<div dir="ltr"><div class="gmail_quote gmail_quote_container"><div dir="ltr"><div class="gmail_quote"><div dir="ltr" class="gmail_attr"><br></div><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div><p class="MsoNormal" style="margin:0cm 0cm 10pt;line-height:115%;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt;line-height:115%">Dear colleagues, </span></p>

<p class="MsoNormal" style="margin:0cm;text-align:justify;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt"> </span></p>

<p class="MsoNormal" style="margin:0cm;text-align:justify;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt">Our next seminar will be
held on Monday, <b>June 9</b>,<b> </b>from <b>3:30
p.m. to 4:30 p.m</b>. (Rio de Janeiro local time).  The meeting will take place at room </span><b><span lang="EN-US" style="font-size:12pt">C116-
Bloco C - CT</span></b><b><span lang="DE" style="font-size:12pt"> – Instituto de Matemática – UFRJ. </span></b><span lang="DE" style="font-size:12pt">There will be no transmission
online. </span><b><span lang="EN-US" style="font-size:12pt"></span></b></p>

<p class="MsoNormal" style="line-height:12.65pt;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;margin:0cm 0cm 10pt;font-size:11pt;font-family:Calibri,sans-serif"><span lang="DE" style="font-size:12pt"> </span></p>

<p class="MsoNormal" style="margin:0cm 0cm 10pt;line-height:115%;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt;line-height:115%;color:black">Speaker: <b>  Daniel Ungaretti (IM-UFRJ)</b></span></p>

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<p class="MsoNormal" style="line-height:normal;margin:0cm 0cm 10pt;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Title: <b>The contact
process on interchange process</b></span></p>

<div>We introduce a model of epidemics among moving particles on any 
locally finite graph. At any time, each vertex is empty, occupied by a 
healthy particle, or occupied by an infected particle. Infected 
particles recover at rate 1 and transmit the infection to healthy 
particles at neighboring vertices at rate <img alt="\lambda" title="\lambda" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Clambda" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.19411311103705708" style="display:inline;vertical-align:-0.667px" height="12" width="8">. In addition, particles perform an interchange process with rate <img alt="v" title="v" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=v" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.7162482997663578" style="display:inline;vertical-align:-0.333px" height="7" width="7">, that is, the states of adjacent vertices are swapped independently at rate <img alt="v" title="v" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=v" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.7520313593697256" style="display:inline;vertical-align:-0.333px" height="7" width="7">, allowing the infection to spread also through the movement of infected particles. On <img alt="\mathbb{Z}^d" title="\mathbb{Z}^d" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cmathbb%7BZ%7D%5Ed" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.2416470046018815" style="display:inline" height="13" width="16">,
 we start with a single infected particle at the origin and with all the
 other vertices independently occupied by a healthy particle with 
probability <img alt="p" title="p" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=p" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.40549950882197083" style="display:inline;vertical-align:-3.333px" height="10" width="8"> or empty with probability <img alt="1-p" title="1-p" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=1-p" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.6123399005565645" style="display:inline;vertical-align:-3.333px" height="14" width="34">. We define the threshold value <img alt="\lambda_c(v,p)" title="\lambda_c(v,p)" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Clambda%5Fc(v,p)" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.38041498216878655" style="display:inline;vertical-align:-4.333px" height="16" width="49"> for <img alt="\lambda" title="\lambda" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Clambda" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.5272228742539378" style="display:inline;vertical-align:-0.667px" height="12" width="8"> above which the infection persists with positive probability and analyze its asymptotic behavior in two regimes: as <img alt="v \to \infty" title="v \to \infty" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=v%09%5Cto%09%5Cinfty" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.5966217622836811" style="display:inline;vertical-align:-0.333px" height="8" width="49"> for fixed p, in which we approach a mean-field behavior for the system, and as <img alt="v \to 0" title="v \to 0" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=v%09%5Cto%090" id="m_-7725463929238569462m_6268708797008612077m_3937129330312543426m_6053380099359632005l0.513654928980337" style="display:inline;vertical-align:-0.333px" height="11" width="41"> for fixed p, in which we approach a system close to the contact process on a static random environment.</div><div><br></div><div><br></div>

<p class="MsoNormal" style="line-height:normal;margin:0cm 0cm 10pt;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt">More complete information
about the seminars can be found at</span></p>

<p class="MsoNormal" style="margin:0cm 0cm 10pt;line-height:115%;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US"><a href="https://ppge.im.ufrj.br/seminarios-de-probabilidade/" target="_blank">https://ppge.im.ufrj.br/seminarios-de-probabilidade/</a></span></p>

<p class="MsoNormal" style="margin:0cm 0cm 10pt;line-height:115%;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt;line-height:115%">Sincerely, </span><span lang="EN-US" style="font-size:12pt;line-height:115%"></span></p>

<p class="MsoNormal" style="margin:0cm 0cm 10pt;line-height:115%;font-size:11pt;font-family:Calibri,sans-serif"><span lang="EN-US" style="font-size:12pt;line-height:115%">Organizers: Giulio Iacobelli and Maria Eulalia
Vares</span></p></div><div><br></div><div><br></div><span class="gmail_signature_prefix">-- </span><br><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr">Maria Eulalia Vares<div>Professora Titular - Instituto de Matemática - UFRJ</div><div>Coordenadora do Programa de Pós-Graduação em Estatística</div><div><a href="https://ppge.im.ufrj.br/" target="_blank">https://ppge.im.ufrj.br/</a></div><div><br></div></div></div></div>
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