solving differential equation (virus released) The virus causing dead people to rise as zombies is released now at time $t = 0$. (No one is infected yet). The virus infects people at a rate proportional to the number of people not yet infected, and after $1$ month, $2/3$ of the world is infected with the zombie virus. The population of Earth is $7*10^9$. How long (in months) until the entire world except one person is infected?

1. Since the rate of spreading is proportional to the current population (denoted as $H$ for Humanity): $$\frac{dH}{dt} = \lambda H$$
2. Separate variables: $$\int_{H_0}^{H}\frac{dH}{H}\ = \int_0^t\lambda dt$$
3. Integrate to get: $$\ln{\frac{H}{H_0}} = \lambda t$$ $$H = H_0 e^{\lambda t}$$
4. Substitute $H(1)$ to get the constant value $\lambda = \ln{\frac{1}{3}}$: $$H(t) = 3^{-t}H_0$$
5. Then solve for time given $H(\tau) = 1$: $$t = \log_3 (H_0)$$