I am reading Karatzas and Shreeve (Brownian Motion and Stochastic Calculus) and on page 128, at the beginning of chapter 3 on Stochastic Integrals, one reads:

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I can't think of any social problem that was treated by Newton and Leibniz. Speaking of differential equations generally financial applications would count as a social problem, but that's only at the turn of the XX century. Are there any earlier applications? Do you know any? Which?

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    $\begingroup$ Look under Lanchester's model of warfare for one classical example. Differential equations can be used to model almost everything:-) $\endgroup$ – Alexandre Eremenko Sep 14 '15 at 18:54
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    $\begingroup$ @AlexandreEremenko, this is not from the period of Leibniz. I was thinking if there were early applications on the social area. If one deals with the Differential equations as a whole, then the financial applications would also count as a social problem beeing dealt within the scope of the theory, However that's only at the turn of the XX century. $\endgroup$ – Conrado Costa Sep 14 '15 at 19:21
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    $\begingroup$ I don't think any "social sciences" existed at the time of Newton-Leibnitz, or even 200 years later:-) $\endgroup$ – Alexandre Eremenko Sep 14 '15 at 19:35

One of the earliest works containing a "mathematical model" in sociology was Malthus's An Essay on the Principle of Population (1798). It did not however contain differential equations, or even formulas for that matter, Malthus expressed that population multiplies geometrically and food arithmetically in words. Malthus's model did influence early work in political economy and sociology, and even Darwin's work. Only in 1838 Comte, the founder of philosophical positivism, declared that "social physics" was ready to join the ranks of empirical sciences, and dubbed it "sociology" (Sieyès used the word in 1780, but not in print).

But like Malthus the early sociologists, Durkheim and Weber, did not use differential equations. One does appear in Verhulst's 1838 work on population growth, which was motivated by Malthus's model, the differential equation for the now ubiquitous logistic growth is derived and solved there. And in Mathematical Researches into the Law of Population Growth Increase (1845) Verhulst gave the logistic its name.

Marginalists applied calculus methods to economics (marginals are economists' name for derivatives). In Cournot's Researches into the Mathematical Principles of Wealth (1838) differentiation is used to find equilibrium quantities by maximizing profit, he also anticipates the idea of non-cooperative games and the Nash equilibruium in his models of oligopoly and duopoly. In Walras's Elements of Pure Economics (1874) Cournot's approach is generalized to multiple market agents and the global equilibrium. Edgeworth modeled market behavior based on utility function in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (1881), and later even criticized others for lack of mathematical rigor. But neither Walras, nor Edgeworth, nor Marshall in Principles of Economics (1890) use differential equations proper in their models. That had to wait until later.

Possibly the earliest entry was Bachelier's PhD thesis Theory of Speculation (1900) that somewhat informally introduced mathematical model that we now call stochastic process of Brownian motion to evaluate stock options even before Einstein and Smoluchowski first studied it in kinetic theory in 1905-06. The model leads to diffusion equations. Poincare, Bachelier's instructor, called the work "very original, and all the more interesting in that Fourier's reasoning can be extended with a few changes to the theory of errors". Bachelier's model of option pricing closely anticipated the Black-Scholes model widely used today, see Schachermayer-Teichmann's How Close Are the Option Pricing Formulas of Bachelier and Black-Merton-Scholes?

EDIT: While epidemiology may be borderline between biology and social sciences I found out that Bernoulli's work, pointed out in skol's answer, led to early applications of differential equations to actuarial science, which clearly belongs to the latter. According to Daw's Smallpox and the Double Decrement Table: a Piece of Actuarial Pre-history Lambert (1772) "gives the earliest application known to me of what would now be called actuarial formulae for dealing with mortality data". Although Lambert's formulae were based on finite differences he was well aware of Bernoulli's work and their relation to the continuous case. In his 1806 book one of the first French actuaries Duvillard combines Bernoulli type differential equations and Lambert's methods to discriminate between effects of different causes of death, his tables came to be used to estimate risk exposure in life insurance calculations. Interestingly, when later the data collection became more systematic calculations based on Bernoulli's equations became ineffective, and new ones were developed starting with Wittstein (1862), see Multiple Decrements or Competing Risks by Seal.

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    $\begingroup$ Thank you! Beyond population, is there any other area explored? For population and even Darwin I would think this is a biological matter, even though it has important connections with politics and economics. $\endgroup$ – Conrado Costa Sep 15 '15 at 0:35
  • $\begingroup$ @Conrado Costa If you are talking about 19th century, not really, see edit. $\endgroup$ – Conifold Sep 15 '15 at 2:14

One of the first appearances of DE in this area must be in D. Bernoulli's study on the impact of variolation:

Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir, Mem. Math. Phys. Acad. R. Sci., 1766

Here is Daniel Bernoulli’s Epidemiological Model Revisited by Dietz and Heesterbeek, which states that "In the second edition of Bailey’s book full tribute is given to Bernoulli’s work and since then he is frequently quoted as originator of the first epidemiological model for an infectious disease".

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    $\begingroup$ The revision does indeed include differential equations, but this is an epidemiological model which would come under the category of a "biological sciences" problem rather than a "social problem." You could argue that epidemiology overlaps both fields. $\endgroup$ – Rory Daulton Sep 19 '15 at 9:10

What about Adolphe Quetelet? Didn't his work with social sciences data inspire Maxwell and Boltzmann's efforts modeling the distribution of velocities in a population of gas particles? See: https://en.wikipedia.org/wiki/Adolphe_Quetelet

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    $\begingroup$ Quetelet's work was statistical and did not involve differential equations or even calculus. Statistical gathering and analysis of social data started in 17th century, e.g. Petty's "political arithmetic" en.wikipedia.org/wiki/William_Petty#Statistician and in simpler forms already in antiquity. $\endgroup$ – Conifold Sep 17 '15 at 22:23

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