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Qualitative analysis and numerical investigations of time-fractional Zika virus model arising in population dynamics

Gaurav Saini, Bappa Ghosh, Sunita Chand

TL;DR

The paper addresses Zika virus transmission between humans and mosquitoes using a time-fractional model with the Caputo derivative $\mathcal{D}_t^{α}$ for $α\in(0,1)$ to incorporate memory effects. It develops existence, uniqueness, and Hyers–Ulam stability results via a fixed-point integral formulation and constructs an efficient L1 discretization paired with Newton–Raphson to simulate the nonlinear system. Key contributions include rigorous contraction-based existence and uniqueness conditions, stability results, and a practical numerical scheme demonstrated on multiple $α$ values to reveal memory-driven dynamics. The findings suggest fractional-order models yield more biologically feasible dynamics and provide a robust computational framework for forecasting and controlling Zika spread with memory effects.

Abstract

Epidemic models play a crucial role in population dynamics, offering valuable insights into disease transmission while aiding in epidemic prediction and control. In this paper, we analyze the mathematical model of the time-fractional Zika virus transmission for human and mosquito populations. The fractional derivative is considered in the Caputo sense of order $α\in(0,1).$ We begin by conducting a qualitative analysis using the stability theory of differential equations. The existence and uniqueness of the solution are established, and the model's stability is examined through Hyers-Ulam stability analysis. Furthermore, an efficient difference scheme utilizing the standard L1 technique is developed to simulate the model and analyze the solution's behavior under key parameters. The resulting nonlinear algebraic system is solved using the Newton-Raphson method. Finally, illustrative examples are presented to validate the theoretical findings. Graphical results indicate that the fractional model provides deeper insights and a better understanding of disease dynamics. These findings aid in controlling the virus through contact precautions and recommended therapies while also helping to predict its future spread.

Qualitative analysis and numerical investigations of time-fractional Zika virus model arising in population dynamics

TL;DR

The paper addresses Zika virus transmission between humans and mosquitoes using a time-fractional model with the Caputo derivative for to incorporate memory effects. It develops existence, uniqueness, and Hyers–Ulam stability results via a fixed-point integral formulation and constructs an efficient L1 discretization paired with Newton–Raphson to simulate the nonlinear system. Key contributions include rigorous contraction-based existence and uniqueness conditions, stability results, and a practical numerical scheme demonstrated on multiple values to reveal memory-driven dynamics. The findings suggest fractional-order models yield more biologically feasible dynamics and provide a robust computational framework for forecasting and controlling Zika spread with memory effects.

Abstract

Epidemic models play a crucial role in population dynamics, offering valuable insights into disease transmission while aiding in epidemic prediction and control. In this paper, we analyze the mathematical model of the time-fractional Zika virus transmission for human and mosquito populations. The fractional derivative is considered in the Caputo sense of order We begin by conducting a qualitative analysis using the stability theory of differential equations. The existence and uniqueness of the solution are established, and the model's stability is examined through Hyers-Ulam stability analysis. Furthermore, an efficient difference scheme utilizing the standard L1 technique is developed to simulate the model and analyze the solution's behavior under key parameters. The resulting nonlinear algebraic system is solved using the Newton-Raphson method. Finally, illustrative examples are presented to validate the theoretical findings. Graphical results indicate that the fractional model provides deeper insights and a better understanding of disease dynamics. These findings aid in controlling the virus through contact precautions and recommended therapies while also helping to predict its future spread.
Paper Structure (8 sections, 4 theorems, 47 equations, 7 figures)

This paper contains 8 sections, 4 theorems, 47 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Existence of the solution
  4. Uniqueness of the solution
  5. Stability of the solution
  6. The discretized problem
  7. Numerical illustration and discussions
  8. Conclusion

Key Result

Theorem 3.1

The kernels $W_1, W_2, W_3, W_4$ satisfy the Lipschitz conditions under the assumptions $(C_1)$, provided that $0 \leq \eta_i < 1$ for all $i = 1, 2, 3, 4$.

Figures (7)

  • Figure 1: Solutions with $\alpha = 0.99$
  • Figure 2: Solutions with $\alpha = 0.95$
  • Figure 3: Solutions with $\alpha = 0.9$
  • Figure 4: Comparison of $S_h(t)$ for different values of $\alpha$
  • Figure 5: Comparison of $I_h(t)$ for different values of $\alpha$
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1: Uniqueness of solution
  • proof
  • Definition 4.2: Hyers-Ulam stability Jung2004
  • ...and 3 more