ch07/seir_stability/SEIR symbolic stability.ipynb

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    "| **Chapter**               | 7:  Temporal dynamics of epidemics                |\n",
    "|---------------------------|---------------------------------------------------|\n",
    "| **Computational Note**    | 7.1: Symbolic identification of equilibria        |\n",
    "|                           | 7.3: Symbolic equilibrium analysis of SEIR models |\n",
    "| **Package prerequisites** | [SymPy](https://www.sympy.org/en/index.html)      |\n",
    "\n",
    "# Symbolic identification of equilibria"
   ],
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    "$$ \\frac{dS}{dt} = \\mu - (\\beta I + \\mu) S $$\n",
    "$$ \\frac{dE}{dt} = \\beta S I - (\\sigma + \\mu) E $$\n",
    "$$ \\frac{dI}{dt} = \\sigma E - (\\gamma + \\mu) I $$\n",
    "$$ \\frac{dR}{dt} = \\gamma I - \\mu R$$\n"
   ],
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     "text": [
      "The watermark extension is already loaded. To reload it, use:\n",
      "  %reload_ext watermark\n",
      "Author: Chris von Csefalvay\n",
      "\n",
      "Github username: chrisvoncsefalvay\n",
      "\n",
      "Python implementation: CPython\n",
      "Python version       : 3.10.2\n",
      "IPython version      : 8.4.0\n",
      "\n",
      "sympy: 1.10.1\n",
      "\n"
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    "%load_ext watermark\n",
    "%watermark --iversions -v -a \"Chris von Csefalvay\" -gu \"chrisvoncsefalvay\" -p sympy"
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   "source": [
    "import sympy\n",
    "\n",
    "sympy.init_printing()"
   ],
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     "name": "#%%\n"
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   "source": [
    "t, beta, sigma, gamma, mu, S, E, I, R = sympy.symbols(\"t beta sigma gamma mu S E I R\")"
   ],
   "execution_count": 23,
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   "source": [
    "solutions = sympy.solve([\n",
    "    mu - (beta * I + mu) * S,\n",
    "    beta * S * I - (sigma + mu) * E,\n",
    "    sigma * E - (gamma + mu) * I,\n",
    "    gamma * I - mu * R,\n",
    "    sympy.Equality(S + E + I + R, 1)\n",
    "], [S, E, I, R])"
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   "execution_count": 24,
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    "solutions"
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   "source": [
    "jac = sympy.Matrix([\n",
    "    mu - (beta * I + mu) * S,\n",
    "    beta * S * I - (sigma + mu) * E,\n",
    "    sigma * E - (gamma + mu) * I,\n",
    "    gamma * I - mu * R,\n",
    "]).jacobian([S, E, I, R])"
   ],
   "execution_count": 26,
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- I \\beta - \\mu & 0 & - S \\beta & 0\\\\I \\beta & - \\mu - \\sigma & S \\beta & 0\\\\0 & \\sigma & - \\gamma - \\mu & 0\\\\0 & 0 & \\gamma & - \\mu\\end{matrix}\\right]$"
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   "source": [
    "substituted_jacobian = jac.subs({\"I\": solutions[1][2],\n",
    "         \"S\": solutions[1][0]})"
   ],
   "execution_count": 28,
   "outputs": [],
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    "substituted_jacobian"
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      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\mu - \\frac{\\mu \\left(\\beta \\sigma - \\gamma \\mu - \\gamma \\sigma - \\mu^{2} - \\mu \\sigma\\right)}{\\left(\\gamma + \\mu\\right) \\left(\\mu + \\sigma\\right)} & 0 & - \\frac{\\left(\\gamma + \\mu\\right) \\left(\\mu + \\sigma\\right)}{\\sigma} & 0\\\\\\frac{\\mu \\left(\\beta \\sigma - \\gamma \\mu - \\gamma \\sigma - \\mu^{2} - \\mu \\sigma\\right)}{\\left(\\gamma + \\mu\\right) \\left(\\mu + \\sigma\\right)} & - \\mu - \\sigma & \\frac{\\left(\\gamma + \\mu\\right) \\left(\\mu + \\sigma\\right)}{\\sigma} & 0\\\\0 & \\sigma & - \\gamma - \\mu & 0\\\\0 & 0 & \\gamma & - \\mu\\end{matrix}\\right]$"
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   "source": [
    "sympy.print_python(substituted_jacobian.eigenvals())"
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   "execution_count": 30,
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     "name": "stdout",
     "text": [
      "beta = Symbol('beta')\n",
      "gamma = Symbol('gamma')\n",
      "mu = Symbol('mu')\n",
      "sigma = Symbol('sigma')\n",
      "e = {-(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)/(3*(27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)) - (27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)/3 - (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(3*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)): 1, -(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)/(3*(Rational(-1, 2) + sqrt(3)*I/2)*(27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)) - (Rational(-1, 2) + sqrt(3)*I/2)*(27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)/3 - (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(3*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)): 1, -(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)/(3*(Rational(-1, 2) - sqrt(3)*I/2)*(27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)) - (Rational(-1, 2) - sqrt(3)*I/2)*(27*beta*mu*sigma/2 - 27*gamma*mu**2/2 - 27*gamma*mu*sigma/2 - 27*mu**3/2 - 27*mu**2*sigma/2 + sqrt(-4*(-3*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**2/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2)**3 + (27*beta*mu*sigma - 27*gamma*mu**2 - 27*gamma*mu*sigma - 27*mu**3 - 27*mu**2*sigma - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2 + 2*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**2)/2 - 9*(beta*gamma*mu*sigma + 2*beta*mu**2*sigma + beta*mu*sigma**2)*(beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(2*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**2) + (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)**3/(gamma*mu + gamma*sigma + mu**2 + mu*sigma)**3)**Rational(1, 3)/3 - (beta*mu*sigma + gamma**2*mu + gamma**2*sigma + 3*gamma*mu**2 + 4*gamma*mu*sigma + gamma*sigma**2 + 2*mu**3 + 3*mu**2*sigma + mu*sigma**2)/(3*(gamma*mu + gamma*sigma + mu**2 + mu*sigma)): 1, -mu: 1}\n"
     ],
     "output_type": "stream"
    }
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  {
   "cell_type": "code",
   "source": [
    "eigs = substituted_jacobian.eigenvals()"
   ],
   "execution_count": 31,
   "outputs": [],
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  {
   "cell_type": "code",
   "source": [
    "real_parts = [sympy.re(i) for i in eigs]"
   ],
   "execution_count": 32,
   "outputs": [],
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  {
   "cell_type": "code",
   "source": [
    "sympy.solve(real_parts[0], [beta, gamma, sigma, mu])"
   ],
   "execution_count": 35,
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "KeyboardInterrupt: ",
     "traceback": [
      "\u001B[0;31m---------------------------------------------------------------------------",
      "Traceback (most recent call last)",
      "    at line 1 in <module>",
      "    at line 1097 in solve(f, *symbols, **flags)",
      "    at line 1788 in _solve_system(exprs, symbols, **flags)",
      "    at line 1131 in as_poly(self, *gens, **args)",
      "    at line 164 in __new__(cls, rep, *gens, **args)",
      "    at line 293 in _from_expr(cls, rep, opt)",
      "    at line 357 in _dict_from_expr(expr, opt)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 3622 in expand(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3555 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 3563 in _expand_hint(expr, hint, deep, **hints)",
      "    at line 961 in _eval_expand_mul(self, **hints)",
      "    at line 926 in _expandsums(sums)",
      "    at line 926 in <listcomp>(.0)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 85 in __new__(cls, evaluate, _sympify, *args)",
      "    at line 524 in flatten(cls, seq)",
      "KeyboardInterrupt: "
     ],
     "output_type": "error"
    }
   ],
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   "source": [
    "trisolutions = sympy.solve([real_parts[0], real_parts[1], real_parts[2]], mu, sigma, gamma, beta)"
   ],
   "execution_count": 33,
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "KeyboardInterrupt: ",
     "traceback": [
      "\u001B[0;31m---------------------------------------------------------------------------",
      "Traceback (most recent call last)",
      "    at line 1 in <module>",
      "    at line 1062 in solve(f, *symbols, **flags)",
      "    at line 1062 in <listcomp>(.0)",
      "    at line 953 in subs(self, *args, **kwargs)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1039 in fallback(self, old, new)",
      "    at line 72 in wrapper(*args, **kwargs)",
      "    at line 1067 in _subs(self, old, new, **hints)",
      "    at line 1040 in fallback(self, old, new)",
      "    at line 1948 in _aresame(a, b)",
      "    at line 2110 in __next__(self)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2083 in _preorder_traversal(self, node, keys)",
      "    at line 2067 in _preorder_traversal(self, node, keys)",
      "KeyboardInterrupt: "
     ],
     "output_type": "error"
    }
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