# Dynamic Infection Spread Model Based Group Testing
## 4 Proposed Algorithms and Analysis
### Dynamic Dorfman Type Group Testing Algorithm  
In dynamic Dorfman type group testing algorithm, we utilize the group testing idea while designing the test matrices at each time instance \(t\geq 1\). At each time instance, dynamic Dorfman type group testing algorithm uniformly randomly partitions the set of all non-isolated individuals to equal sized \(T/2\) disjoint sets (with possible 1 unequal sized set if the total number of non-isolated individuals is not divisible by \(T/2\)). Then, test samples of the individuals are mixed with others in the same group: \(T/2\) group tests are performed, and positive and negative groups are determined. Then, among the positive groups, one group (or multiple groups if the sizes of the groups are less than \(T/2\), depending on the system parameters) is uniformly randomly selected to be individually tested. \(T/2\) individuals from the selected group are uniformly randomly selected and individually tested; here depending on the parameters, some individuals from the selected group may not be tested, as well as individuals from multiple positive groups may be selected. Detected infections are isolated and at the next time instance, the whole process is repeated with uniform random selections. Since the partition selection and individuals within group selection are uniformly random at each time instance, dynamic Dorfman type group testing algorithm is symmetric. Similar to Section 4.1, we proceed by showing that the dynamic Dorfman type group testing algorithm satisfies the convergence criterion in (4) as well. **Lemma 7**: _For constant \(T\) and \(n\), dynamic Dorfman testing algorithm satisfies the convergence criterion_  
\[\lim_{t\to\infty}P(U_{i}(t)=1)=0,\quad i\in[n] \tag{42}\]  
The probability that an individual is identified at a time instance \(t\), which is \(1-p^{\prime}(t)\), satisfies the following \[1-p^{\prime}(t)\geq\frac{T}{2n} \tag{43}\]  
since \(T/2\) individuals are individually tested at each time instance and due to the symmetry of the infection status in the system and the fact that the individuals are selected from a positive group (or from multiple positive groups), the probability of detection for the dynamic Dorfman type group testing algorithm, at each time instance, must be higher than uniformly randomized testing of \(T/2\) individuals. Now, since the events of identification of individuals are independent across time due to the uniform random selection of tested individuals at each time instance, and the fact that  
\[\sum_{i=1}^{\infty}(1-p^{\prime}(t))\geq\sum_{i=1}^{\infty}\frac{T}{2n} \tag{44}\]  
grows to infinity, we conclude that \(\lim_{t\rightarrow\infty}P(U_{i}(t)=1)=0\), from the second Borel-Cantelli lemma as in Lemma 6. Similar to the dynamic individual testing case, we focus on a weak version of the dynamic Dorfman type group testing algorithm to provide a lower bound for the expected number of susceptible individuals in the system at the steady state. In the weak version of the dynamic Dorfman type group testing algorithm, the results from the \(T/2\) group tests are discarded and it is basically equivalent to the uniformly random individual testing of \(T/2\) individuals. Furthermore, the isolated individuals are also included in the testing procedure: \(n\) individuals are divided into groups and then tested at each time instance, rather than only non-isolated individuals as in the original dynamic Dorfman type group testing algorithm. The probability of identification at time \(t\) for the weak dynamic Dorfman type group testing algorithm, given by \(1-p^{\prime}(t)\), is always less than the original dynamic Dorfman type group testing algorithm, due to the discarded \(T/2\) group tests and included isolated individuals to the tests. Note that the weak dynamic Dorfman type group testing algorithm is also symmetric and satisfies the convergent criterion (4), since Lemma 7 still holds; the lower bound in (44) is the detection probability of the weak algorithm.