## Model Initialization Strategies

Simulation is a first step in after the model development to verify convergence, validate the model response to input changes, and manually adjust parameters to fit an expected response. This section demonstrates how to set up and solve a dynamic model.

Although a problem may be written correctly, sometimes the solver fails to find a solution or requires excessive time to find a solution. Initialization strategies are critical in these situations to find a nearby solution that seeds the optimization problem with a starting point that allows convergence^{1}.

#### Exercise

**Objective:** Simulate a highly nonlinear system, using initialization strategies to find a suitable approximation for a future parameter estimation exercise. Create a MATLAB or Python script to simulate and display the results. *Estimated Time: 2 hours*

The spread of HIV in a patient is approximated with balance equations on (H)ealthy, (I)nfected, and (V)irus population counts^{2}.

Initial Conditions H = healthy cells = 1,000,000 I = infected cells = 0 V = virus = 100 LV = log virus = 2 Equations dH/dt = kr_{1}- kr_{2}H - kr_{3}H V dI/dt = kr_{3}H V - kr_{4}I dV/dt = -kr_{3}H V - kr_{5}V + kr_{6}I LV = log_{10}(V)

There are six parameters (kr_{1..6}) in the model that provide the rates of cell death, infection spread, virus replication, and other processes that determine the spread of HIV in the body.

Parameters kr_{1}= new healthy cells kr_{2}= death rate of healthy cells kr_{3}= healthy cells converting to infected cells kr_{4}= death rate of infected cells kr_{5}= death rate of virus kr_{6}= production of virus by infected cells

The following data is provided from a virus count over the course of 15 years. Note that the virus count information is reported in log scale.

With guess values for parameters (kr_{1..6}), approximately match the laboratory data for this patient. A subsequent section introduces methods for parameter estimation by minimizing an objective function.

#### Solution

#### References

- Safdarnejad, S.M., Hedengren, J.D., Lewis, N.R., Haseltine, E., Initialization Strategies for Optimization of Dynamic Systems, Computers and Chemical Engineering, 2015, Vol. 78, pp. 39-50, DOI: 10.1016/j.compchemeng.2015.04.016. Article
- Nowak, M. and May, R. M.
*Virus dynamics: mathematical principles of immunology and virology: mathematical principles of immunology and virology*. Oxford university press, 2000. - Lewis, N.R., Hedengren, J.D., Haseltine, E.L., Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities, Special Issue on Algorithms and Applications in Dynamic Optimization, Processes, 2015, 3(3), 701-729; doi:10.3390/pr3030701. Article