Report on a Delayed Fractional-Order COVID-19 Model for Malaysia: Emphasizing Sustainable Development Goals

Abstract

This report details the development and analysis of a delayed fractional-order SEIHR-M (Susceptible, Exposed, Infected, Hospitalized, Recovered, Media) model to investigate the transmission dynamics of COVID-19 in Malaysia. This research directly supports the achievement of several Sustainable Development Goals (SDGs), primarily SDG 3 (Good Health and Well-being), by providing a sophisticated tool for understanding and managing infectious disease outbreaks. By integrating fractional-order dynamics to capture memory effects and a time-delay to model media influence lag, the model offers a more accurate representation of real-world epidemic scenarios. Theoretical analysis confirmed the model’s biological feasibility. The basic reproduction number (R₀) was derived, serving as a critical threshold for disease control strategies aligned with SDG 3. The analysis revealed that the disease-free state is stable when R₀ 1, the system’s stability is dependent on the media response delay (τ), with a critical threshold (τ₀) beyond which the system exhibits periodic oscillations, mirroring recurrent epidemic waves. These findings underscore the importance of timely information dissemination for maintaining public health stability. Numerical simulations and model fitting with Malaysian data validated the theoretical framework, demonstrating the model’s applicability. Sensitivity analysis highlighted that media interventions are paramount in controlling epidemic spread, providing actionable insights for public health policies that contribute to resilient societies (SDG 11) and effective institutions (SDG 16).

1. Introduction: Epidemic Modeling in the Context of Sustainable Development

The COVID-19 pandemic presented an unprecedented global challenge, profoundly impacting public health and global economies, thereby threatening progress on SDG 3 (Good Health and Well-being) and SDG 8 (Decent Work and Economic Growth). In Malaysia, government interventions such as Movement Control Orders (MCO) were necessary to curb viral spread but caused significant economic disruption, particularly in vital sectors like tourism. In this context, mathematical modeling emerges as a critical tool for developing evidence-based public health strategies that balance health security with economic stability, a core tenet of sustainable development.

While traditional integer-order models like SIR and its variants (SEIR, SEIHR) have been foundational, they often fail to capture the complex memory effects and behavioral response delays inherent in real-world epidemics. This report presents a novel approach by:

• Incorporating Fractional-Order Calculus: To better model the hereditary properties of disease transmission, where past events influence future spread.
• Integrating Media Influence with Time-Delay: To quantify the impact of public information campaigns on behavior and the critical lag between information dissemination and public response. This directly relates to SDG 16 (Peace, Justice and Strong Institutions) by examining the effectiveness of institutional communication.

This advanced modeling framework provides a more nuanced understanding of epidemic dynamics, offering critical support for public health decision-making and contributing to the creation of resilient and sustainable communities (SDG 11).

1.1. Primary Contributions to Sustainable Development

1. Advanced Model Construction: A delayed fractional-order SEIHR-M model was developed. Its structure captures the memory effects of disease transmission and the lagged behavioral response to media, providing a robust tool for planning interventions under SDG 3.
2. Theoretical Validation for Policy Reliability: The model’s mathematical properties, including the existence and stability of solutions, were rigorously analyzed. The derivation of the basic reproduction number (R₀) and the identification of Hopf bifurcation conditions offer a solid theoretical foundation for evidence-based policymaking, strengthening institutional capacity as per SDG 16.
3. Simulation of Real-World Scenarios: Numerical simulations demonstrated how fractional order and time delays influence epidemic trends, including the potential for recurrent outbreaks (periodic oscillations). This foresight is crucial for building resilient healthcare systems (SDG 3) and stable economies (SDG 8).
4. Application and Sensitivity Analysis: The model was successfully fitted to early COVID-19 data from Malaysia, confirming its practical applicability. Sensitivity analysis revealed that media-related parameters are highly influential, emphasizing that effective and timely communication is a powerful tool for achieving public health goals under SDG 3.

2. Model Formulation and Theoretical Analysis

2.1. The SEIHR-M Model Structure

A time-delay epidemic model using the Caputo fractional derivative was formulated to describe the dynamics among six compartments: Susceptible (S), Exposed (E), Infectious (I), Hospitalized (H), Recovered (R), and Media (M). The model is designed to reflect how media coverage, driven by the number of active cases, influences public behavior and thus reduces the rate of transmission. This feedback loop is critical for understanding how information can be leveraged as a public health tool, a key component of building strong, responsive institutions (SDG 16).

2.2. Foundational Analysis for Model Reliability

A thorough theoretical analysis was conducted to ensure the model’s credibility as a tool for guiding policy. Key findings include:

• Existence, Non-Negativity, and Boundedness: The model’s solutions were proven to be well-posed, ensuring that the simulations produce biologically meaningful results (e.g., non-negative population sizes).
• Basic Reproduction Number (R₀): An explicit formula for R₀ was derived. This threshold is fundamental for public health, as it determines whether an epidemic will spread (R₀ > 1) or die out (R₀ SDG 3).
• Equilibrium Points: The model contains a disease-free equilibrium (eradication) and, under the condition R₀ > 1, a unique endemic equilibrium (persistent disease). Analyzing these states is essential for long-term strategic planning.

2.3. Stability Analysis and the Role of Time Delay

Stability analysis provides insight into the long-term behavior of the epidemic, which is crucial for sustainable planning.

• Disease-Free Equilibrium (R₀ This state was found to be locally asymptotically stable, regardless of the media response delay (τ). This confirms that if a disease is not highly transmissible, it will naturally decline.
• Endemic Equilibrium (R₀ > 1): The stability of this state is critically dependent on the time delay τ.
  • Without Delay (τ = 0): The endemic state is stable, meaning the disease persists at a constant level.
  • With Delay (τ > 0): A critical delay (τ₀) exists. If the media response is faster than this threshold (τ τ₀), the system undergoes a Hopf bifurcation, leading to periodic oscillations. These oscillations represent recurrent waves of the epidemic, which place sustained pressure on health systems (SDG 3) and create economic uncertainty (SDG 8).

3. Numerical Simulation and Validation

Numerical simulations were performed to validate the theoretical findings and explore the model’s dynamics under various conditions, providing a visual understanding of its implications for the SDGs.

3.1. Scenario 1: Disease Control (R₀

Simulations confirmed that when R₀ SDG 3 and SDG 8.

3.2. Scenario 2: Endemic Disease and Recurrent Waves (R₀ > 1)

When R₀ > 1, the simulations vividly illustrated the impact of the media delay parameter (τ):

• Short Delay (τ The number of infected individuals stabilizes at an endemic level. This scenario requires long-term management strategies to mitigate the ongoing health burden.
• Long Delay (τ > τ₀): The system exhibits sustained periodic oscillations. This finding is critical, as it models the recurrent waves of infection seen in many countries. Such waves pose a significant threat to the sustainability of healthcare infrastructure and can lead to repeated economic lockdowns, hindering progress on SDG 3, SDG 8, and SDG 11. The simulations show that a longer delay leads to higher and more frequent peaks, exacerbating the crisis.

4. Model Fitting and Policy Implications for Sustainable Development

4.1. Validation with Malaysian Data

The model was fitted to active COVID-19 case data from Malaysia (March-May 2020). The fractional-order delay model provided a significantly better fit compared to a corresponding integer-order model, particularly in capturing the peak and decline of the initial wave. This demonstrates its value as a more accurate and reliable tool for real-time analysis and forecasting, supporting the development of effective institutions (SDG 16) through evidence-based policy.

4.2. Sensitivity Analysis: Pinpointing Effective Interventions

A local sensitivity analysis was conducted to identify the most influential parameters on epidemic outcomes. The results strongly reinforce the importance of strategic communication in public health crises:

• Media Parameters are Critical: The parameters governing media influence—the rate of information dissemination (σ₁), the rate of information decay (σ₂), and the response delay (τ)—were found to be highly sensitive. This provides quantitative evidence that investing in robust public health communication infrastructure is a highly effective strategy for achieving SDG 3.
• Timeliness is Paramount: The delay parameter (τ) was shown to be a key determinant of the epidemic’s peak and duration. A short delay leads to a lower, earlier peak, while a long delay results in a higher, later peak and potential for recurrent waves. This highlights that rapid, transparent communication can prevent a health crisis from escalating into a prolonged socio-economic one.

4.3. Policy Recommendations for Sustainable Health Security

Based on the model’s findings, the following policy actions are recommended to build resilience against future health threats, in alignment with the SDGs:

1. Strengthen Public Health Communication: To advance SDG 3 and SDG 16, governments should invest in agile and multi-channel communication systems to ensure that accurate health information reaches the public with minimal delay.
2. Promote Evidence-Based Decision-Making: Utilize advanced modeling tools like the one presented here to forecast epidemic trajectories and assess the potential impact of different interventions, thereby optimizing resource allocation and minimizing negative impacts on economic activity (SDG 8).
3. Build Resilient Communities: Foster public trust and engagement through sustained, credible information campaigns. An informed and responsive public is a cornerstone of a resilient community (SDG 11) capable of weathering public health crises effectively.

5. Conclusion

This report has detailed a delayed fractional-order SEIHR-M model that provides significant insights into the transmission dynamics of COVID-19, with a strong emphasis on the role of media. The research makes a direct contribution to the Sustainable Development Agenda by demonstrating how mathematical modeling can inform policies that protect public health (SDG 3), maintain economic stability (SDG 8), build resilient communities (SDG 11), and foster effective governance (SDG 16).

The key takeaway is the critical importance of timely and effective communication. The model’s demonstration of a Hopf bifurcation triggered by delays in media response provides a stark warning: information lags can lead to recurrent, damaging waves of infection. By prioritizing rapid and sustained media interventions, policymakers can flatten the curve, reduce the overall burden on society, and steer their nations towards a more sustainable and resilient future.

Analysis of Sustainable Development Goals (SDGs) in the Article

1. Which SDGs are addressed or connected to the issues highlighted in the article?

• SDG 3: Good Health and Well-being

  The article is fundamentally centered on public health. It investigates the transmission dynamics of COVID-19, a major global health crisis. The entire study, from modeling the disease spread (SEIHR-M model) to analyzing the impact of interventions, directly contributes to understanding and combating communicable diseases, which is a core component of SDG 3.

• SDG 8: Decent Work and Economic Growth

  The article explicitly mentions the economic consequences of the pandemic. It states that COVID-19 resulted in “unprecedented impacts on the global economy” and specifically notes that in Malaysia, control measures “caused significant disruptions to economic activities.” It highlights the severe impact on the tourism industry, a key economic sector for Malaysia, stating that “international tourist arrivals plummeting, leaving related businesses and workers struggling to survive.” This directly connects the health crisis to its effects on economic growth and employment.

• SDG 16: Peace, Justice and Strong Institutions

  The article’s focus on the role of “media reporting” and “information dissemination” in controlling the epidemic relates to SDG 16. It analyzes how media interventions, as a form of public information access, influence public behavior and the effectiveness of health policies. The study concludes that “timely and effective information dissemination plays a crucial role in reducing the peak of infections,” which aligns with the goal of ensuring public access to information, a key element for responsive and effective institutions.

2. What specific targets under those SDGs can be identified based on the article’s content?

• Target 3.3: End epidemics of communicable diseases

  The article’s primary focus is on modeling the transmission of COVID-19, a communicable disease. The introduction states that the pandemic has become “one of the most pressing public health crises in recent years.” The development of the SEIHR-M model is a direct effort to understand and predict the spread of this epidemic, which is the first step toward controlling and eventually ending it.

• Target 3.d: Strengthen capacity for early warning, risk reduction, and management of health risks

  The study aims to provide “valuable theoretical support for public health decision-making.” By creating a mathematical model that simulates epidemic dynamics and the effects of interventions (like media coverage), the research contributes to strengthening Malaysia’s capacity to manage national health risks. The model itself is a tool for early warning and risk reduction, allowing policymakers to understand potential outcomes of different strategies.

• Target 8.9: Promote sustainable tourism

  The article directly identifies the negative impact of the pandemic on tourism in Malaysia, a crucial industry for the country’s economy. It states, “Tourism, one of Malaysia’s key industries, was hit particularly hard, with international tourist arrivals plummeting.” This highlights the vulnerability of this sector and the challenges to achieving sustainable tourism in the face of global crises, making the target highly relevant.

• Target 16.10: Ensure public access to information

  The article extensively discusses the role of media as a “key channel for information dissemination” and models its impact on public behavior. The “media compartment (M(t))” in the model is designed to quantify the intensity of media reporting and public attention. The conclusion that “timely and effective information dissemination plays a crucial role” directly underscores the importance of public access to information for managing a public crisis, which is the essence of this target.

3. Are there any indicators mentioned or implied in the article that can be used to measure progress towards the identified targets?

• Indicator for Target 3.3 (Implied): Incidence of communicable diseases

  The article models and fits data on the number of active COVID-19 cases in Malaysia. The variables in the model, such as Infected (I), Exposed (E), and Hospitalized (H), are direct measures of the disease’s prevalence and incidence. The fitting of the model to “Malaysia’s active COVID-19 case data” and the calculation of error metrics (RMSE, MAE, MAPE) serve as a way to track and predict the incidence of the disease, which is analogous to official indicators like 3.3.1 (Number of new HIV infections).

• Indicator for Target 3.d (Implied): Health emergency preparedness capacity

  The development and application of the sophisticated “delayed fractional-order SEIHR-M model” itself can be seen as an indicator of a country’s growing capacity for health emergency preparedness. The article’s purpose is to provide “valuable theoretical support for public health decision-making,” which enhances risk management capabilities. This aligns with the spirit of Indicator 3.d.1 (International Health Regulations (IHR) capacity), as such modeling is a key component of a robust public health response system.

• Indicator for Target 8.9 (Mentioned): Impact on the tourism sector

  The article explicitly mentions a key metric related to tourism’s economic contribution: “international tourist arrivals plummeting.” This serves as a direct, though informal, indicator of the severe disruption to the tourism industry, which would be formally measured by indicators like 8.9.1 (Tourism direct GDP as a proportion of total GDP).

• Indicator for Target 16.10 (Implied/Modeled): Effectiveness and timeliness of public information

  The article proposes a way to measure the impact of public access to information through its model parameters. The “media coverage follow-up rate (σ1),” the “dissipation rate of media information (σ2),” and the “time delay (τ)” are quantifiable variables used to assess the effectiveness and timeliness of information dissemination. The sensitivity analysis, which shows how changes in these parameters affect the number of active cases, provides a modeled indicator of how well public access to information is functioning to mitigate the crisis.

4. Table of Findings

Source: nature.com