**The COVID-19 emergency highlighted the importance of simulation techniques.**

**To understand and optimize the performances of a complex system, characterized by mutual-influence of several variables, in the presence of phenomena not entirely definable a priori but subjected to randomness, requires specifics models, algorithms drawing from the decision sciences field. Pandemic, like other catastrophic events, requires systems and organizations to respond appropriately. This paper explains the scientific system design approaches to make this possible.**

Why is the simulation so important in pandemics and exceptional events? The quick answer is: because **it permits us to study the dynamic behavior of complex systems under various hypotheses** and to design the facilities, the organization, and the procedures. The simulator is a **virtual copy of the real system**, with the possibility to perform experiments that are difficult, costly, or even impossible, to do in the real world.

**Simulation and decision making**

Simulation brings to the decision-maker the knowledge about how the system will work under different circumstances. During pandemics, or in general, when exceptional and unexpected situations could occur, the importance of such virtual-copy of the system assumes even more significance. It enables the decision-makers to **explore in-depth the behavior, test, compare different possibilities, and find appropriate intervention strategies.**

When it comes to extraordinary circumstances, simulation becomes a method to answer questions that would be almost impossible to address in other manners. This is even true for other advanced analytics techniques like forecasting and artificial intelligence in general. These tools, being exceptional methods in standard conditions, are by nature influenced by the past data from which they learn.

As a consequence, when a situation is extremely new, they necessarily face the risk of providing unreliable results. This is an important issue also for simulation, which, however, presents the advantage of allowing the explicit modeling of hypothesized dynamics, without recurring to past information.

Of course, it is crucial that the simulator faithfully represents the real world, giving the right answers to the decision-makers. The right approach, the right way to model the system, and the data are essential success factors. To be mentioned that artificial intelligence and simulation can also be combined, but we do not touch such a topic in this paper.

**Different simulation paradigms**

Let’s recall what simulation is and what the differences are between different simulation paradigms. Wikipedia says: Â«A **simulation** is an **approximate ****imitation** of the operation of a process or system [1] **that represents its operation over time**.Â»

Simulation techniques can be applied to any system at a different level of deepness. Some examples are a supply chain, a factory, a specific work center, but also a big plant, chemical reactions, robots, mechanical systems, or hospitals, for instance, to evaluate how they will support the COVID-19 crisis, etc.

The above examples intuitively make evident that the nature of the model could be different for the different types of systems to simulate. An important distinction is between the **continuous and discrete processes**. In continuous processes, the variables vary continuously over time (like the temperature in a reactor), according to specific equations.

In the case of discrete processes, the status of the system can be modeled by discrete instantaneous changes, like, for example, the state of a machine switching from the status of “working” to “idle”. Hybrid situations exist in the real world, where part of the process belongs to the first category and part of the process to the second category.

Discrete models well represent
several systems. In these cases, **the paradigm used is called Discrete Event
Simulation (DES)**.

**Agent-Based Simulation (ABS)** represents another essential approach to simulate complex dynamic and adaptive systems. ABS is often used to simulate the behavior of humans, such for example, to study evacuation in case of emergency or the movement of flows of people in constrained space (an airport, a store, etc.). The dynamic evolution of a virus is also another good example where ABS helps to understand how the infection evolves under different conditions.

**Expected and unexpected conditions**

In a simulation model, the variables and their cross-relations are represented over time, permitting the decision-makers to understand and study the system under different conditions.

Such conditions could be “the expected conditions”, for example, the expected demand. In other cases, such conditions could represent exceptional situations, like for example, the failure of important machines or the absence of operators, for instance, due to the COVID-19, or any other improbable, but possible, circumstances.

**It’s evident that during an unexpected situation**, the traditional capabilities, by humans &/or even algorithms, might fail. The status and the dynamic of the system could be dramatically different from the usual. Some resources and components work at their limit, without necessarily reaching stationary conditions.

**Interconnection brings complexity**

**The interconnection of sub-systems and sub-processes might bring the system to a dangerous, unknown status**. In these cases, the simulation becomes a powerful method to study strategies, to better design components or a process, to predict the crises situation permitting to anticipate actions.

There are multiple ways to assess performance over time. For example, suppose a planner wants to evaluate, over a quarter, the impact of the demand. The planner could create a spreadsheet calculating, month by month, the expected demand, and the effect on the resources and productivity, for month 1, month 2, month3 (three pictures giving the idea of what will happen month by month).

This is a simplified way to evaluate impact over time, sufficient in uncomplicated cases, but not in-depth enough in more complex cases. For example, it does not permit to evaluate peaks within the month and the implication, over time, of such peaks. In other words, **to understand the system behavior is needed to evaluate the time-dependence of the variables and cross-time-dependence of multiple variables. Here is where the simulation plays a significant role**.

Let’s use the example of the hospitals during the COVID-19 crises. A peak in the emergency rooms after hours becomes a peak of saturation of intensive care units that persist for days. The impact depends on how patients’ conditions evolve. The dynamic of the peaks is affected by multiple factors; some could be, at least partially controlled, others no. The involved variables are many, variables related to operators variable related to materials, to the infrastructure, the rate of arrivals of patients. It is the interconnection of such variables that leads to the whole final performance.

**Deterministic phenomena and randomness**

In some cases, the model
variables can be represented in a deterministic way, like, for example, if we
are studying the dynamics of a generic thermo-mechanical process**. In many
other cases, the variables are significantly affected by randomness**. For
instance, the number of COVID patients arriving at an emergency room hour by
hour and day by day, the possibility that some operators will be infected,
causing a crisis within the crises. How data are modeled and used within the
model is a crucial success factor and is not always straightforward as it could
appear.

Often in business is used the average to represent a phenomenon, like the average of orders per week, the average of arrival patients, the average time to complete an activity, etc. The **dynamic behavior of a system could be dramatically different in relation to the variabilities around those average values**.

The decisions on how many ambulances to keep available in a city, how many operators to staff in a roadside assistance call center, the optimal sizing of an electrical system, or of a production interoperations buffer only make sense by considering the natural randomness of real events. In these situations, the decision-makers should completely ignore the most likely values and look with attention at possible spikes. The aim is not anymore, the overall balancing of a system, but rather the right dimensioning to guarantee a proper service in the vast majority of cases, taking into account the related costs.

**An approximation to reality**

A** simulation model is not a perfect copy of reality but always an approximation of the real systems**.

Consequently, an essential
question arises: **what is the appropriate level of approximation? **To
answer this question, we must be considering which decisions the model must
support, which questions the model must answer, which data are available.
Expert modelers are needed to correctly evaluate the appropriate level of
approximation of the model and how to prepare the input data.

Specific competences and significant modeling experiences are needed to crate reliable simulators. **Any deviation from the appropriate methods exposes to dangerous, not robust, or even false answers, that the simulator provides to decision-makers.**

**The COVID-19 case**

Let’s consider a simplified example to study the capability of a COVID-19 hospital with intensive therapy facing the pandemic crisis.

Let’s assume the following data:

- Number of
patients requiring intensive care:
- 2/day (on average) due to the epidemic.

- 1/day (on average) due to other causes.

- Expected
hospitalization period of a patient in intensive care:
- between 2 and 14 (average 8) days for â€śpandemicâ€ť patients.

- between 4 and 12 (average 8) days for â€śstandardâ€ť patients.

Key questions:

How many beds will be necessary to support a crisis of 6 weeks?

How many beds will be necessary to support a crisis of 10 weeks?

The following table, for example, shows the results for different scenarios with 20, 24, and 28 beds.

Scenarios 1, 2, and 3 refer to a crisis of 6 weeks, scenarios 4, 5, 6 refer to a crisis of 10 weeks.

From the above table we see that none of the scenarios permits to accept patients directly with no waiting time, the scenario 3, relevant of a 6 weeks crisis, and the 6, related to a 10 weeks crisis, are (apparently) close to be acceptable, having a waiting time for about 1 patient of few hours.

**When patients arrive**

In the above scenarios, we have considered a time-constant stochastic distribution representing the arrival rate of patients (average of 2 patients per day).

We know that a typical pandemic curve has a shape like the one shown in the below chart:

How would the performance of the system change if – maintaining the same average over the period – the arrival rate followed this curve?

The answer, for the case of 10 weeks crisis, is shown in the following table:

To obtain performance comparable to the previous scenario 6 with 28 beds, and patients arrival-rate represented in a simplified manner, are needed 11 additional beds as shown by the following table:

A similar simulation-based approach could be applied to the supply chain, to study impacts on manufacturing facilities, etc.

**Ludovica Maccarrone:** M.Sc. in Management Engineering and Ph.D. in Operations Research, is the head of the Simulation department in ACT OR, now part of Spindox. Ludovica has significant experience in Optimization and Discrete Event and Stochastic Simulation.

**Federica Mo** works as an Algorithm Engineer at ACT Operations Research, now part of Spindox. She has a master’s degree in Management Engineering, achieved with honors and academic excellence. Federica developed simulation models and algorithms to support decision-making in different industrial areas, from distribution centers to fleet management in the automotive sector.

**Raffaele Maccioni** is the CEO and R&D Executive of ACT Operations Research, now part of Spindox. After the masterâ€™s degree at the Polytechnic of Milan, Raffaele spent his professional career working on advanced analytics & modeling to support complex decisions. Raffaele led the team reaching, in 2018, the prestigious Franz Edelman award by Informs for Achievement in Advanced Analytics, Operations Research, and Management Science.