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Research article

Atangana-Baleanu fractional dynamics of dengue fever with optimal control strategies

  • Received: 04 January 2023 Revised: 23 March 2023 Accepted: 03 April 2023 Published: 26 April 2023
  • MSC : 34H05, 49K15, 49K40

  • Dengue fever, a vector-borne disease, has affected the whole world in general and the Indian subcontinent in particular for the last three decades. Dengue fever has a significant economic and health impact worldwide; it is essential to develop new mathematical models to study not only the dynamics of the disease but also to suggest cost-effective mechanisms to control disease. In this paper, we design modified facts about the dynamics of this disease more realistically by formulating a new basic ShEhIhRh host population and SvIv vector population integer order model, later converting it into a fractional-order model with the help of the well-known Atangana-Baleanu derivative. In this design, we introduce two more compartments, such as the treatment compartment Th, and the protected traveler compartment Ph in the host population to produce ShEhIhThRhPh. We present some observational results by investigating the model for the existence of a unique solution as well as by proving the positivity and boundedness of the solution. We compute reproduction number R0 by using a next-generation matrix method to estimate the contagious behavior of the infected humans by the disease. In addition, we prove that disease free and endemic equilibrium points are locally and globally stable with restriction to reproduction number R0. The second goal of this article is to formulate an optimal control problem to study the effect of the control strategy. We implement the Toufik-Atangana scheme for the first time to solve both of the state and adjoint fractional differential equations with the ABC derivative operator. The numerical results show that the fractional order and the different constant treatment rates affect the dynamics of the disease. With an increase in the fractional order and the treatment rate, exposed and infected humans, as well as the infected mosquitoes, decrease. However, the optimal control analysis reveals that the implemented optimal control strategy is very effective for disease control.

    Citation: Asma Hanif, Azhar Iqbal Kashif Butt. Atangana-Baleanu fractional dynamics of dengue fever with optimal control strategies[J]. AIMS Mathematics, 2023, 8(7): 15499-15535. doi: 10.3934/math.2023791

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  • Dengue fever, a vector-borne disease, has affected the whole world in general and the Indian subcontinent in particular for the last three decades. Dengue fever has a significant economic and health impact worldwide; it is essential to develop new mathematical models to study not only the dynamics of the disease but also to suggest cost-effective mechanisms to control disease. In this paper, we design modified facts about the dynamics of this disease more realistically by formulating a new basic ShEhIhRh host population and SvIv vector population integer order model, later converting it into a fractional-order model with the help of the well-known Atangana-Baleanu derivative. In this design, we introduce two more compartments, such as the treatment compartment Th, and the protected traveler compartment Ph in the host population to produce ShEhIhThRhPh. We present some observational results by investigating the model for the existence of a unique solution as well as by proving the positivity and boundedness of the solution. We compute reproduction number R0 by using a next-generation matrix method to estimate the contagious behavior of the infected humans by the disease. In addition, we prove that disease free and endemic equilibrium points are locally and globally stable with restriction to reproduction number R0. The second goal of this article is to formulate an optimal control problem to study the effect of the control strategy. We implement the Toufik-Atangana scheme for the first time to solve both of the state and adjoint fractional differential equations with the ABC derivative operator. The numerical results show that the fractional order and the different constant treatment rates affect the dynamics of the disease. With an increase in the fractional order and the treatment rate, exposed and infected humans, as well as the infected mosquitoes, decrease. However, the optimal control analysis reveals that the implemented optimal control strategy is very effective for disease control.



    The polynomials of special kinds in finite fields have attracted a lot of research interest (see, e.g., [1,2,3,6,9]). Let Fq be the finite field of q elements with characteristic p and n1 be an integer. Let Iq(n) denote the number of monic irreducible polynomials of degree n over Fq and μ() the M¨obius function. Gauss [4] found that

    Iq(n)=d|nμ(d)qnd/n. (1.1)

    Let f(x)=xn+a1xn1++anFq[x] be an irreducible polynomial. Suppose that α is a root of f(x), then all the roots of f(x) are αqi,i=0,1,,n1. The trace function of α is defined to be Tr(α)=α+αq++αqn1. So a1=Tr(α) and therefore a1 is also called the trace of f(x). Let Nq(n) denote the number of monic irreducible polynomials of degree n over Fq with nonzero traces. Suppose n=mpe with pm. Carlitz [1] and Ruskey, Miers and Sawada [12] obtained that:

    Theorem 1.1. The number of monic irreducible polynomials of degree n over Fq with nonzero traces is given by

    Nq(n)=(q1)d|mμ(d)qnd/qn.

    An irreducible polynomial in Fq[x] is called a normal polynomial if its roots are linearly independent over Fq. All the roots of a normal polynomial of degree n over Fq form a normal basis of Fqn over Fq. Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography (see, e.g., [6] and [7]). It is not easy to determine whether a given irreducible polynomial is normal or not. Let ϕ() denote the Euler totient function. Using Theorem 1.1 and other results, Huang, Han and Cao [5] showed that:

    Theorem 1.2. The following inequality holds

    qnmd|m(qτ(d)1)ϕ(d)/τ(d)(q1)d|mμ(d)qn/d/q, (1.2)

    where τ(d) is the order of q modulo d. Furthermore, the following statements are equivalent:

    (i) Inequality (1.2) becomes an equality.

    (ii) n=pe, or n is a prime different from p and q is a primitive root modulo n.

    (iii) Every irreducible polynomial of degree n over Fq with nonzero trace is a normal polynomial.

    Let f(x) be an irreducible polynomial of degree n over Fq as before. For a positive divisor n1 of n, define the n1-traces of f(x) to be Tr(α;n1)=α+αq++αqn11 where α's are the roots of f(x). Let Nq(n;n1) denote the number of monic irreducible polynomials of degree n over Fq with nozero n1-traces. Obviously, the function Tr(α;n1) generalizes the usual trace function Tr(α). Gauss's formula (1.1) gives the formula for Nq(n;1) for n2 and Theorem 1.1 gives the formula for Nq(n;n). In this paper, we obtain the explicit formula for Nq(n;n1) in the general case, including a new proof to Theorem 1.1.

    To state our result, we need to introduce more notation and convention.

    ● Assume that n=mpe and n1=m1pe1 with n1n and pmm1.

    ● For a positive integer d, let P(d) denote the set of all distinct positive prime divisors of d. Assume that P(n)={p1,p2,,pk} and pk=p if e1.

    ● Without loss of generality, assume that {aP(n):aP(m1),aP(mm1)}={p1,,pt} with tk. Write m1=p1pt.

    The following is the main theorem of this paper.

    Theorem 1.3 (Main Theorem). With the above notation and convention, we have

    Nq(n;n1)={[d|nμ(d)qndd|m1μ(d)(qn1d1qn1pd)]/n if e1=e>0 and k=t+1,(d|nμ(d)qndd|m1μ(d)qn1d1)/nif e1=e=0 and k=t,d|nμ(d)qnd/notherwise.

    In particular, for n1=1,

    Nq(n;1)={Iq(n)1=q1if n=1,Iq(n)=d|nμ(d)qnd/nif n>1,

    and for n1=n,

    Nq(n;n)=Nq(n)=(q1)d|mμ(d)qnd/qn.

    Let N0q(n;n1) denote the number of monic irreducible polynomials of degree n over Fq with zero n1-traces. The following corollary is a direct consequence of Theorem 1.3.

    Corollary 1.4. With the above notation and convention, we have

    N0q(n;n1)={[d|m1μ(d)(qn1d1qn1pd)]/nif e1=e>0 and k=t+1,(d|m1μ(d)qn1d1)/n if e1=e=0 and k=t,0otherwise.

    Since our method is based on the properties of linearized polynomials in finite fields, we will introduce them in the next section. The proof of Theorem 1.3 will be given in Section 3.

    Linearized polynomials are also called q-polynomials in the literature. Many definitions and results of this section go back to the fundamental papers of Ore [8,9,10,11], and we just list them without examples and proofs for the sake of brevity; see [6] for further details.

    Let r be a positive integer. A polynomial of the form L(x)=ni=0cixqi with coefficients in an extension filed Fqr of Fq is called a linearized polynomial over Fqr. The polynomials l(x)=ni=0cixi and L(x)=ni=0cixqi over Fqr are called q-associates of each other. More specially, l(x) is the conventional q-associate of L(x) and L(x) is the linearized q-associate of l(x). Given two linearized polynomials L1(x) and L2(x) over Fqr, we define symbolic multiplication by L1(x)L2(x)=L1(L2(x)). The ordinary product of linearized polynomials need not to be a linearized polynomial. And the set of linearized polynomials over Fq forms an integral domain under the operations of symbolic multiplication and ordinary addition.

    If L1(x) and L2(x) are two linearized polynomials over Fq, we say that L1(x) symbolically divides L2(x) (or that L2(x) is symbolically divisible by L1(x)) if L2(x)=L1(x)L(x) for some linearized polynomial L(x) over Fq. Similarly, one can define symbolic factorization and symbolic irreducibility for linearized polynomials.

    Lemma 2.1. ([6,Lemma 3.59]) Let L1(x) and L2(x) be linearized polynomials over Fq with conventional q-associates l1(x) and l2(x). Then l(x)=l1(x)l2(x) and L(x)=L1(x)L2(x) are q-associates of each other.

    The following criterion is an immediate consequence of the lemma above.

    Corollary 2.2. Let L1(x) and L(x) be linearized polynomials over Fq with conventional q-associates l1(x) and l(x). Then L1(x) symbolically divides L(x) if and only if l1(x) divides l(x). In particular, L(x) is symbolically irreducible over Fq if and only if l(x) is irreducible over Fq.

    Let L(x) be a linearized polynomial over Fq with conventional q-associate l(x). We write L(x)e:=L(x)L(x)e for short. Let L(x)=ri=1Li(x)ei be the symbolic factorization with distinct symbolically irreducible linearized polynomials Li(x) over Fq, then by Lemma 2.1, l(x)=ri=1li(x)ei is the canonical factorization of l(x) in Fq[x], where li(x) is the conventional q-associate of Li(x).

    For a positive integer d, let φd(x) denote the dth cyclotomic polynomial of degree ϕ(d) where ϕ(d) is the Euler totient function, and let Ψd(x) denote the linearized q-associate of φd(x). The following lemma is well known.

    Lemma 3.1. ([6,Theorems 2.45 and 2.47])

    xn1=(xm1)pe=d|m(φd(x))pe.

    As a direct consequence of Lemmas 2.1 and 3.1, we have

    Corollary 3.2.

    xqnx=d|mΨd(x)pe.

    Definition 3.3. We define

    A={αFqn:α+αq++αqn11=0},

    and for each piP(n), we define

    Api=Fqn/pi={αFqn:αqnpiα=0}.

    Remark 3.4. (i) From the definition of Api, one easily knows that it is just the finite field of size qn/pi. However, since the sets A and Api will be considered together later, we adopt this notation to keep consistency.

    (ii) Let αFqn(piP(n)ApiA). Then from the definition above, we know that the n1-trace of α over Fq is nonzero, and that the degree of the minimal polynomial of α over Fq is just n. So α and all its conjugates in Fqn form the roots of an irreducible polynomial of degree n over Fq with nonzero n1-traces.

    To calculate the intersection of A and Api's, we need to factorize the two polynomials x+xq++xqn11 and xqnpix into irreducible linearized polynomials.

    Lemma 3.5.

    A={αFqn:(αqα)pe111<d|m1Ψd(α)pe1=0}, (3.1)

    and

    Api={αFqn:d|npiΨd(α)=0}, (3.2)

    for piP(n). In particular, if pip, then

    Api={αFqn:(αqα)pe1<d|mpiΨd(α)pe=0}, (3.3)

    and

    Ap={αFqn:(αqα)pe11<d|mΨd(α)pe1=0}. (3.4)

    Proof. Using Corollary 3.2 and the factorization

    xn11=(x1)(x1)pe111<d|m1(φd(x))pe1

    that comes from Lemma 3.1, we get (3.1). Similarly, we can get (3.2), (3.3) and (3.4).

    Suppose that J={pj1,,pjr}P(n). Without loss of generality, assume that {aJ:aP(m1),aP(mm1)}={pj1,,pjt}, and p=pjr if pJ. Observe that if pJ, then

    gcd(m1,mpj1pjr)=gcd(m1,m1pj1pjtmm1pjt+1pjr)=m1pj1pjt, (3.5)

    and if p=pjrJ, then

    gcd(m1,mpj1pjr1)=gcd(m1,m1pj1pjtmm1pjt+1pjr1)=m1pj1pjt. (3.6)

    We will use the two observations (3.5) and (3.6) in the lemma below.

    Lemma 3.6. (i) If  pJ or p=pjrJ and e1<e, then

    (jJAj)A={αFqn:(αqα)pe111<d|m1pj1pjtΨd(α)pe1=0}.

    (ii) If p=pjrJ and e1=e, then

    (jJAj)A={αFqn:(αqα)pe111<d|m1pj1pjtΨd(α)pe11=0}.

    Proof. To calculate (jJAj)A, we first calculate (jJAj). By (3.2),

    jJAj={αFqn:d|npj1pjrΨd(α)=0}. (3.7)

    If pJ, then by (3.3) and (3.7),

    jJAj={αFqn:(αqα)pe1<d|mpj1pjrΨd(α)pe=0}. (3.8)

    If p=pjrJ, then by (3.4) and (3.7),

    jJAj={αFqn:(αqα)pe11<d|mpj1pjr1Ψd(α)pe1=0}. (3.9)

    Now we calculate (jJAj)A. Recall the observations (3.5) and (3.6) and that 0e1e which will be used below. If  pJ, then by (3.1) and (3.8),

    (jJAj)A={αFqn:(αqα)min{pe,pe11}1<d|gcd(m1,mpj1pjr)Ψd(α)min{pe,pe1}=0}={αFqn:(αqα)pe111<d|m1pj1pjtΨd(α)pe1=0}.

    If p=pjrJ, then by (3.1) and (3.9),

    (jJAj)A={αFqn:(αqα)min{pe1,pe11}1<d|gcd(m1,mpj1pjr1)Ψd(α)min{pe1,pe1}=0}={{αFqn:(αqα)pe111<d|m1pj1pjtΨd(α)pe11=0}if e1=e,{αFqn:(αqα)pe111<d|m1pj1pjtΨd(α)pe1=0}if e1<e.

    This finishes the proof.

    The following lemma plays a vital role in the proof of our main theorem.

    Lemma 3.7. (i) |A|=qn11.

    (ii) |jJAj|=qnpj1pjr.

    (iii) |(jJAj)A|=qn1ppj1pjt if pJ and e1=e.

    (iv) |(jJAj)A|=qn1pj1pjt1 if pJ or e1<e.

    Proof. Since (ⅰ) and (ⅱ) are trivial, we only need to prove (ⅲ) and (ⅳ). First consider the case pJ. Set

    G(x)=(xqx)pe111<d|m1pj1pjtΨd(x)pe1.

    By Lemma 3.6 (ⅰ),

    (jJAj)A={αFqn:G(α)=0}.

    Notice that the degree of G(x) is

    qpe11+pe1(dm1pj1pjtϕ(d)1)=qm1pe1pj1pjt1=qn1pj1pjt1,

    where we use the fact that dm1pj1pjtϕ(d)=m1pj1pjr. So G(x) has qn1pj1pjt1 simple roots. Similarly for the case pJ, and we can get

    |(jJAj)A|={qm1pe11pj1pjt=qn1ppj1pjt if  p=pjrJ and e1=e,qm1pe1pj1pjt1=qn1pj1pjt1 if  pJ or e1<e.

    The result follows.

    Now we are in the position to prove our main theorem.

    Proof of Theorem 1.3. Write Ai=Api(i=1,,k) for short. Let αFqn(ki=1AiA). By Remark 3.4 (ii), we know that Tr(α;n1)0 and that the degree of the minimal polynomial of α over Fq is n. Since all the conjugates of α have the same property, namely, Tr(αqi;n1)0 and the degree of the minimal polynomial of αqi over Fq is n for i=0,1,,n1, we have

    Nq(n;n1)=|Fqn(ki=1AiA)|/n. (3.10)

    We first consider the simplest two cases that e1<e and e1=e=0, in which Lemma 3.7 (iii) is not used that makes the calculation relatively easy. By the inclusion-exclusion principle and Lemma 3.7,

    \begin{eqnarray*} && \big|\mathbb{F}_{q^{n}}\backslash(\bigcup\limits_{i = 1}^{k}A_{i}\bigcup A)\big| \nonumber \\ & = & q^{n}+\sum\limits_{l = 1}^{k}(-1)^l\sum\limits_{|I| = l}\big|\bigcap\limits_{i\in I\subseteq \{1, \dots, k\}}A_i\big|+\sum\limits_{l = 1}^{k+1}(-1)^l\sum\limits_{l_1+l_2 = l-1}\big|\bigcap\limits_{i\in I_1\subseteq \{1, \dots, t\}\atop |I_1| = l_1}A_i\bigcap\bigcap\limits_{j\in I_2\subseteq \{t+1, \dots, k\}\atop |I_2| = l_2}A_j\bigcap A\big|\nonumber \\ & = & q^{n}+\sum\limits_{l = 1}^{k}(-1)^l\underset{1\leq i_{1} \lt \cdots \lt i_{l}\leq k}\sum q^{\frac{n}{p_{i_{1}}\cdots p_{i_{l}}}}+\sum\limits_{l = 1}^{k+1}(-1)^l\sum\limits_{l_1+l_2 = l-1}{k-t \choose l_2}\underset{1\leq i_{1} \lt \cdots \lt i_{l_1}\leq t}\sum q^{\frac{n_{1}}{p_{i_{1}}\cdots p_{i_{l_1}}}-1}. \nonumber \end{eqnarray*}

    For a positive integer, let \omega(d) denote the number of distinct prime factors of d . Recall \mathfrak{P}(n) = \{p_{1}, p_{2}, \ldots, p_{k}\} and m_1' = p_{1}\cdots p_{t} . Hence the above equation can be rewritten as

    \begin{equation} \big|\mathbb{F}_{q^{n}}\backslash(\bigcup\limits_{i = 1}^{k}A_{i}\bigcup A)\big| = \sum\limits_{d|n}\mu(d)q^{\frac{n}{d}}-\sum\limits_{l = 0}^{k}(-1)^l\sum\limits_{d|m_1'}{k-t \choose l-\omega(d)}q^{\frac{n_{1}}{d}-1}. \end{equation} (3.11)

    Note that

    \begin{equation} \begin{aligned}[t] &\sum\limits_{l = 0}^{k}(-1)^l\sum\limits_{d|m_1'}{k-t \choose l-\omega(d)}q^{\frac{n_{1}}{d}-1} \\ = &q^{n_{1}-1}-\sum\limits_{i = 1}^{t}q^{\frac{n_{1}}{p_{i}}-1}-{k-t \choose 1}q^{n_{1}-1}+\underset{1\leq j_{1} \lt j_{2}\leq t}\sum q^{\frac{n_{1}}{p_{j_{1}}p_{j_{2}}}-1}+{k-t \choose 1}\sum\limits_{i = 1}^{t}q^{\frac{n_{1}}{p_{i}}-1} \\ &+{k-t \choose 2}q^{n_{1}-1}-\underset{1\leq j_{1} \lt j_{2} \lt j_{3}\leq t}\sum q^{\frac{n_{1}}{p_{j_{1}}p_{j_{2}}p_{j_{3}}}-1}-{k-t \choose 1}\underset{1\leq j_{1} \lt j_{2}\leq t}\sum q^{\frac{n_{1}}{p_{j_{1}}p_{j_{2}}}-1} \\ &-{k-t \choose 2}\sum\limits_{i = 1}^{t}q^{\frac{n_{1}}{p_{i}}-1}-{k-t \choose 3}q^{n_{1}-1} +\cdots \\ = & \underset{d|m_1'}\sum\mu{(d)}q^{\frac{n_1}{d}-1}-{k-t \choose 1}\underset{d|m_1'}\sum\mu{(d)}q^{\frac{n_1}{d}-1}+{k-t \choose 2}\underset{d|m_1'}\sum\mu{(d)}q^{\frac{n_1}{d}-1}+\cdots \\ = & \left\{ \begin{array}{ll} \underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{d}-1} & \hbox{if} \ k = t, \\ 0 & \hbox{if}\ k \gt t. \end{array} \right. \end{aligned} \end{equation} (3.12)

    By (3.10), (3.11) and (3.12), we get

    \begin{equation} N_q^*(n;n_1) = \left\{ \begin{array}{ll} \underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}/n & \hbox{if } e_1 \lt e \hbox{ or } e_1 = e = 0 \hbox{ and } k \gt t, \\ (\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{d}-1})/n & \hbox{if } e_1 = e = 0 \hbox{ and } k = t. \end{array} \right. \end{equation} (3.13)

    Next we consider the more complicated case that e_1 = e > 0 , in which Lemma 3.7 (ⅲ) is used that makes the calculation relatively lengthy. We omit some detail due to the similar deduction and notation as above. Note that p_k = p in this case by assumption. By the inclusion-exclusion principle,

    \begin{eqnarray} && \big|\mathbb{F}_{q^{n}}\backslash(\bigcup\limits_{i = 1}^{k}A_{i}\bigcup A)\big| \\ & = & q^{n}+\sum\limits_{l = 1}^{k}(-1)^l\sum\limits_{|I| = l}\big|\bigcap\limits_{i\in I\subseteq \{1, \dots, k\}}A_i\big| \\ & & +\sum\limits_{l = 1}^{k}(-1)^l\sum\limits_{l_1+l_2 = l-1}\big|\bigcap\limits_{i\in I_1\subseteq \{1, \dots, t\}\atop |I_1| = l_1}A_i\bigcap\bigcap\limits_{j\in I_2\subseteq \{t+1, \dots, k-1\}\atop |I_2| = l_2}A_j\bigcap A\big| \\ & & +\sum\limits_{l = 2}^{k+1}(-1)^l\sum\limits_{l_1+l_2 = l-2}\big|\bigcap\limits_{i\in I_1\subseteq \{1, \dots, t\}\atop |I_1| = l_1}A_i\bigcap\bigcap\limits_{j\in I_2\subseteq \{t+1, \dots, k-1\}\atop |I_2| = l_2}A_j\bigcap A_k\bigcap A\big|. \end{eqnarray} (3.14)

    By Lemma 3.7,

    \begin{eqnarray} && \sum\limits_{l = 1}^{k}(-1)^l\sum\limits_{l_1+l_2 = l-1}\big|\bigcap\limits_{i\in I_1\subseteq \{1, \dots, t\}\atop |I_1| = l_1}A_i\bigcap\bigcap\limits_{j\in I_2\subseteq \{t+1, \dots, k-1\}\atop |I_2| = l_2}A_j\bigcap A\big| \\ & = & \sum\limits_{l = 1}^{k}(-1)^l\sum\limits_{l_1+l_2 = l-1}{k-t-1 \choose l_2}\underset{1\leq i_{1} \lt \cdots \lt i_{l_1}\leq t}\sum q^{\frac{n_{1}}{p_{i_{1}}\cdots p_{i_{l_1}}}-1} \\ & = &-\sum\limits_{l = 0}^{k-1}(-1)^l\sum\limits_{d|m_1'}{k-t-1 \choose l-\omega(d)}q^{\frac{n_{1}}{d}-1}, \end{eqnarray} (3.15)

    and

    \begin{eqnarray} && \sum\limits_{l = 2}^{k+1}(-1)^l\sum\limits_{l_1+l_2 = l-2}\big|\bigcap\limits_{i\in I_1\subseteq \{1, \dots, t\}\atop |I_1| = l_1}A_i\bigcap\bigcap\limits_{j\in I_2\subseteq \{t+1, \dots, k-1\}\atop |I_2| = l_2}A_j\bigcap A_k\bigcap A\big| \\ & = &\sum\limits_{l = 2}^{k+1}(-1)^l\sum\limits_{l_1+l_2 = l-2}{k-t-1 \choose l_2}\underset{1\leq i_{1} \lt \cdots \lt i_{l_1}\leq t}\sum q^{\frac{n_{1}}{pp_{i_{1}}\cdots p_{i_{l_1}}}}. \\ & = &\sum\limits_{l = 0}^{k-1}(-1)^l\sum\limits_{d|m_1'}{k-t-1 \choose l-\omega(d)}q^{\frac{n_{1}}{pd}}. \end{eqnarray} (3.16)

    It follows from (3.14), (3.15) and (3.16) that

    \begin{equation} \big|\mathbb{F}_{q^{n}}\backslash(\bigcup\limits_{i = 1}^{k}A_{i}\bigcup A)\big| = \sum\limits_{d|n}\mu(d)q^{\frac{n}{d}}-\sum\limits_{l = 0}^{k-1}(-1)^l\sum\limits_{d|m_1'}{k-t-1 \choose l-\omega(d)}(q^{\frac{n_{1}}{d}-1}-q^{\frac{n_{1}}{pd}}). \end{equation} (3.17)

    Similar to (3.12), we have

    \begin{align} & \sum\limits_{l = 0}^{k-1}(-1)^l\sum\limits_{d|m_1'}{k-t-1 \choose l-\omega(d)}(q^{\frac{n_{1}}{d}-1}-q^{\frac{n_{1}}{pd}})\quad\quad\quad\quad\quad\quad\quad\quad \\ = & \left\{ \begin{array}{ll} \underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{d}-1}-\underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{pd}} & \hbox{if}\ k = t+1, \\ 0 & \hbox{if} \ k \gt t+1, \end{array} \right. \end{align} (3.18)

    By (3.10), (3.17) and (3.18), we get

    \begin{equation} N_q^*(n;n_1) = \left\{ \begin{array}{ll} \underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}/n & \hbox{if } e_1 = e \gt 0 \hbox{ and } k \gt t+1, \\ \big[\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}(q^{\frac{n_1}{d}-1}-q^{\frac{n_1}{pd}})\big]/n & \hbox{if } e_1 = e \gt 0 \hbox{ and } k = t+1. \end{array} \right. \end{equation} (3.19)

    Putting (3.13) and (3.19) together, we obtain

    \begin{equation} N_q^*(n;n_1) = \left\{ \begin{array}{ll} \big[\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}(q^{\frac{n_1}{d}-1}-q^{\frac{n_1}{pd}})\big]/n & \hbox{ if } e_1 = e \gt 0\ \hbox{ and } k = t+1, \\ (\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{d}-1})/n & \hbox{ if } e_1 = e = 0 \hbox{ and } k = t, \\ \underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}/n & \hbox{ otherwise}, \end{array} \right. \end{equation} (3.20)

    as desired.

    Now suppose that n_1 = 1 in which e_1 = 0 . So the case e_1 = e > 0 is excluded. For the case e_1 = e = 0 and k = t , since t = 1 , we must have k = 1 and hence n = 1 . By (3.20),

    \begin{equation*} N_q^*(1;1) = \underset{d|1}\sum\limits\mu{(d)}q^{\frac{1}{d}}-\underset{d|1}\sum\limits\mu{(d)}q^{\frac{1}{d}-1} = q-1 = I_q(1)-1. \end{equation*}

    For the other cases, by (3.20) we get N_q^*(n; 1) = \sum_{d|n}\mu(d)q^{\frac{n}{d}}/n = I_q(n) .

    Finally suppose that n_1 = n in which m_1 = m and e = e_1 . There are two subcases: e = e_1 = 0 and e = e_1 > 0 .

    ● If e = e_1 = 0 , then n = m = n_1 = m_1 , and by (3.20) it is easy to verify that

    N_q^*(n;n) = (\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}q^{\frac{n_1}{d}-1})/n = (q-1)\sum\limits\limits_{d|m}\mu(d)q^{\frac{n}{d}}/qn.

    ● If e = e_1 > 0 , then k = t+1 and hence by (3.20),

    \begin{eqnarray*} N_q^*(n;n) & = & \big[\underset{d|n}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m_1'}\sum\limits\mu{(d)}(q^{\frac{n_1}{d}-1}-q^{\frac{n_1}{pd}})\big]/n \\ & = & \big[\underset{d|m}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m}\sum\limits\mu{(d)}q^{\frac{n}{pd}}-\underset{d|m}\sum\limits\mu{(d)}(q^{\frac{n}{d}-1}-q^{\frac{n}{pd}})\big]/n \\ & = & \big(\underset{d|m}\sum\limits\mu{(d)}q^{\frac{n}{d}}-\underset{d|m}\sum\limits\mu{(d)}q^{\frac{n}{d}-1}\big)/n\\ & = & (q-1)\sum\limits\limits_{d|m}\mu(d)q^{\frac{n}{d}}/qn. \end{eqnarray*}

    The proof is complete.

    The main contribution of this paper is to provide a new proof to Theorem 1.1 and its generalizations. To be precise, based on the properties of linearized polynomials, we count the monic irreducible polynomials of degree n over \mathbb{F} _q with nozero n_1 -traces, where n_1 is a divisor of n . Since n_1 -traces have the close relationship with the roots (and hence the coefficients) of the associated polynomial, this approach may be adopted to deal with the other enumeration problems concerning the irreducible polynomials with the restricted coefficients in finite fields.

    As one reviewer pointed out, the first two cases (i.e. e_1 = e > 0 and k = t+1 , respectively, e_1 = e = 0 and k = t ) in Theorem 1.3 only occur for n = n_1 . Thus it remains to consider the case n < n_1 , which can be deduced from the additive version of Hilbert's Theorem 90. However, if we drop out the restriction that the polynomials must be irreducible, i.e., allowing the degree of \alpha in \mathrm{Tr}(\alpha; n_1) to be less than n , our approach still works while Hilbert's Theorem 90 may not be valid again.

    The authors sincerely thank the referees for their helpful comments which led to a substantial improvement of this paper. This work was jointly supported by the National Natural Science Foundation of China (Grant No. 11871291) and Ningbo Natural Science Foundation (Grant No. 2019A610035), and sponsored by the K. C. Wong Magna Fund in Ningbo University.

    The authors declare that there is no conflict of interest.



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