### Instruction:
Design a JavaScript/TypeScript web application that integrates with the Google Maps API to display markers for user-specified locations and calculates the shortest route between these locations using the Dijkstra's algorithm.

### Response:

To create a web application that integrates with the Google Maps API and calculates the shortest route between user-specified locations using Dijkstra's algorithm, follow these steps:

1. Set up an HTML file for your application

Create a new HTML file called `index.html` and add the following code to set up the basic structure of the page:

```html
<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Shortest Route Finder</title>
  <script src="https://maps.googleapis.com/maps/api/js?key=YOUR_API_KEY&callback=initMap" async defer></script>
  <script src="app.js"></script>
</head>
<body>
  <div id="map" style="height: 100vh; width: 100vw;"></div>
</body>
</html>
```

Replace `YOUR_API_KEY` with your actual Google Maps API key.

2. Create a JavaScript/TypeScript file for your application logic

Create a new JavaScript or TypeScript file called `app.js` (or `app.ts`, if you prefer TypeScript) and add the following code to initialize the map:

```javascript
let map;
let markers = [];

function initMap() {
  map = new google.maps.Map(document.getElementById('map'), {
    center: { lat: -34.397, lng: 150.644 },
    zoom: 8,
  });

  map.addListener('click', (event) => {
    addMarker(event.latLng);
  });
}

function addMarker(location) {
  const marker = new google.maps.Marker({
    position: location,
    map: map,
  });

  markers.push(marker);
}
```

This code initializes the map and adds a click event listener to it. When the user clicks on the map, a new marker is added to the clicked location.

3. Implement Dijkstra's algorithm

Add the following code to your `app.js` or `app.ts` file to implement Dijkstra's algorithm:

```javascript
function dijkstraAlgorithm(startIndex) {
  const distances = [];
  const visited = [];

  for (let i = 0; i < markers.length; i++) {
    distances[i] = Infinity;
    visited[i] = false;
  }

  distances[startIndex] = 0;

  for (let i = 0; i < markers.length - 1; i++) {
    const minDistanceIndex = findMinDistance(distances, visited);
    visited[minDistanceIndex] = true;

    for (let j = 0; j < markers.length; j++) {
      if (!visited[j]) {
        const distance = google.maps.geometry.spherical.computeDistanceBetween(markers[minDistanceIndex].getPosition(), markers[j].getPosition());
        if (distances[minDistanceIndex] + distance < distances[j]) {
          distances[j] = distances[minDistanceIndex] + distance;
        }
      }
    }
  }

  return distances;
}

function findMinDistance(distances, visited) {
  let minValue = Infinity;
  let minIndex = -1;

  for (let i = 0; i < distances.length; i++) {
    if (!visited[i] && distances[i] < minValue) {
      minValue = distances[i];
      minIndex = i;
    }
  }

  return minIndex;
}
```

This code implements Dijkstra's algorithm using Google Maps' spherical geometry library to calculate the distance between two points on Earth.

4. Calculate and display the shortest route

To calculate and display the shortest route between all the user-specified locations, add the following code to your `app.js` or `app.ts` file:

```javascript
function calculateShortestRoute() {
  const distances = dijkstraAlgorithm(0);
  const shortestDistanceIndex = findMinDistance(distances.slice(1), Array(markers.length - 1).fill(false)) + 1;

  const shortestPath = new google.maps.Polyline({
    path: [markers[0].getPosition(), markers[shortestDistanceIndex].getPosition()],
    geodesic: true,
    strokeColor: '#FF0000',
    strokeOpacity: 1.0,
    strokeWeight: 2,
  });

  shortestPath.setMap(map);
}

// Add a button to the HTML page for calculating the shortest route
const button = document.createElement('button');
button.textContent = 'Calculate Shortest Route';
button.addEventListener('click', calculateShortestRoute);
document.body.appendChild(button);
```

This code adds a button to the web page that, when clicked, calculates the shortest route between all the user-specified locations and displays it on the map using a polyline.

Now you have a basic JavaScript/TypeScript web application that integrates with the Google Maps API, allows users to add markers by clicking on the map, and calculates the shortest route between these locations using Dijkstra's algorithm.