In this paper, we extend these investigations and consider for the first time the network decontamination problem when the parameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1500373X&_mathId=si60.gif&_user=111111111&_pii=S0166218X1500373X&_rdoc=1&_issn=0166218X&md5=e5cd44ce8abb429003339a42bb882465" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode"> is an arbitrary integer value class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1500373X&_mathId=si63.gif&_user=111111111&_pii=S0166218X1500373X&_rdoc=1&_issn=0166218X&md5=0a24872bfb1dedb40657327214e2fadd" title="Click to view the MathML source">m≥1class="mathContainer hidden">class="mathCode">. We direct our study to widely used interconnection networks, namely meshes, tori, and trees. For each of these classes of networks, we present decontamination algorithms with threshold class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1500373X&_mathId=si60.gif&_user=111111111&_pii=S0166218X1500373X&_rdoc=1&_issn=0166218X&md5=e5cd44ce8abb429003339a42bb882465" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">; these algorithms work even in asynchronous setting, either directly or with a simple modification requiring one additional agent. We also establish general lower bounds on the number of agents necessary for decontamination with immunity class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1500373X&_mathId=si60.gif&_user=111111111&_pii=S0166218X1500373X&_rdoc=1&_issn=0166218X&md5=e5cd44ce8abb429003339a42bb882465" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">; these bounds are tight in the case of trees, while large gaps still exist in the case of meshes and tori.