A simple model for the dynamics of malaria due to Ross (1911) and Macdonald (1952) is $$ \begin{aligned} &\dot{x}=\left(\frac{a b M}{N}\right) y(1-x)-r x \\ &\dot{y}=a x(1-y)-\mu y \end{aligned} $$ where: \(x, y\) are the infected proportions of the human host, female mosquito populations, \(N, M\) are the numerical sizes of the human, female mosquito populations, \(a\) is the biting rate by a single mosquito, \(b\) is the proportion of infected bites that result in infection, \(r, \mu\) are per capita rates of recovery, mortality for humans, mosquitoes, respectively. Show that the disease can maintain itself within these populations or must die out according as $$ R=\frac{M}{N} \frac{a^{2} b}{\mu r}>1 \text { or }<1. $$

In the field of epidemiology, the basic reproduction number, commonly denoted as R or R0, is a crucial concept that helps predict the spread of an infectious disease within a population. It represents the average number of secondary cases produced by a single infectious individual in a completely susceptible population. Think of it as a measure of the potential for contagion: if one person infected with a disease can, on average, infect more than one other person (R > 1), the disease is likely to spread rapidly. On the other hand, if each infected individual spreads the disease to less than one person on average (R < 1), the disease transmission will likely decline and eventually die out.

In the context of the exercise, R is computed using specific parameters pertinent to the dynamics of malaria transmission between humans and mosquitoes. By determining the value of this reproduction number, we can conclude whether malaria is likely to maintain itself in the population or not. The fact that R depends on various factors such as the mosquito biting rate, the rate of recovery from malaria, and the relative sizes of human and mosquito populations underscores the interconnectedness of disease dynamics and highlights the importance of each variable in modeling the outcome of an epidemic.

When analyzing a system of differential equations that describe the spread of an infectious disease, the Jacobian matrix is a powerful tool. The matrix contains the partial derivatives of the function that represents the system's rate of change with respect to its variables. In the language of mathematics, it encapsulates the local dynamics of a system near a specific point. By evaluating the Jacobian matrix at the disease-free equilibrium point (x = 0, y = 0), we can learn about the stability of this equilibrium and the system's behavior in the absence of the disease.

The nature of the eigenvalues of the Jacobian matrix provides insights into whether small perturbations away from equilibrium will die out or lead to larger outbreaks, which is directly tied to whether the disease ultimately vanishes or persists within the population. In our exercise, the Jacobian matrix calculated at the disease-free equilibrium helps us draw conclusions about the disease's fate based on the provided parameters and the computed basic reproduction number.

Equilibrium points are steady states where the variables in a set of differential equations do not change over time, indicating that the system is in balance. In our study of malaria dynamics, identifying the equilibrium points allows us to understand the scenarios where the disease either persists or is eradicated. These points are found by setting the rate of change of infected individuals in the human and mosquito population to zero. Once identified, these points act as benchmarks to analyze the stability of the system and predict its long-term behavior.

For example, the disease-free equilibrium (DFE) represents a state where both human and mosquito populations are free from infection. The stability of this point tells us whether an introduced infection will spread or die out. If the DFE is stable, any small number of infections introduced into the system will eventually subside, leading to the disease's eradication. However, if the DFE is unstable, even a few cases can grow into an outbreak or endemic situation. Equilibrium points thus serve as a foundation for understanding the outcomes of infection spread and strategizing public health interventions.