Time-dependent probability distribution for number of infection in a stochastic SIS model: case study COVID-19

Abstract

We derive the time-dependent probability distribution for the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness, and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.

Keywords: COVID-19; Hypergeometric; Infection; Kummer; Laguerre function; Probability density function; Stochastic differential equation; Whittaker function.

Fig. 1

Comparison of the exact density functions $p_{\mathcal{I}}\left(\mathcal{I}\right|t=2,{\mathcal{I}}_0)$ and $p_{\mathcal{I}}^s\left(I\right)$ with simulated distribution of ${\left\{{\mathcal{I}}_{20}^l\right\}}_{l=1}^L$ and ${\left\{{\mathcal{I}}_{1000}^l\right\}}_{l=1}^L,$ respectively.

Fig. 2

Graphs of the probability density function $p_{\mathcal{I}}(\mathcal{I},t|{\mathcal{I}}_0=0.05)$ and $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$ as $\beta$ increases.

Fig. 3

Stationary density function $p_{\mathcal{I}}(\mathcal{I},)$ for the case where $R_0\to 1^{+}$ and ${\mathcal{R}}_0\to 1^{+},$ respectively.

Fig. 4

Effect of changing $\gamma$ on the stationary density function $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$.

Fig. 5

Probability and stationary density functions $p_{\mathcal{I}}\left(\mathcal{I}\right|t,{\mathcal{I}}_0=0.05)$ and $p_{\mathcal{I}}^s\left(\mathcal{I}\right),$ respectively, for the case where $\overline{\mathcal{A}}\approx 0$.

Fig. 6

Probability density function $p_{\mathcal{I}}\left(\mathcal{I}\right|t,{\mathcal{I}}_0=0.05)$ for the case where $E=\mu +\gamma \approx 0$.

Fig. 7

Effect of noise intensity $\sigma$ on the stationary density function $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$.

Fig. 8

Graphs of the mean, variance, skewness, kurtosis function with time.

Fig. 9

Graphs of the mean, variance, skewness, kurtosis function with time.

Fig. 10

Graphs of the mean, variance, skewness, kurtosis function with time.

Fig. 11

Real and simulated aggregate counts of COVID-19 infection for the states and territories: AK, AR, AZ, CA, CO, CT, DC, DE, GA, GU, HI, IA, ID, IL, IN, KS in the United States.

Fig. 12

Real and simulated aggregate counts of COVID-19 infection for the states and territories: KY, LA, MD, ME, MI, MN, MO, MP, MS, MT, NC, ND, NE, NH, NJ, NM in the United States.

Fig. 13

Real and simulated aggregate counts of COVID-19 infection for the states and territories: NV, OH, OK, OR, PA, RI, SC, SD, TN, TX, UT, VA, VT, WI, WV, WY in the United States.

Fig. 14

Probability distribution of aggregate number of counts that occurred on March 23, 2021.

Fig. 15

Comparison of the probability density function $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$ and $\underset{t\to \infty }{lim}{p}_{\mathcal{I}}\left(\mathcal{I}\right|t,{\mathcal{I}}_0)$.

Fig. 16

Distribution $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$ of the final size of the aggregate infection as the infection growth rate $\beta$ increases for the states: AK, ID, KY, ND, SD, WI, WV, WY in the United States.

Fig. 17

Distribution $p_{\mathcal{I}}^s\left(\mathcal{I}\right)$ of the final size of the aggregate infection as the aggregate harvesting rate $\mu +\gamma$ reduces with large noise $\sigma$ for the states: AK, ID, KY, ND, SD, WI, WV, WY in the United States.