How can I maintain my store's state while using react-router and redux?

I am building an app prototype that essentially simulates ecommerce. I have components that each have different items that can be added to a cart(below I just show an example of how one would basically work). These components are accessed via different routes using the react-router. There is a header component that displays the number of items currently in the cart. The header gets the number of items in the cart from the state in the redux store. However, if I navigate to a new route, the store goes back to the default state. I need the the store to keep its state when a new route is navigated to. For example, if I go to the ShoppingPage, add an item to the cart, and then go back to the Home page, I need the cart to still have an item in it.

actions.js

export const actionTypes = Object.freeze({
    UPDATE_CART: Symbol('UPDATE_CART'),
});

export const updateCart = (payload) => {
    return {
        type: actionTypes.UPDATE_CART,
        payload,
    };
};

export default actionTypes;

reducer.js

import actions from './actions';

export const INITIAL_STATE = {
    cart: [],
};

export default (state = INITIAL_STATE, action) => {
    switch (action.type) {
        case actions.UPDATE_CART: {
            return {
                ...state,
                cart: action.payload,
            };
        }
        default: {
            return state;
        }
    };
};

index.js

import React from 'react';
import ReactDOM from 'react-dom';
import App from './App';
import * as serviceWorker from './serviceWorker';
import { BrowserRouter } from 'react-router-dom';
import { Provider } from 'react-redux';
import { createStore } from 'redux';
import reducer, { INITIAL_STATE } from './reducer';

const store = createStore(reducer, INITIAL_STATE);
console.log(store.getState());
ReactDOM.render(
<Provider store ={store}>
    <BrowserRouter>
        <App />
    </BrowserRouter>
</Provider>
, document.getElementById('root'));

serviceWorker.unregister();

ShoppingPage.js

import React, { Component } from 'react';
import { connect } from 'react-redux';
import { updateCart } from './actions';

class ShoppingPage extends Component {
    addToCart = () => {
        const cart = [...this.props.cart];
        cart.push('new item');
        this.props.modifyCart(cart);

    render() {
        return(
            <div>
                <button onClick={addToCart}>
                    Add To Cart
                </button>
            </div>
        )
    }
}

const mapDispatchToProps = dispatch => ({
    modifyCart: payload => dispatch(updateCart(payload)),
});

const mapStateToProps = state => ({
    cart: state.cart,
});

export default connect(
    mapStateToProps,
    mapDispatchToProps,
)(ShoppingPage);

Home.js

import React, { Component } from 'react';
import { ListGroup, ListGroupItem } from 'reactstrap';

class Home extends Component {
    render() {
        return(
            <div>
                <ListGroup>
                    <ListGroupItem><a href='/ShoppingPage'>ShoppingPage</a></ListGroupItem>
            </div>
        )
    }
}

export default Home;

Header.js

import React, { Component } from 'react';
import { Navbar, NavbarBrand } from 'reactstrap';
import { connect } from 'react-redux';

class Header extends Component {
    render() {
        return(
            <Navbar sticky='top' className='nav'>
                <NavbarBrand href='/'>Online Shopping</NavbarBrand>
                    <span>{'Items in Cart: '}{this.props.cart.length}</span>
            </Navbar>
        )
    }
}

const mapStateToProps = state => ({
    cart: state.cart,
});

export default connect(
    mapStateToProps
)(Header);

Routes.js

import React from 'react';
import { Switch, Route } from 'react-router-dom';
import Home from './Home';
import ShoppingPage from './ShoppingPage';

const Routes = () => (
    <Switch>
        <Route exact path='/' component={Home} />
        <Route exact path='/ShoppingPage' component={ShoppingPage} />
    </Switch>
);

export default Routes;

App.js

import React from 'react';
import Routes from './Routes';
import Header from './Header';

function App() {
  return (
    <div>
        <Header />
        <Routes />
    </div>
  );
}

export default App;