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Optimal control for dengue eradication program under the media awareness effect

  • Dipo Aldila EMAIL logo
Veröffentlicht/Copyright: 2. Dezember 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 24 Heft 1

Abstract

In this article, a mathematical model is proposed to assess the effects of media awareness on dengue eradication programs. First, the existence and local stability of equilibrium points are discussed using the concept of the basic reproduction number. Using the center-manifold theorem, it is shown that the proposed model always undergoes a forward bifurcation at the basic reproduction number equal to unity. It is observed that the high-intensity media awareness could reduce the size of the endemic equilibrium. Based on local sensitivity analysis, we identify the three most sensitive parameters, namely the natural death rate of mosquito (μ v ), infection rates (β h1, β v1), and hospitalization rate (η). Hence, control variables need to be introduced to increase/reduce these parameters. In this article, we use three different control variables, namely the media campaign, (u 1(t)), to reduce infection rates, additional hospitalization rate, (u 2(t)), and fumigation rate, (u 3(t)), to increase mosquitoes death rate. Pontryagin’s maximum principle is used to determine the optimal conditions. Some numerical simulations are performed to describe a possible scenario in the field. Cost effectiveness analysis is then conducted to determine the best strategy for the dengue eradication program. We conclude that a combination of media campaigns and fumigation is the most effective strategy to prevent a significant increase in the number of infected individuals.

Keywords: basic reproduction number; center-manifold; cost effectiveness; dengue; media awareness; optimal control
Corresponding author: Dipo Aldila, Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia, E-mail:

Funding source: Kementerian Riset Teknologi Dan Pendidikan Tinggi Republik Indonesia

Award Identifier / Grant number: NKB-2803/UN2.RST/HKP.05.00/2020

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This research was financially supported by Universitas Indonesia, under the PPI-Q2 research grant scheme 2021 (ID number: NKB-590/UN2.RST/HKP.05.00/2021).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Proof of Lemma 1

From dengue model in (2), since

d S d t S = 0 , I ≥ 0 , H ≥ 0 , R ≥ 0 , U ≥ 0 , V ≥ 0 = Λ h + δ R ,

we have that the rates are non-negative on the boundary planes of R + 6 . Hence, we have that all vector field are directed inward from the boundary. Using similar approach, we have that d I d t > 0 , d H d t > 0 , d R d t > 0 , d U d t > 0 , d V d t > 0 in the boundary planes of R + 6 . Hence, whenever the systems start with a non-negative initial conditions in R + 6 , all solutions of system (2) will be positive. Hence, the proof is completed.

Appendix B: Proof of Lemma 2

Summing up human population in system (2), we have

d N d t = d S d t + d I d t + d H d t + d R d t = Λ h − μ h ( S + I + H + R ) = λ h − μ h N .

Solving above equation yields

N ( t ) = N 0 e − μ h t + Λ h μ h 1 − e − μ h t ,

where N 0 = S 0 + I 0 + H 0 + R 0. If N 0 < Λ h μ h , then d N d t > 0 which means that N(t) is increasing, and it tends to N ( t ) = Λ h μ h . On the other hand, if N 0 > Λ h μ h , then d N d t < 0 which means that N(t) is decreasing to Λ h μ h for t → ∞. Lastly, if N 0 = Λ h μ h , then d N d t = 0 which means that the total of population remain constant in time at N ( t ) = Λ h μ h . A similar approach can be used to prove M ( t ) → Λ v μ v for t → ∞. Hence, the region Ω attracts all solution in R + 6 .

Appendix C: Proof of Theorem 1

The Jacobian matrix of system (2) at E 0 is given by

J 0 = − μ h 0 0 δ 0 − Λ h β h 1 μ h 0 − η − α 0 − μ h 0 0 0 Λ h β h 1 μ h 0 η − α 1 − μ h 0 0 0 0 α 0 α 1 − δ − μ h 0 0 0 − Λ v β v 1 μ v 0 0 − μ v 0 0 Λ v β v 1 μ v 0 0 0 − μ v .

We have four negative eigenvalues of J 0, λ 1 = −μ v , λ 2 = −μ v , λ 3 = −(α 1 + μ h ), λ 4 = −(μ h + δ), while the other two are taken from the root of the polynomial a 2 λ 2 + a 1 λ + a 0 = 0, where a 2 = μ h μ v , a 1 = μ h μ v (η + α 0 + μ h + μ v ), and a 0 = μ h μ v 2 ( η + α o + μ h ) ( 1 − R 0 ) . It is evident that the last two eigenvalues have negative real components if a 0 < 0 ⟺ R 0 < 1 . Hence, E 0 is locally asymptotically stable if and only if R 0 < 1 .

Appendix D: The endemic equilibrium

The endemic equilibrium of dengue model in system (2) is given by

E 1 = S , I , H , R , U , V = S † , I † , H † , R † , U † , V † ,

where

S † = μ v m + I † I † 2 ( β v 1 − β v 2 ) + I † m β v 1 + I † μ v + m μ v η + α 0 + μ h Λ v I † ( β v 1 − β v 2 ) + m β v 1 I † ( β h 1 − β h 2 ) + m β h 1 , H † = η I † α 1 + μ h , R † = I † η α 1 + α 0 α 1 + α 0 μ h α 1 + μ h δ + μ h , U † = Λ v m + I † I † 2 ( β v 1 − β v 2 ) + I † m β v 1 + I † μ v + m μ v , V † = I † Λ v I † β v 1 − β v 2 + m β v 1 μ v I † 2 β v 1 − β v 2 + I † m β v 1 + I † μ v + m μ v .

Appendix E: Proof of Theorem 2

From the form of E 1, it is evident that S †, H †, R †, U †, V † will be positive if I † is positive. Therefore, E 1 exists if P 1(I) has a positive root, I. Since b 3 < 0, we have that lim I → − ∞ P 1 ( I ) = − ∞ , and lim I → ∞ P 1 ( I ) = ∞ . Let us assume that b 0 = 0 ⟺ R 0 = 1 . In this case, we also assume that there are no positive roots of P 1(I), and that one of its roots is zero. Therefore, in the case that b 0 < 0 ⟺ R 0 > 1 , P 1(I) will be shifted in y-positive direction, which will produce at least one positive root. Therefore, we have at least one dengue-endemic equilibrium when R 0 > 1 . For an illustration of the proof, please consult Figure 1. This completes the proof.

Appendix F: Proof of Theorem 3

For the sake of written simplification, let us assume the following:

S = x 1 , I = x 2 , H = x 3 , R = x 4 , U = x 5 , V = x 6 , d S d t = g 1 , d I d t = g 2 , d H d t = g 3 , d R d t = g 4 , d U d t = g 5 , d V d t = g 6 .

Hence, the Dengue model in system (2) now read as

(17) g 1 = Λ h − β h 1 − β h 2 x 2 + x 3 m + x 2 + x 3 x 1 x 6 − μ h x 1 + δ x 4 , g 2 = β h 1 − β h 2 x 2 + x 3 m + x 2 + x 3 x 1 x 6 − α 0 x 2 − η x 2 − μ h x 2 , g 3 = η x 2 − α 1 x 3 − μ h x 3 , g 4 = α 0 x 2 + α 1 x 3 − μ h x 4 − δ x 4 , g 5 = Λ v − β v 1 − β v 2 x 2 + x 3 m + x 2 + x 3 x 5 x 2 − μ v x 5 , g 6 = β v 1 − β v 2 x 2 + x 3 m + x 2 + x 3 x 5 x 2 − μ v x 6 .

Next, we choose β 1 as the bifurcation parameter, then setting R 0 2 = 1 gives

β h 1 = β h 1 * = R 0 μ h μ v 2 η + α 0 + μ h Λ h Λ v β v 1 .

The Jacobian matrix of system (17) at the dengue-free equilibrium is given as,

(18) J ( E 0 ) = − μ h 0 0 δ 0 − Λ h β h 1 μ h 0 − η − α 0 − μ h 0 0 0 Λ h β h 1 μ h 0 η − α 1 − μ h 0 0 0 0 α 0 α 1 − δ − μ h 0 0 0 − Λ v β v 1 μ v 0 0 − μ v 0 0 Λ v β v 1 μ v 0 0 0 − μ v .

The system (17) at the dengue-free equilibrium E 0 evaluated for β h 1 = β h 1 * has a simple zero eigenvalues λ 1 = 0, whereas the other five eigenvalues are λ 2 = −μ v , λ 3 = −μ h , λ 4 = −(α 1 + μ h ), λ 5 = −(δ + μ h ), and δ 6 = −(α 0 + η + μ h + μ v ) which all negative. Hence, we apply the Center Manifold Theorem to analyze system (17) near β h 1 = β h 1 * , which represent the analysis in the neighborhood of R 0 = 1 .

The Jacobian matrix J(E 0) at β h 1 = β h 1 * has a right eigenvector, which associated with λ 1 = 0, given by w ⃗ = ( w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) . The right eigenvector w ⃗ is obtained as follows:

(19) w 1 = − δ η + δ α 1 + δ μ h + η α 1 + η μ h + α 0 α 1 + α 0 μ h + α 1 μ h + μ h 2 η δ + μ h , w 2 = α 1 + μ h η , w 3 = 1 , w 4 = η α 1 + α 0 α 1 + α 0 μ h η δ + μ h , w 5 = − α 1 + μ h Λ v β v 1 η μ v 2 , w 6 = α 1 + μ h Λ v β v 1 η μ v 2 .

In a similar way, Jacobian matrix J(E 0) at β h 1 = β h 1 * also has left eigenvector that associated with λ 1 = 0, given by v ⃗ = ( v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) , where

(20) v 1 = 0 , v 2 = 1 , v 3 = 0 , v 4 = 0 , v 5 = 0 , v 6 = μ v η + α 0 + μ h Λ v β v 1 .

To determine the bifurcation type of system (17) at R 0 = 1 , we will calculate the following second order partial derivatives of g i at the dengue-free equilibrium E 0. Since the eigenvector v 1 = 0, v 3 = 0, v 4 = 0, and v 5 = 0, we do not need to look for a partial derivative of g 1, g 3, g 4 and g 5. Therefore, we only calculate the second derivatives of g 2 and g 6. From non-zero g 2 and g 6, the second order derivatives are as follows:

∂ 2 g 2 ∂ x 1 ∂ x 2 = ∂ 2 g 2 ∂ x 2 ∂ x 1 = 1 Λ μ + ξ + η 3 μ + ξ + η 2 + γ 2 − Λ β 3 + μ 2 + μ ξ + μ η 3 γ 2 + μ + ξ + η 3 − Λ β 2 + μ 2 + μ ξ + μ η 2 δ + μ + ξ + η 2 + γ 2 − Λ β 3 + μ 2 + μ ξ + μ η 3 γ 1 + μ η 1 + μ μ + ξ + η 3 , ∂ 2 g 2 ∂ x 1 ∂ x 3 = ∂ 2 g 2 ∂ x 3 ∂ x 1 = β 2 , ∂ 2 g 2 ∂ x 1 ∂ x 4 = ∂ 2 g 2 ∂ x 4 ∂ x 1 = β 3 , ∂ 2 g 2 ∂ x 2 ∂ β 1 = ∂ 2 g 2 ∂ β 1 ∂ x 2 = Λ μ . ∂ 2 g 2 ∂ x 1 ∂ x 6 = ∂ 2 g 2 ∂ x 6 ∂ x 1 = β h 1 , ∂ 2 g 2 ∂ x 2 ∂ x 6 = ∂ 2 g 2 ∂ x 6 ∂ x 2 = − Λ h β h 2 μ h m , ∂ 2 g 2 ∂ x 3 ∂ x 6 = ∂ 2 g 2 ∂ x 6 ∂ x 3 = − Λ h β h 2 μ h m , ∂ 2 g 6 ∂ x 2 ∂ x 3 = ∂ 2 g 6 ∂ x 3 ∂ x 2 = − Λ v β v 2 μ v m , ∂ 2 g 6 ∂ x 2 ∂ x 5 = ∂ 2 g 6 ∂ x 5 ∂ x 2 = β v 1 , ∂ 2 g 2 ∂ x 6 ∂ β h 1 = ∂ 2 g 2 ∂ β h 1 ∂ x 6 = Λ h μ h ,

Now, let us calculate the coefficient A and B in Theorem 4.1 [54] of Castillo–Chavez and Song as follows.

A = ∑ k , i , j = 1 3 v k w i w j ∂ 2 g k ∂ x i ∂ x j ( 0,0 ) = v 2 w 1 w 6 ∂ 2 g 2 ∂ x 1 ∂ x 6 + v 2 w 2 w 6 ∂ 2 g 2 ∂ x 2 ∂ x 6 + v 2 w 3 w 6 ∂ 2 g 2 ∂ x 3 ∂ x 6 + v 2 w 6 w 1 ∂ 2 g 2 ∂ x 6 ∂ x 1 + v 2 w 6 w 2 ∂ 2 g 2 ∂ x 6 ∂ x 2 + v 2 w 6 w 3 ∂ 2 g 2 ∂ x 6 ∂ x 3 + v 6 w 2 w 3 ∂ 2 g 6 ∂ x 2 ∂ x 3 + v 6 w 2 w 5 ∂ 2 g 6 ∂ x 2 ∂ x 5 + v 6 w 3 w 2 ∂ 2 g 6 ∂ x 3 ∂ x 2 + v 6 w 5 w 2 ∂ 2 g 6 ∂ x 5 ∂ x 2 = − 2 α 1 + μ h η 2 δ + μ h μ v 2 μ h m β v 1 δ η m Λ v β h 1 β v 1 2 μ h + δ η m α 1 β v 1 2 μ h μ v + δ η m β v 1 2 μ h 2 μ v + δ m Λ v α 1 β h 1 β v 1 2 μ h + δ m Λ v β h 1 β v 1 2 μ h 2 + δ m α 0 α 1 β v 1 2 μ h μ v + δ m α 0 β v 1 2 μ h 2 μ v + δ m α 1 β v 1 2 μ h 2 μ v + δ m β v 1 2 μ h 3 μ v + η m Λ v α 1 β h 1 β v 1 2 μ h + η m Λ v β h 1 β v 1 2 μ h 2 + η m α 1 β v 1 2 μ h 2 μ v + η m β v 1 2 μ h 3 μ v + m Λ v α 0 α 1 β h 1 β v 1 2 μ h + m Λ v α 0 β h 1 β v 1 2 μ h 2 + m Λ v α 1 β h 1 β v 1 2 μ h 2 + m Λ v β h 1 β v 1 2 μ h 3 + m α 0 α 1 β v 1 2 μ h 2 μ v + m α 0 β v 1 2 μ h 3 μ v + m α 1 β v 1 2 μ h 3 μ v + m β v 1 2 μ h 4 μ v + δ η 2 β v 2 μ h μ v 2 + δ η Λ h Λ v β h 2 β v 1 2 + δ η α 0 β v 2 μ h μ v 2 + δ η β v 2 μ h 2 μ v 2 + δ Λ h Λ v α 1 β h 2 β v 1 2 + δ Λ h Λ v β h 2 β v 1 2 μ h + η 2 β v 2 μ h 2 μ v 2 + η Λ h Λ v β h 2 β v 1 2 μ h + η α 0 β v 2 μ h 2 μ v 2 + η β v 2 μ h 3 μ v 2 + Λ h Λ v α 1 β h 2 β v 1 2 μ h + Λ h Λ v β h 2 β v 1 2 μ h 2 < 0

B = ∑ k , i = 1 3 v k w i ∂ 2 g 2 ∂ x 6 ∂ β h 1 ( 0 , 0 ) = α 1 + μ h Λ v β v 1 Λ h η μ h μ v 2 > 0 .

It can be seen that A < 0 and B > 0 . Hence, we establish the theorem, which states that model (2) always exhibit a forward bifurcation phenomenon at R 0 = 1 .

Appendix G: Proof of Theorem 4

The existence of optimal control of dengue control model in system (10) can be analyzed using the results by authors in [55]. Using a similar approach as in Lemma 2, it can be shown that system (10) is bounded above which allow us to use the result in [55] to show the existence of optimal control on our problem.

The integrand of the objective function which is given by

I ( t ) + w 1 u 1 2 ( t ) + w 2 u 2 2 ( t ) + w 3 u 3 2 ( t )

is convex on a convex and closed set of U . Furthermore, the state variables on human and mosquito, also control variables are non-negative. The compactness of the system is coming from the boundedness of the control system. Last, there exist a constant ζ > 1 and positive number k 1, k 2 such that

I ( t ) + w 1 u 1 2 ( t ) + w 2 u 2 2 ( t ) + w 3 u 3 2 ( t ) ≥ k 1 + k 2 ( | u 1 | 2 + | u 2 | 2 + | u 3 | 2 ) ζ ,

where k 1 is the lower bound of I(t), and k 2 = min(w 1, w 2, w 3).

Appendix H: Proof of Theorem 5

The adjoint variables z i  (i = 1, 2, 3, 4, 5, 6) satisfy the following system of ordinary differential equations.

(21) d z 1 d t = − ∂ L ∂ S , = β h 1 − β h 2 I + H ( 1 − u 1 ) m + I + H V z 1 − z 2 + μ h z 1 , d z 2 d t = − ∂ L ∂ I , = − 1 + β h 2 I + H ( ( 1 − u 1 ) m + I + H ) 2 − β h 2 1 ( 1 − u 1 ) m + I + H S V ( z 1 − z 2 ) + α 0 ( z 2 − z 1 ) + ( η + u 2 ) ( z 2 − z 3 ) + μ h z 2 + β v 1 − β v 2 I + H ( 1 − u 1 ) m + I + H U ( z 5 − z 6 ) + β v 2 I + H ( ( 1 − u 1 ) m + I + H ) 2 − β v 2 1 ( 1 − u 1 ) m + I + H U I ( z 5 − z 6 ) , d z 3 d t = − ∂ L ∂ H , = β h 2 I + H ( ( 1 − u 1 ) m + I + H ) 2 − β h 2 1 ( 1 − u 1 ) m + I + H S V ( z 1 − z 2 ) + μ h z 3 , + α 1 ( z 3 − z 4 ) + β v 2 I + H ( ( 1 − u 1 ) m + I + H ) 2 − β v 2 1 ( 1 − u 1 ) m + I + H U I ( z 5 − z 6 ) ,

d z 4 d t = − ∂ L ∂ R , = δ ( z 4 − z 1 ) + μ h z 4 , d z 5 d t = − ∂ L ∂ U , = β v 1 − β v 2 I + H ( 1 − u 1 ) m + I + H I ( z 5 − z 6 ) + ( μ v + u 3 ) z 5 , d z 6 d t = − ∂ L ∂ V , = β h 1 − β h 2 I + H ( 1 − u 1 ) m + I + H S z 1 − z 2 + ( μ v + u 3 ) z 6 ,

supplemented with the transversality condition z i (T) = 0 for i = 1, 2, 3, 4, 5, 6. Taking the derivative of L with respect to u i yields

∂ L ∂ u 1 = 2 w 1 u 1 + β h 2 ( H + I ) m S V ( ( 1 − u 1 ) m + H + I ) 2 ( z 1 − z 2 ) + β v 2 ( H + I ) m U I ( ( 1 − u 1 ) m + H + I ) 2 ( z 5 − z 6 ) , ∂ L ∂ u 2 = 2 w 2 u 2 + I ( z 3 − z 2 ) , ∂ L ∂ u 3 = 2 w 3 u 3 − U z 5 − V z 6 .

Solving ∂ L ∂ u i = 0 with respect to all control variables yields

u 1 = 1 2 w 1 β h 2 ( H + I ) m S V ( ( 1 − u 1 ) m + H + I ) 2 ( z 2 − z 1 ) + β v 2 ( H + I ) m U I ( ( 1 − u 1 ) m + H + I ) 2 ( z 6 − z 5 ) , u 2 = 1 2 w 2 I ( z 2 − z 3 ) , u 3 = 1 2 w 3 ( U z 5 + V z 6 ) .

Considering that the admissible control should lie between 0 and its upper bound, ( u i max ) , i = 1, 2, 3, the optimal u i * is given by

u 1 * = min max 0 , 1 2 w 1 β h 2 ( H + I ) m S V ( ( 1 − u 1 ) m + H + I ) 2 ( z 2 − z 1 ) + β v 2 ( H + I ) m U I ( ( 1 − u 1 ) m + H + I ) 2 ( z 6 − z 5 ) , u 1 max , u 2 * = min max 0 , 1 2 w 2 I ( z 2 − z 3 ) , u 2 max , u 3 * = min max 0 , 1 2 w 3 ( U z 5 + V z 6 ) , u 3 max .

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Received: 2020-07-01
Accepted: 2021-11-04
Published Online: 2021-12-02
Published in Print: 2023-02-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
  15. Convergence analysis of online learning algorithm with two-stage step size
  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
  21. Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
  23. Construction of complexiton-type solutions using bilinear form of Hirota-type
  24. Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
https://doi.org/10.1515/ijnsns-2020-0142
Schlagwörter für diesen Artikel
basic reproduction number; center-manifold; cost effectiveness; dengue; media awareness; optimal control

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
  15. Convergence analysis of online learning algorithm with two-stage step size
  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
  21. Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
  23. Construction of complexiton-type solutions using bilinear form of Hirota-type
  24. Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
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