Deterministic models are constructed without probabilities whereas in stochastic models we include uncertainty (probabilities).

These links may help: https://www.researchgate.net/post/What_is_the_difference_between_the_mechanistic_and_statistical_modeling

http://math.oregonstate.edu/~gibsonn/Teaching/MTH323-010S17/Supplements/IntroToModel/IntroToMathModel.pdf

https://jvanderw.une.edu.au/Lecture1_IntroToMathModelling.pdf

Deterministic-no probabilities, stochastic- has probabilities

Deterministic vs. stochastic models

• In deterministic models, the output of the model is fully determined by the parameter values and the initial conditions.

• Stochastic models possess some inherent randomness. The same set of parameter values and initial conditions will lead to an ensemble of different outputs.

• Obviously, the natural world is buffeted by stochasticity stochasticity. But, stochastic stochastic models are considerably considerably more complicated. When do deterministic models provide a useful approximation to truly stochastic processes? Demographic vs. environmental stochasticity

• Demographic stochasticity describes the randomness that results from the inherently discrete nature of individuals individuals. It has the largest largest impact on small populations.

• Environmental stochasticity describes the randomness resulting from any change that impacts an entire p po ulation (such as changes in the environment). Its impact does not diminish as populations become large. Stochastic models, brief mathematical considerations • There are many different ways to add stochasticity to the same deterministic skeleton.

• Stochastic models in continuous time are hard.

Adding to previous answers:

For stochastic models we must differentiate between 'stationary' and 'ergodic' systems. In stationary systems the statistics (e.g. mean and standard deviation) are fixed, whereas, in ergodic systems the statistics can vary over time and the statistics are estimated continuously from an ensemble of data.

Graham W Griffiths

What are the different types of mathematical models?

There are some main types of methods which are based on different mathematical means and tool – analytical, simulation, and empirical (statistical). Each type of methods has sub-types of approaches for model designing. Different program tools (software) exist for realization, execution and investigation of each mathematical model and for obtaining experimental results for analysis. A classification of these models is presented below.

A. Homogeneous models

A1: Formal (abstract) models – abstract description of the investigated process or system based on the apparatus of discrete mathematics (sets with ordered couples and relations, graph formalizations, finite automata, etc.).

A2: Functional models – description of the functionality of the investigated object (algorithms investigation by D-carts, ordered logical schemes of algorithms, E-R diagrams, IDEF standards, Data Flow diagrams, Dialog transition networks, etc.).

A3: Analytical models – realization of an analytical description of the object by using deterministic or stochastic mathematical apparatus. Determined analytical models are designed on the base of means of discrete mathematics (analytical equations, graph theory, Petri nets, etc.). Stochastic analytical models are designed on the base of the theory of probabilities, random processes, Markov random processes (Markov chains), Queueing theory, etc.).

A4: Simulation models which are executed in the time and could be realized as a discrete, continuous or hybrid simulation. These models can use generators of random numbers, queueing theory, etc. There are over 4000 software products for supporting different types of simulation modelling.

A5: Empirical (statistical) models based on the empirical data which are processed by using methods and tools of the Mathematical Statistics (correlation, regression, dispersion, covariation) for one-factor experiments and/or multi-factor experiments

B. Heterogeneous models (combination of selected homogeneous models)

B1: Hybrid models

B2: Hierarchical models

Additional information can be find in:

R. Romansky. Technology of computer modelling, Sofi, 2008 (108 p.) – in Bulgarian

R. Romansky. Information Servicing in Distributed Learning Environments. Formalization and Model Investigation (ISBN 978-3-330-02932-3), LAP LAMBERT Academic Publishing, Saarbrüken, Germany, 2017 (100 p.) – in English

Romansky, R. Formalization and Discrete Modelling of Communication in the Digital Age by Using Graph Theory, Chapter 13 in the book “Handbook of Research on Advanced Applications of Graph Theory in Modern Society“ (ed. Madhumangal Pal, Sovan Samanta & Anita Pal), IGI Global, USA, ISBN 9781522593805, еISBN: 9781522593829, August 2019 (601 pages), pp. 320-353; DOI: 10.4018/978-1-5225-9380-5 – in English

In a deterministic model, the parameters and variable states are known. Examples of deterministic models include timetables, pricing structures, and maps.

Probabilistic models incorporate random variables and probability distributions. The random variables represent the potential outcomes of an uncertain event, whereas the probability distributions assign probabilities to the various potential outcomes. Examples of probabilistic models include regression models, Monte Carlo simulation, and Markov models.

A deterministic model is a set of fixed relationships between the input and output of a system. These relationships may or may not vary over time. Example x_2dot=f(t,x) where f is a fixed function of t and x. A stochastic model is a set of relationships between the input and output of a system where both the inputs and outputs are random variables and the relationships can also be random variables.

These relationships often involve "infinitesimal changes" in the input output/relations which are known as "stochastic differential equations." A good example is the Black-Scholes model in economics.

https://en.wikipedia.org/wiki/Stochastic_differential_equation

In stochastic models there are various conditions that can be placed on the underlying process. These conditions make the analysis of the model tractable to analysis. Examples of the properties are ergodicity, strongly stationary, wide-sense stationary and cyclostationary.

With deterministic processes one looks to solve the output of a system as a function of a given input. With stochastic models, one looks to derive that relationship between the underlying probability distortions of input and output. For example Fokker-Planck (a.k.a. the forward Kolmogorov equation in statistical mechanics).

https://en.wikipedia.org/wiki/Fokker–Planck_equation

The backward Kolmogorov equation equation plays a central role in the theory of may biological processing and diffusion processes.

https://en.wikipedia.org/wiki/Kolmogorov_equations#The_modern_view