Assuming a homogeneous population, we apply the mass action law for rate of new infections and a second-order gamma distribution for removal probability to model spread of an epidemic. In numerical examinations of higher-order gamma distributions for removal probability, we discover an unexpected pattern in maximum fraction of population infected. We develop from first principles of probability an eighth-order system of ordinary differential equations to model effects of isolation and quarantine. We derive analytical expressions for reproduction numbers modeling isolation and quarantine when applied separately and together and verify them numerically. We quantify strength and speed required of these interventions to contain epidemics of varying severity and examine how their effectiveness depends on when they begin. We find that effectiveness is highly sensitive to small changes of intervention strength in a critical region. Finally, adding two more differential equations to capture natural population dynamics, we calculate endemic disease equilibria when affected by isolation and examine dynamics of coming to an equilibrium state.

假设人口同质，我们将新感染率的质量作用定律和移除概率的二阶伽马分布应用于流行病的传播模型。在移除概率的高阶伽马分布的数值检查中，我们发现了最大感染人口比例的意外模式。我们从概率的第一原理发展了一个八阶常微分方程组，以模拟隔离和隔离的影响。我们推导出模拟隔离和隔离的繁殖数的解析表达式，当单独和一起应用时，并对其进行数值验证。我们量化了这些干预措施所需的强度和速度，以遏制不同严重程度的流行病，并检查其有效性如何取决于它们何时开始。我们发现有效性对关键区域干预强度的微小变化高度敏感。最后，再添加两个微分方程来捕捉自然人口动态，我们计算受隔离影响时的地方病平衡，并检查达到平衡状态的动态。