# Dynamic SAFFRON: Disease Control Over Time via Group Testing
## 3 Proposed Algorithms and Analysis
### Related Prior Results  
For the sake of completeness, here we present a revised version of the results from [38], that we use for the analysis in this paper. To keep it compact, we only include the main results that we use in the analysis in this paper.  
We consider symmetric and converging dynamic testing algorithms, which must satisfy the symmetry criterion:  
\[P(U_{i}(t)=k)=P(U_{j}(t)=k),\quad k\in\{0,1,2\} \tag{2}\]  
for each time instance \(t\geq 0\) and for all \(i,j\in[n]\) pair. Moreover, they must satisfy the convergence criterion:  
\[\lim_{t\to\infty}P(U_{i}(t)=1)=o(1/n),\quad i\in[n] \tag{3}\]  
Let \(p^{\prime}(t)\) denote the probability of an individual not being identified during the testing phase at time \(t\). Note that, since we consider symmetric and converging dynamic testing algorithms, this probability is the same for all individuals. We consider dynamic testing algorithms where \(p^{\prime}(t)\) only depends on the testing capacity \(T\), \(\alpha(t)\), \(\lambda(t)\) and \(\gamma(t)\). We assume the infection spread probability \(q\) scales as \(o(1/n)\). Since the infection spread is realized for every susceptible and non-isolated infected individual pair and there can be \(O(n^{2})\) such pairs at each time instance, assuming \(q\) to be \(o(1/n)\) is a mild assumption.  
In the remainder of this subsection, we present our prior result from [38] that we use in our analysis in this paper, regarding symmetric and converging dynamic testing algorithms.  
We start with the following lemma, where we prove that symmetric and converging dynamic testing algorithms guarantee the disease to be controlled eventually.  
Lemma 1: _When a symmetric and converging dynamic testing algorithm is implemented, \(\lim_{t\to\infty}E[\lambda(t)]=o(1)\) and thus, the system approaches the steady state, in the mean sense._  
The statements of Lemma 2 and Theorem 1 give improved characterizations of the approximation terms compared to the original statements in [38]. The approximations are characterized by \(o(1)\) terms in this paper, which follow from the linear approximations in Taylor series expansions for exponential terms and hold since \(q=o(1/n)\). Moreover, arbitrarily small variance conditions in [38] are also characterized to scale as \(o(1)\) here as well. In the following lemma, we characterize an approximation of the probability of an individual being infected and non-isolated at time \(t\), i.e., probability of the event of \(U_{i}(t)=1\).  
**Lemma 2**.: _When a symmetric and converging dynamic testing algorithm is implemented, we have_  
\[P(U_{i}(t)=1)=p((1+nq(1-p)))^{t}\prod_{j=1}^{t}p^{\prime}(j)+o(1) \tag{4}\]  
_where the conditional probability of an individual not being identified in the tests at time \(t\) given \(\lambda(t-1)\) is denoted by \(p^{\prime}_{\lambda(t-1)}\) and when \(cov\Big{(}P(U_{i}(t)=0|\lambda(t)),p^{\prime}_{\lambda(t)}(t+1)\Big{)}\) and \(cov\Big{(}P(U_{i}(t)=1|\lambda(t)),p^{\prime}_{\lambda(t)}(t+1)\Big{)}\) both scale as \(o(1)\) with respect to \(n\) for all \(t\geq 0\)._  
In the following theorem, we present a general result that characterizes the approximation of the expectation of \(\alpha(t)\). This result holds for all symmetric and converging dynamic testing algorithms that satisfy the small covariance conditions that we state in Lemma 2.  
**Theorem 1**.: _When a symmetric and converging dynamic testing algorithm is implemented and vanishing covariance constraints in Lemma 2 are satisfied for all \(t\geq 0\), we have_  
\[E[\alpha(t)]=n(1-p)(1-q)^{np\prod\limits_{i=0}^{t-1}\left((1+nq(1-p))^{i} \prod\limits_{j=1}^{i}p^{\prime}(j)\right)}+o(1) \tag{5}\]  
We use Theorem 1 to characterize the expected number of susceptible individuals when the disease is brought under control, which is one of the two performance metrics that we consider. On the other hand, for the characterization of the disease control time \(\bar{t}\), our analysis is based on Lemma 2.