Namesma 226 Project 2 Due April 25 Consider The Classical Si

Namesma 226 Project 2due April 25consier The Classical Sir Model For

Consider the classical SIR model for the spread of an epidemic. A population is divided into three groups: susceptible (S), infected (I), and recovered (R). The model assumes that once a person has had the disease, their immune system prevents re-infection, and the total population remains constant at 1, so S + I + R = 1 at all times. The disease spreads via interactions proportional to both the susceptible and infected populations, with a fraction leading to new infections, and infected individuals recover at a rate proportional to the infected population. The model is described by the differential equations:

• dS/dt = -μSI
• dI/dt = μSI - γI
• dR/dt = γI

Considering only S and I (since the total remains 1), the system reduces to:

dS/dt = -μSI

dI/dt = μSI - γI

In a scenario where a virus (Coda virus) causes infecteds to actively infect susceptibles, the model is modified by replacing I with pI in the infection term, resulting in:

• dS/dt = -μS p I
• dI/dt = μS p I - γI

Similarly, when modeling zombie spread with human survivors fighting zombies, the equations are adjusted to include effects of humans eliminating zombies, with infected zombies progressing to recovered at a rate proportional to the human population.

Paper For Above instruction

The classical SIR model provides a foundational framework for understanding the dynamics of infectious diseases within a fixed population. By dividing the population into susceptible (S), infected (I), and recovered (R) compartments, the model captures the progression of diseases such as influenza, measles, or other contagious illnesses. The basic differential equations, where the total population remains constant, describe the flows between these compartments based on transmission rates and recovery rates, respectively. Specifically, the model hinges on several assumptions: individuals can only be infected once, disease spread reduces the susceptible pool, and recoveries confer immunity, preventing re-infection.

Analyzing the equilibrium points involves setting the derivatives to zero, indicating a static state in the population. For the classic SIR model, the disease-free equilibrium occurs at (S, I) = (1, 0), assuming no ongoing infection, representing a population with everyone either recovered or never infected. The endemic equilibrium, characterized by a persistent level of infection, exists when the basic reproduction number R0 = μ/γ exceeds one, leading to positive I values. Solving the equilibrium equations allows us to find thresholds for disease persistence and conditions under which an epidemic occurs.

Understanding the phase plane and the region where dI/dt > 0 informs epidemiologists about potential outbreak growth. The condition dI/dt > 0 simplifies to μSI > γI, revealing that infection grows when the product of the transmission rate, susceptible fraction, and infected fraction surpasses the recovery rate. Graphing this region illuminates the initial conditions conducive to epidemic expansion. As the infection spreads, the susceptible pool diminishes, and the infection peaks before declining as recoveries increase, illustrating typical epidemic curves.

Extending the model to include the Coda virus, where infecteds actively infect susceptibles with a stronger and more aggressive process represented by the modification I to pI, introduces a 'zombie' dynamic. In this context, the recovered individuals can be analogous to zombies—those who are dead or turned but still influence the dynamics by actively infecting others. In this scenario, the 'recovered' state might be reinterpreted as zombies, beings that no longer recover but instead perpetuate infection actively. The model predicts that if the infection coefficient μp is high enough, the outbreak will lead to widespread zombification. The phase portrait illustrates this, with different initial conditions possibly leading to varying outcomes—either containment of the outbreak or catastrophic spread—highlighting sensitivity to initial populations.

Adjusting parameters such as μ and γ affects the model's predictions. For example, increased transmission rates (μ) or higher infectivity p lead to rapid epidemic spread or zombie proliferation. Conversely, increasing the recovery or removal rate (γ) reduces the total number of infected individuals over time, potentially controlling the outbreak. Technological tools like phase plane analysis and numerical integration (using software such as MATLAB, Python, or specialized applications) generate phase portraits illustrating these dynamics. For the zombie model, simulations often show initial growth of zombies before plateauing or declining based on parameters, emphasizing the importance of early intervention or enhanced defensive measures.

In a further extension, especially when considering a zombie outbreak, incorporating human efforts such as humans eliminating zombies (via grenade belts or other weapons) modifies the equations to include a term representing human removal of zombies. When humans actively destroy zombies, the rate of increase in recovered (dead or neutralized zombies) becomes proportional to the human population, contributing to containment. Adjusting the rate parameters to reflect better equipment or teamwork shows that the outbreak can be slowed or halted, as the effective infection rate decreases, resulting in fewer zombies over time.

Connecting this modeling framework to popular media, such as George Romero's zombie films or "The Walking Dead," the parameters can be tuned to reflect different scenarios: high contact rates among zombies (high μ), slow human response (low removal or recovery rate), or rapid zombie movement (high p). For example, in "Resident Evil," the rapid spread due to pharmaceutical mutation can be modeled with a high μ and p, leading to swift, uncontrollable outbreaks in the phase space. Conversely, stories depicting militarized responses, such as "World War Z," could be modeled with increased human removal rates, showing different trajectories in the phase portrait. These models serve as a valuable tool for exploring different narrative outcomes and understanding the implications of disease or zombie spread dynamics.

In conclusion, the SIR model and its modifications provide essential insights into epidemic and outbreak behaviors, whether dealing with biological diseases or fictional zombie pandemics. Through equilibrium analysis, phase portraits, and parameter sensitivity studies, we gain a comprehensive understanding of how initial conditions and intervention strategies can influence the course of an outbreak. These models underscore the importance of early detection, prompt response, and effective removal efforts. While simplified, they offer an invaluable foundation for understanding complex real-world processes and for designing effective control measures in public health or emergency scenarios.

References

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• Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599–653.
• Allen, L. J. S. (2008). An Introduction to Mathematical Biology. Pearson Education.
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• Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
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