∞.

(11.6) Therefore, J is bounded. So, R has at least one fixed point which is only the solution of the model (3.3) . Hence the result is obtained. Appendix IV On adding the first two eqs. of the co-infection model (3. 3), the total mosquitoes population, is given by

The solution of the model (3.3) is given by

where E α,β is known as the Mittag-Leffler function. On concerning [15] the point that the behaviour of this function is an asymptotic, so we've E α,β (g) ∼ − µ j=1 g −j Γ(β − αj) + O |j| −1−µ , |g| → ∞, |arg(g)| ∈ (απ/2, π] . (11.8) Therefore, from the eqn. (11.7) we've Z h (t) → ∆ 1 /d m as t → ∞.

Further, the proof in the case of the human population is completely similar to the vector population and hence that's why it is omitted. Thus, for all positive values of time, all the solutions of the Caputo fractional derivative the model (3.3) with initial conditions of Π remain in Π. Therefore, the region Π is +vely invariant fr the model (3.3) and attracts all solutions in R 6 + . Appendix V In order to explore the non-negativity solution, it is require to show that on every hyperplane bounding of R 6 + . From the model (3.3), we have:

Thus, by the corollary 4.6, the above target set has been achieved that the solution will stay in R 6 + and thus we have the following biologically feasible region: The characterstic values of the Jacobian matrix J E 0 are λ 1 = −d m , λ 2 = −d h , λ 3 = −(d h + φ h ), & λ 4 = −d h , and the other two roots are find out from the quadratic equation

where

. and is called a basic reproduction number. Hence E 0 is locally asymptotically stable if R 0 is less than 1 and is unstable if R 0 is greater than 1.

Appendix VII Let us consider the following Lyapnuov function:

where, a 1 = 1 d m and a 2 = 1 d h + γ h . Therefore, the Lyapunov function V is well defined, positive definite and continuous for all I m (t) & I h (t) > 0. By the eqn. (2.1), we have

From the Caupto fractional order dengue fever and covid-19 model (3.3), we have

So, we have

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Credit Author Statement Attiq-Ul-Rehman: Conceptualization, Methodology, Writing-Original draft, Software Ram Singh: Conceptualization, Methodology, Data Analysis, Software, Supervision Praveen Agarwal: Methodology, Data Analysis

The Cauchy problem is given as CF 0 D α t g(t) =f (t, g(t)), g(0) = g 0 , (7.20) where the derivative is Caupto fabrizio derivative. Newton method is used to solve the eqn. (7.20) . For this, we first transform the eqn. (7.20) intof (χ, g(χ))dχ. (7.21) Since at t i+1 = (i + 1)∆t, we havet i+1 0 f (χ, g(χ))dχ. (7.22) At t i = i∆t, we havet i 0 f (χ, g(χ))dχ. (7.23) From eqns. (7.25 ) and (7.24), we getandUsing the Newton poly. we can write the approx. of the function f (t, g(t)) as follows). Thus, using eqn. (7.26) into (7.25), we getand more simplified as(7.28)The calculations for the above integrals (7.28) is asNow by using eqns. (7.29) and (7.30) into (7.28) we gettn f (t n−2 , g n−2 ) + G 4 5 2 + G 5 23 6 ∆t, (7.31) and we can arrange asAfter applying the C-F derivative, we havewhere,

Here we take the following Cauchy problem.ABC 0 D α t g(t) =f (t, g(t)), g(0) = g 0 . (7.39)In this subsection, we provide a numerical scheme to solve the eqn. (7.39). Applying A-B integral, we convert the eqn. (7.39) intoHere, for the approx. of the function f (t, g(t)) we apply the Newton poly. asUsing eqn. (7.43) into (7.42) we getThis implies,Now the integral of eqn. (7.45) as×(i−n) + 2α 2 + 9α + 12 − (i − n) α (i − n + 1) α 2(i − n) 2 + (5α + 10)(i − n) + 6α 2 + 18α + 12 . Since, the function R is well defined and the unique of the model (3.3) is just the fixed point of the function. Now, let us take,Therefore, it is enough to show that RB k ⊂ B k , where B k = {ϕ ∈ E : ϕ ≤ k} is convex and closed set. Now, for any ϕ in B k , it givesThus, the target is followed. Also, for any ϕ 1 , ϕ 2 ∈ E, we haveThis implies that (Rϕ 1 )(t) − (Rϕ 2 )(t) ≤ ℵF K |ϕ 1 − ϕ 2 |. Hence, by the consquence of the principle of Banach contraction. The model (3.3) has a unique solution.Appendix II Since, the continuity of K =⇒ R. So, for any ϕ in B k , we haveAs,Since all the biological parameters of the model (3.3) are nonnegative, it follows that C 0 D α t V ≤ 0 f or R 0 < 1 with 

The author(s) declare(s) that there has been no conflict of interest.