由於沒有提供相關頁面html我簡單寫一個範例，用純粹前端js處理，機制是一直循環，因為我分不出來你的頁面執行機制，如果是每次重整頁面，使用setTimeout就可以。

html

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/bootstrap@4.6.1/dist/css/bootstrap.min.css" integrity="sha384-zCbKRCUGaJDkqS1kPbPd7TveP5iyJE0EjAuZQTgFLD2ylzuqKfdKlfG/eSrtxUkn" crossorigin="anonymous">

<nav aria-label="Page navigation example">
  <ul class="pagination">
    <li class="page-item active"><a class="page-link" href="#">1</a></li>
    <li class="page-item"><a class="page-link" href="#">2</a></li>
    <li class="page-item"><a class="page-link" href="#">3</a></li>
    <li class="page-item"><a class="page-link" href="#">4</a></li>
    <li class="page-item"><a class="page-link" href="#">5</a></li>
  </ul>
</nav>

javascript

$(function() {
       
    function initCurrent(){
    	const items = [...document.querySelectorAll('body > nav > ul > li')];
    	const selectItem = document.querySelector('body > nav > ul > li.page-item.active');
    	const idx = items.indexOf(selectItem);    
      callNext(items,idx);
    }
    
    function callNext(items,idx){
    	const next = items.length-1 === idx ? 0 : idx+1;
			$(items[idx]).removeClass('active');
      $(items[next]).addClass('active');   
    }
        
    function initCycle(){
    	const clock =  setInterval(initCurrent , 1000);
      //clearInterval(clock); //stop setInterval
    }
      
    initCycle();   
});