1. A certain bacteria population \((\mathrm{P})\) is known to demonstrate exponential growth in the human body with an exponent of \( r \) in the absence of any antibiotics treatment. If some antibiotics (ABs) are introduced to the patients metabolism, AB -induced death of bacteria are observed. The rate of such deaths is linearly proportional (\( m \)) to the AB concentration (C) in the patient's body. Since this bacterium is known to be quite aggressive, the preferred treatment approach is to provide ABs via intravenous injection, rather than oral pills. During intravenous administration of drugs, it is assumed that the ABs are injected at a constant rate (Dose). The decay (metabolism) rate of the drug in the body is known to be proportional \((k)\) to the drug level in the body. a. Write down the differential equations that represent the bacteria population and drug concentration dynamics in a patient's body. Conduct a parametric analysis on the plausible dynamics b. Comment on the impact of dosing decisions on the dynamics. What would you expect with an insufficient dose? What would you expect with a more than sufficient dose? c. Consider a patient who is receiving such an intravenous antibiotics treatment. However, the administration is not continuous. The drugs are injected twice a day (i.e. at 08:00 and 20:00), and each injections takes about 2 hours. During this time interval, the injection rate of the drug is constant. Plot the plausible dynamics of drug level in the patient's body during a 4day treatment, assuming that initial level of the drug concentration is 0.